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The Target Study: A Conceptual Model and Framework for Measuring Disparity

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Title: The Target Study: A Conceptual Model and Framework for Measuring Disparity
Language: English
Authors: John W. Jackson (ORCID 0000-0002-1528-7003), Yea-Jen Hsu, Raquel C. Greer, Romsai T. Boonyasai, Chanelle J. Howe (ORCID 0000-0001-5379-472X)
Source: Sociological Methods & Research. 2026 55(2):405-458.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 54
Publication Date: 2026
Sponsoring Agency: National Heart, Lung, and Blood Institute (NHLBI) (DHHS/NIH)
Contract Number: K01HL145320
Document Type: Journal Articles
Reports - Research
Descriptors: Models, Differences, Measurement, Disadvantaged, Health, Sampling
DOI: 10.1177/00491241251314037
ISSN: 0049-1241
1552-8294
Abstract: We present a conceptual model to measure disparity--the target study--where social groups may be similarly situated (i.e., balanced) on allowable covariates. Our model, based on a sampling design, does not intervene to assign social group membership or alter allowable covariates. To address nonrandom sample selection, we extend our model to generalize or transport disparity or to assess disparity after an intervention on eligibility-related variables that eliminates forms of collider-stratification. To avoid bias from differential timing of enrollment, we aggregate time-specific study results by balancing calendar time of enrollment across social groups. To provide a framework for emulating our model, we discuss study designs, data structures, and G-computation and weighting estimators. We compare our sampling-based model to prominent decomposition-based models used in healthcare and algorithmic fairness. We provide R code for all estimators and apply our methods to measure health system disparities in hypertension control using electronic medical records.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1502013
Database: ERIC
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  Value: <anid>AN0192656324;som01may.26;2026Apr02.02:51;v2.2.500</anid> <title id="AN0192656324-1">The Target Study: A Conceptual Model and Framework for Measuring Disparity </title> <p>We present a conceptual model to measure disparity—the target study—where social groups may be similarly situated (i.e., balanced) on allowable covariates. Our model, based on a sampling design, does not intervene to assign social group membership or alter allowable covariates. To address nonrandom sample selection, we extend our model to generalize or transport disparity or to assess disparity after an intervention on eligibility-related variables that eliminates forms of collider-stratification. To avoid bias from differential timing of enrollment, we aggregate time-specific study results by balancing calendar time of enrollment across social groups. To provide a framework for emulating our model, we discuss study designs, data structures, and G-computation and weighting estimators. We compare our sampling-based model to prominent decomposition-based models used in healthcare and algorithmic fairness. We provide R code for all estimators and apply our methods to measure health system disparities in hypertension control using electronic medical records.</p> <p>Keywords: equity; disparity; ethics; fairness; target study emulation; conceptual model; framework</p> <hd id="AN0192656324-2">Introduction</hd> <p>Measuring disparity is a key step in making progress toward health equity. Disparity measures underlie descriptive reports and trends and serve as benchmarks for evaluating the effects of interventions and policies ([<reflink idref="bib13" id="ref1">13</reflink>]). Although the measurement of disparity is critical and there has been much discussion and debate about what constitutes a disparity ([<reflink idref="bib36" id="ref2">36</reflink>]; [<reflink idref="bib4" id="ref3">4</reflink>]; [<reflink idref="bib22" id="ref4">22</reflink>]), there has been limited discussion about best practices and principles for measurement of disparity, especially when using secondary data not collected for research purposes.</p> <p>Conceptual models serve as important guides for the analysis and interpretation of secondary data. For example, consider the target trial framework ([<reflink idref="bib32" id="ref5">32</reflink>]), which lays out the hypothetical randomized controlled trial one would conduct if the goal were to estimate the effect of a treatment strategy to inform clinical decision-making. The elements of the trial (eligibility, treatment strategies, outcome follow-up) guide the design and analysis of a study based on secondary data help ensure that the measure of association has a causal interpretation that applies to (a) the population of interest, (b) treatment strategies of interest, and (c) outcomes of interest, all of which are critical for informing treatment policy decisions.</p> <p>The target trial framework cannot guide a descriptive measurement of disparity where there is no intervention. Still, without a conceptual guide, the population of interest and the follow-up period that pertain to unjust processes or outcomes may be unclear which can impede appropriate policymaking. Without a conceptual model it is difficult to justify and interpret covariate adjustment in health disparities research ([<reflink idref="bib41" id="ref6">41</reflink>]). Causal models have been used to define disparities ([<reflink idref="bib21" id="ref7">21</reflink>]), but they have stringent assumptions and abstract away important realities. Meanwhile, there are intense discussions about nonrandom sample selection and its impact on related concepts such as discrimination ([<reflink idref="bib46" id="ref8">46</reflink>]; [<reflink idref="bib26" id="ref9">26</reflink>]). Outlining the hypothetical study one could do in the real world to measure disparity will provide clarity on these issues.</p> <p>We present a novel conceptual model—the target study—to address these issues and provide a framework for emulating it. The paper is organized as follows. We begin by introducing our motivating example. The section "Conceptual Issues in Measuring Disparity" reviews key issues in disparity measurement. The section "A Target Study Conceptual Model for Measuring Disparity" presents our model under the case where investigators wish to capture all effects of nonrandom sample selection. The section "Extension of the Target Study to Address Non-Random Sample Selection" expands the model to address nonrandom sample selection by generalizing to a broader population, transporting to a different population, or estimating disparity in a counterfactual population where certain consequences of non-random sample selection are absent. The section "Emulation of the Target Study with Secondary Data" proposes data structures and estimators to emulate the target study. The section "Contributions and Comparison to Existing Literature" outlines our contributions and compares our model to others widely used to study disparity in healthcare and algorithmic fairness. The section "Discussion" discusses strengths and limitations. To aid readability, we use modular sections with ample cross-referencing so readers may skip directly to sections of interest.</p> <hd id="AN0192656324-3">Motivating Example</hd> <p>Consider the measurement of racial disparities in hypertension outcomes of primary care patients diagnosed with hypertension who receive care at a large regional health system in the USA. The outcomes of interest are a health-related quantity <emph>Y</emph> (e.g., hypertension control) or a healthcare decision <emph>D</emph> made by a clinician (e.g., to intensify hypertension treatment). We are concerned with average outcomes across a categorical social grouping, such as race <emph>R</emph>, where a socially disadvantaged (henceforth referred to as marginalized) group (e.g., Black persons) is denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> and a socially advantaged (henceforth referred to as privileged) group (e.g., White persons) is denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>0</mn></math> </ephtml> . The available data are electronic medical records stamped at time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="script">h</mi></math> </ephtml> (e.g., in minutes, hours, seconds) from office-based primary care visits over multiple years that include measures of prior hypertension, demographics <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">X</mi></mrow><mrow><mi mathvariant="script">h</mi></mrow></msub></math> </ephtml> (e.g., age and sex assigned at birth), comorbidity and socioeconomic status (SES) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">L</mi></mrow><mrow><mi mathvariant="script">h</mi></mrow></msub></math> </ephtml> , hypertension control as of that visit, denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi mathvariant="script">h</mi></mrow></msub></math> </ephtml> (1: yes, 0: no), antihypertensive treatment intensification (i.e., initiation, change in dose, or change in class) within the 14 days after the visit, denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>D</mi><mrow><mi mathvariant="script">h</mi></mrow></msub></math> </ephtml> (1: yes, 0: no), and time-stamped enrollment in an electronic patient portal program (EPPP) for care management. The notation <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">h</mi></mrow></math> </ephtml> refers to the timing of a variable's measurement (e.g., the date/time of the visit) rather than the time at which its value is realized (e.g., some date/time before the visit). Later on, we will coarsen measurement time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">h</mi></mrow></math> </ephtml> into some chosen unit of calendar time <emph>k</emph> (e.g., months). Finally, note that persons may have multiple visits per day or week.</p> <hd id="AN0192656324-4">Conceptual Issues in Measuring Disparity</hd> <p></p> <hd id="AN0192656324-5">Defining Disparity</hd> <p>In medicine and public health, the definition of disparity depends on whether the outcome is a health status (e.g., hypertension control <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> </ephtml> ) or a healthcare commodity (e.g., treatment intensification <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math> </ephtml> ). For health status outcomes, the Healthy People 2020 report committee defined disparities as "systematic, plausibly avoidable health differences adversely affecting socially disadvantaged groups" ([<reflink idref="bib5" id="ref10">5</reflink>]). This builds upon a World Health Organization definition ([<reflink idref="bib83" id="ref11">83</reflink>]) and relates to the National Institute of Minority Health and Health Disparities (NIMHD) definition ([<reflink idref="bib22" id="ref12">22</reflink>]): "a health difference that adversely affects disadvantaged populations, based on one or more health outcomes" where outcomes range from health behaviors, to diagnosis- or stage-specific-clinical endpoints or self-reported measures, to overall mortality. For healthcare outcomes (e.g., treatment), the Institute of Medicine (IOM) report "Unequal Treatment" ([<reflink idref="bib36" id="ref13">36</reflink>]) defines disparities as</p> <p>differences in the quality of care that are <emph>not due to</emph> access-related factors or clinical needs, preferences, and appropriateness of intervention...[where] analysis is focused at two levels: 1) the operation of the health systems and the legal and regulatory climate...; 2) discrimination at the individual, patient-provider level. (emphasis added)</p> <p>Disparity reflects society's failure to achieve equity in health, defined as "everyone having a fair and just opportunity to be as healthy as possible" ([<reflink idref="bib83" id="ref14">83</reflink>]; [<reflink idref="bib5" id="ref15">5</reflink>]).[<reflink idref="bib8" id="ref16">8</reflink>]</p> <hd id="AN0192656324-6">Temporal Framing</hd> <p>To aid decision-makers, community members, and other stakeholders, disparity refers not to a universal, general phenomenon, but to outcomes among people nested in a particular context at a point in (or span of) calendar time. For example, we could describe the disparity in prevalent uncontrolled hypertension for primary care visits during each month during the peak of the COVID-19 pandemic in 2020–2022. If we include one care episode per person per month, we can meaningfully summarize disparity over the entire period by averaging over the month-specific estimates of disparity. Such a summary measure would be interpreted as an average disparity over populations indexed by calendar months. By accounting for calendar time when producing such summary estimates, we avoid confounding by time-specific trends in enrollment and the outcome. To obtain a summary estimate of disparity between social groups with the same person-time experience of the health system, the summary must properly account for calendar time.</p> <hd id="AN0192656324-7">Allowability</hd> <p>In the IOM definition, disparity compares groups who are similarly situated (i.e., balanced) on "allowable" covariates. Allowable covariates are those whose differential distribution does not lead to inequitable outcomes. For a distributed good outcome <emph>D</emph> (e.g., healthcare) they are factors that, on moral arguments, are appropriate for determining allocation ([<reflink idref="bib37" id="ref17">37</reflink>]). For example, disparities in healthcare treat clinical need as allowable based on clinical guidelines ([<reflink idref="bib54" id="ref18">54</reflink>]; [<reflink idref="bib12" id="ref19">12</reflink>]). For a state outcome <emph>Y</emph> (e.g., health), the differential distribution of allowable covariates does not contribute to worse outcomes among the marginalized group ([<reflink idref="bib37" id="ref20">37</reflink>]). For example, if the marginalized group is younger and increased age predicts worse hypertension control, the younger age of the Black population does not contribute to the disparate distribution of hypertension control at the population-level. Not treating age as allowable could mask disparity from barriers to hypertension control that the Black population disproportionately faces (e.g., neighborhood disadvantage and limited options for healthy diet, physical activity, and pharmacies; [<reflink idref="bib57" id="ref21">57</reflink>]).</p> <hd id="AN0192656324-8">Is a Causal Framing of Disparity Necessary?</hd> <p>A fundamental question in conceiving of disparity is how the groups come to be similarly situated (i.e., balanced) on the allowables. Many authors conceive of disparity as comparing populations that are similarly situated through an intervention where an external actor makes existing groups similar by changing the value(s) of each person's allowable covariate(s)[<reflink idref="bib9" id="ref22">9</reflink>] ([<reflink idref="bib54" id="ref23">54</reflink>]; [<reflink idref="bib21" id="ref24">21</reflink>]; [<reflink idref="bib12" id="ref25">12</reflink>]; [<reflink idref="bib41" id="ref26">41</reflink>]). For example, disparate pulse oximeter performance is assessed in desaturation studies where hypoxia is induced among healthy volunteers ([<reflink idref="bib25" id="ref27">25</reflink>]). Disparate healthcare utilization is assessed in statistical analyses that hypothetically modify individuals' health and project utilization after this modification. This causal reading of the IOM definition is justified by its phrase "not due to," interpreted as "not caused by," where disparity compares social groups who are made similar (on the allowables) by intervention, to isolate the mediating role of inappropriate factors (e.g., SES) in producing differences in healthcare utilization ([<reflink idref="bib54" id="ref28">54</reflink>]). But the phrase "not due to" also permits a noncausal framing where, by design, disparity compares social groups who are already alike on the allowables. Early work that applied the IOM definition of disparity was motivated by non-causal studies where patients of different social groups with the same underlying need for medical treatment are compared in their distribution of appropriate medical treatment received, and actually framed such studies as IOM concordant ([<reflink idref="bib12" id="ref29">12</reflink>]: 2–3). We argue that approaches that balance allowables by design (e.g., our model) align with the IOM definition. More broadly, arguments about the exact causes of disparity are not needed to view disparity with concern ([<reflink idref="bib4" id="ref30">4</reflink>]). Moral concern may arise by the impact that disparity has on the human rights of marginalized groups ([<reflink idref="bib35" id="ref31">35</reflink>]).</p> <hd id="AN0192656324-9">Non-Random Sample Selection</hd> <p>Some frameworks for health equity acknowledge that non-random sample selection may impact a disparity measure ([<reflink idref="bib43" id="ref32">43</reflink>]). Consider the causal graph of Figure 1a, where variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> establishing eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> (eligibility-related variables) are prior hypertension, established care in the health system, enrollment in the electronic portal program (EPPP), and current visit, with <emph>k</emph> indexing the calendar time at which a variable is measured, and <emph>J</emph> indexing the outcome's follow-up time.</p> <p>Graph: Figure 1. Causal directed acyclic graphs depicting causal relationships between historical processes H , race R , demographics age and sex Xk, and comorbidities and adult socioeconomic status Lk, hypertension control Yk+J, and eligibility-related variables: Wk (prior hypertension, established care, electronic patient portal program [EPPP] enrollment, current visit [in (a) ]), Wk‡ (prior hypertension and established care [in (c) ], and additionally current visit [in (b) ]), Wk† (EPPP enrollment [in (b) and (c) ], and Wk≀ (current visit [in (c) ]), all measured at calendar time k. Full eligibility Qk is based on all eligibility-related variables Wk. Similarly, indicators of partial eligibility Qk‡, Qk†, Qk≀ are based on their corresponding subsets of eligibility-related variables Wk‡, Wk†, Wk≀. The subscript k marks calendar time and J the interval of time until Yk+J is measured. Dashed lines emphasize selective paths that become unblocked when conditioning on the indicator of eligibility Qk or all indicators of partial eligibility (Qk‡, Qk†, and Qk≀). Note that the subscript k refers to the calendar time at which a variable (e.g., a persons level of SES in adulthood) is measured (e.g., the calendar time of the visit) rather than the time at which its value was realized (e.g., some time before the visit).</p> <p>Lack of generalizability occurs when a disparity measure is unbiased for the study sample (e.g., those enrolled in the EPPP) but biased for the broader population of interest (e.g., the entire health system). Lack of transportability occurs when the disparity measure is unbiased for the study sample but biased for a different population of interest (e.g., not enrolled in EPPP) ([<reflink idref="bib72" id="ref33">72</reflink>]). Either scenario arises when (i) a risk factor <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> (e.g., SES) has different associations with the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) across social groups <emph>R</emph> and (ii) the risk factor <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> 's distribution depends on eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., it differs by EPPP enrollment), even with no difference in eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> across social groups <emph>R</emph>.[<reflink idref="bib10" id="ref34">10</reflink>] They also arise when the effect of eligibility-related variables on the outcome differs across social groups.</p> <p>Collider stratification creates an association between social group <emph>R</emph> and the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> among those who are eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> . ([<reflink idref="bib79" id="ref35">79</reflink>]) It can occur when eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., EPPP enrollment) are affected by (i) social group <emph>R</emph> and (ii) a risk factor <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> (e.g., SES) for the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> ([<reflink idref="bib30" id="ref36">30</reflink>]; [<reflink idref="bib23" id="ref37">23</reflink>]). Recent work ([<reflink idref="bib70" id="ref38">70</reflink>]; [<reflink idref="bib60" id="ref39">60</reflink>]) implies that collider stratification occurs if the probability ratio of being eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (comparing levels of social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> </ephtml> ) varies across levels of a risk factor <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> for the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> , even if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> has homogeneous associations with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> across social groups <emph>R</emph>.</p> <p>Because collider stratification due to non-random sample selection can induce an association between social group and the outcome among the sample (e.g., those enrolled in the EPPP) that is not present among the broader population (e.g., irrespective of EPPP enrollment), it is often viewed as a bias ([<reflink idref="bib79" id="ref40">79</reflink>]; [<reflink idref="bib46" id="ref41">46</reflink>]; [<reflink idref="bib69" id="ref42">69</reflink>]). There are reasons to include contributions of collider stratification to disparity. First, if disparity is measured in a meaningfully defined population of interest,[<reflink idref="bib11" id="ref43">11</reflink>] the contributions are substantively grounded as they reflect that population of interest ([<reflink idref="bib79" id="ref44">79</reflink>]). Consider when eligibility is defined by a condition (e.g., hypertension) that gives meaning to the outcome (e.g., hypertension control). For example, persons without history of hypertension can have elevated blood pressure due to hypertension onset or due to exercise, but these reasons do not represent uncontrolled hypertension. The disparity is only defined among eligible persons. Second, for a meaningful population of interest, when collider stratification disadvantages the marginalized group on baseline covariates leading to a worse outcome distribution compared to the privileged group, this aligns with definitions of disparity (see the subsection "Defining Disparity"). Third, the contribution of collider stratification is amenable to intervention by changing how covariates affect eligibility or the outcome.[<reflink idref="bib12" id="ref45">12</reflink>] However, when collider stratification advantages the marginalized group, it may mask disparity from other sources and investigators may choose to exclude it from disparity.</p> <hd id="AN0192656324-10">A Target Study Conceptual Model for Measuring Disparity</hd> <p></p> <hd id="AN0192656324-11">Overview</hd> <p>We now describe the elements of our conceptual model for measuring disparity, the target study. In this heuristic, an eligible population [denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> (1: eligible, 0: otherwise) ] from two or more social groups (e.g., Black <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> and White persons <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>0</mn></math> </ephtml> ) are selected from an eligible source population (e.g., established care in system, prior hypertension, enrolled in EPPP, current visit) within a given moment or span of coarsened time <emph>k</emph> representing the enrollment period (e.g., the month of January 2023). This selection occurs at the end of the enrollment window through a two-stage sampling strategy. The first stage of sampling addresses non-random selection into the study. The second stage of sampling similarly situates (i.e., balances) the social groups on the allowable covariates (if any are chosen) so that for both social groups, the allowables follow a distribution from a within-sample standard population (chosen by the investigator). Thus, the disparity estimate in the final sample is not due to differences in the allowable distributions. After both stages of sampling, those enrolled are followed for a specified period of time. A specified statistical comparison of outcomes across social groups provides the measure of disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> which is indexed by the enrollment period <emph>k</emph>.[<reflink idref="bib13" id="ref46">13</reflink>] The additive and ratio disparity are contrasts of mean outcomes (prevalence or risk with binary outcomes): <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup><mo>=</mo><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width=".1em" /><mrow><mi mathvariant="normal">and</mi></mrow><mspace width=".1em" /><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup><mo>=</mo><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> denotes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold">Ω</mi></mrow></mrow></msub><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> , the average outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi><mspace width="0.25em" /></mrow></msub></math> </ephtml> in social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> who are eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (based on criteria for eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> ) and enrolled in the target study sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="bold">Ω</mi></mrow></mrow></math> </ephtml> at calendar time <emph>k</emph> with follow-up time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>Under this conceptual model, the target population (in which inference is made) operationally consists of the source population that, within the enrollment period, is eligible, sampled, and enrolled. That is, in real life, if one wanted to make inferences about disparity in a population that exists within a certain span of time, one would carry out the protocol of the target study. At any calendar time unit <emph>k</emph>, a person only enrolls once into a target study. (Each unit of calendar time is of equal length). Results of studies <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> carried out at distinct calendar times <emph>k</emph> can be aggregated into a summary measure of disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Ψ</mi></mrow></math> </ephtml> . A weighted average of disparity measures <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> indexed at each calendar time <emph>k</emph> can be taken as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Ψ</mi></mrow><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mo movablelimits="false">∑</mo></mrow><mi>k</mi></msub><mspace width="0.2em" /><msub><mi>γ</mi><mi>k</mi></msub><msub><mi>ψ</mi><mi>k</mi></msub></mrow><mrow><msub><mrow><mo movablelimits="false">∑</mo></mrow><mi>k</mi></msub><mspace width="0.2em" /><msub><mi>γ</mi><mi>k</mi></msub></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub></math> </ephtml> is a weight specific to calendar time <emph>k</emph>.</p> <p>We discuss the choice of the weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub></math> </ephtml> in the subsection "Statistical Analysis" where we discuss the analysis of target studies.</p> <p>We begin with our default model (Design 1) where we choose to enroll all eligible persons (or a simple random sample of eligible persons) during the first stage of sampling. Recall that eligibility criteria cause persons to be non-randomly selected from the source population, so our default model includes all contributions of this non-random selection of persons to disparity. Adaptations to deal with such non-random sample selection (i.e., Designs 2, 3, or 4) are discussed in the section "Extension of the Target Study to Address Non-Random Sample Selection". Design 1 has minimal structural constrains on the underlying causal relations between all relevant variables.[<reflink idref="bib14" id="ref47">14</reflink>]</p> <hd id="AN0192656324-12">Enrollment Window(s)</hd> <p>To conduct a target study, we first choose a specific moment or narrow span in calendar time, denoted by <emph>k</emph>, to enroll persons. This requires choosing a level of granularity for calendar time (e.g., hours, days, months, years) and a specific moment <emph>k</emph> as the enrollment period (e.g., the month of January 2023). For each person, all eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., prior diagnosis of hypertension, established care in the health system, with a current visit in the window) and all allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., age, sex) are defined and measured at or before the end of this period. Thus, relative to the end of the enrollment window, eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> is based on having acceptable current or prior values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">w</mi></mrow></mrow></mrow><mi>k</mi></msub></math> </ephtml> of the eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> . At any calendar time <emph>k</emph>, a person may only enroll in one study.</p> <hd id="AN0192656324-13">Enrollment Groups</hd> <p>The definitions of disparity in the subsection "Defining Disparity" compare groups with persistently different levels of social advantage, privilege, power, wealth, or prestige because of their position in society ([<reflink idref="bib4" id="ref48">4</reflink>]). Within the USA, the NIMHD's concept of a disparity specifies social groups such as racial and ethnic minoritized groups (versus majoritized groups), underserved rural residents (versus urban residents), lower socioeconomic status (versus higher socioeconomic status), and sexual and gender minorities (versus sexual and gender majorities) ([<reflink idref="bib22" id="ref49">22</reflink>]). The National Institute of Mental Health (NIMH) further specifies groups with serious mental illness (versus those without) who have experienced long-standing stigmatization, discrimination, social exclusion, and loss of agency in society. Reflecting an intersectional perspective that mechanisms of social injustice combine to uniquely shape experience ([<reflink idref="bib11" id="ref50">11</reflink>]), social groups may be defined by joint membership along multiple axes (e.g., Black women versus White men) ([<reflink idref="bib40" id="ref51">40</reflink>]). This list is not exhaustive, and our model accommodates categorical[<reflink idref="bib15" id="ref52">15</reflink>] and time-varying definitions[<reflink idref="bib16" id="ref53">16</reflink>] of social groups.</p> <hd id="AN0192656324-14">Eligibility Criteria</hd> <p>The eligibility criteria can define the population of interest, reflecting issues of scope, societal level, and timing. In terms of scope, the criteria can restrict to places (e.g., the Mid-Atlantic region), institutions (e.g., a particular health system), or shared experiences or conditions (e.g., diagnosis of hypertension) that define a meaningful population. In terms of societal level, the criteria can focus on persons under the purview of a specific decision-maker (e.g., a clinical provider), facility (e.g., a clinic), or institution (e.g., a health system). In terms of timing, the criteria can focus on critical life stages, such as birth or a milestone event (e.g., myocardial infarction) where outcomes (e.g., appropriate medical treatment) are given meaning by that event. From here, we will use the following criteria: prior hypertension, established care in the health system, EPPP enrollment before calendar time <emph>k</emph>, and a recent primary care visit within calendar time <emph>k</emph>.</p> <hd id="AN0192656324-15">Allowable Covariates</hd> <p>We choose the covariates that the social groups are to be similarly situated (i.e., balanced) on by the end of the enrollment process. These allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are ones not implicated in generating disparate outcomes among the marginalized group, as described in the subsection "Allowability". Because the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are used to guide the enrollment process by defining the sampling fractions, they must be defined and measured by the end of the enrollment window when enrollment occurs. No allowable covariates may be chosen at all (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><mo>⊘</mo></math> </ephtml> ), as discussed the subsection "Enrollment Process (for Design 1)"). Although the eligibility variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (used to form the sampling frame) are separate from the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (used to define the sampling fractions), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are conceptually deemed allowable as the enrollment process similarly situates social groups on them.</p> <hd id="AN0192656324-16">Standard Distribution</hd> <p>During enrollment we sample individuals so that the distribution of allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> is the same for social groups, following a <emph>within-sample</emph> standard distribution chosen by the investigator. If the disparity varies across the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., age is considered to be allowable and disparity is higher in mid-life) the choice of the standard population, denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> , that defines the standard distribution may impact the magnitude and direction of the disparity measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> . The choice may be motivated by normative, theoretical, or practical concerns. If the marginalized group is the standard, its experience is emphasized ([<reflink idref="bib76" id="ref54">76</reflink>]). Then, the disparity measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> compares the experience of the marginalized group (enrolled through simple random sampling) to the experience of a privileged group (enrolled through stratified sampling) that shares the marginalized group's distribution of allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., its age structure).</p> <p>To balance the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> across social groups <emph>R</emph>, the values of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> found among the standard population must be within those of each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> at each time <emph>k</emph>, which is an overlap assumption: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml></p> <p>Graph</p> <p>for all values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml> and all <emph>k</emph>.</p> <p>The overlap assumption (<reflink idref="bib3" id="ref55">3</reflink>) requires that at each time <emph>k</emph> we look among the standard population (denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> ) and note the pattern of allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> covariate values. Then for each of those strata, we need to find (among eligible persons) members of each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . Otherwise, the sampling strategies we now describe will not be able to balance the allowables according to the standard distribution.</p> <hd id="AN0192656324-17">Enrollment Process (for Design 1)</hd> <p>At any time <emph>k</emph>, each person enrolls (once) into one study through multiple stages of sampling.[<reflink idref="bib17" id="ref56">17</reflink>] Limiting participation to a single enrollment in a single study per unit of calendar time maps inference to well-defined populations at each unit of calendar time. There is a pre-stage where eligible individuals are selected, a first stage that addresses the contribution of selective mechanisms to disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> , and a second stage that balances allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> in the final sample. For any target study design <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">D</mi></mrow></math> </ephtml> , at each stage <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi></math> </ephtml> , a person is selected via a known sampling fraction <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow><mrow><mi mathvariant="script">D</mi></mrow></msubsup></math> </ephtml> defined as the ratio of sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math> </ephtml> 's size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub></math> </ephtml> to the sampling frame <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> 's size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></mrow></msub></math> </ephtml> ([<reflink idref="bib50" id="ref57">50</reflink>]). The sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow><mrow><mi mathvariant="script">D</mi></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> may be stratified by and vary across covariate levels <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold-italic">V</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow></math> </ephtml> .</p> <p>We assume that the sampling is process is innocuous with respect to the outcome: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>f</mi><mi>Y</mi><mrow><mi>s</mi><mi>a</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mi>d</mi></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mrow><msub><mi mathvariant="bold-italic">V</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>f</mi><mi>Y</mi><mrow><mi>u</mi><mi>n</mi><mi>s</mi><mi>a</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>e</mi><mi>d</mi></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mrow><msub><mi mathvariant="bold-italic">V</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is the conditional probability mass function for discrete outcomes (conditional density for continuous outcomes), for all <emph>k</emph>.</p> <p>In words, the conditional distribution of the outcome given that persons are eligible and have covariate values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold-italic">V</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow></math> </ephtml> is the same for those enrolled and those not enrolled. Sampling does not affect the outcome. A person is randomly selected (without replacement) using a probability equal to their rescaled sampling fraction <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow><mrow><mo>*</mo><mo>,</mo><mrow><mi mathvariant="script">D</mi></mrow></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">v</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> that is bounded between zero and one.[<reflink idref="bib18" id="ref58">18</reflink>] We draw a uniformly distributed random number bounded between zero and one (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>U</mi><mo stretchy="false">[</mo><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mo stretchy="false">]</mo></math> </ephtml> ) and, if it is equal to or less than a person's rescaled sampling fraction, they are included ([<reflink idref="bib75" id="ref59">75</reflink>]). The sampling process is separate for each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> .</p> <p>The pre-stage <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi><mo>=</mo><mn>0</mn></math> </ephtml> is at the end of the enrollment window for time <emph>k</emph>. All eligibility criteria <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and all covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold-italic">V</mi><mi mathvariant="bold-italic">k</mi></msub></mrow></math> </ephtml> used in the enrollment process are measured by this point. From the source population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">P</mi></mrow><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , we select the eligible population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of chosen size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> . Full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> is defined as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>∈</mo><mrow><mrow><mi mathvariant="double-struck">w</mi></mrow></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">w</mi></mrow></mrow></math> </ephtml> represents the eligible values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold-italic">w</mi><mi mathvariant="bold-italic">k</mi></msub></mrow></math> </ephtml> of the eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> .</p> <p>In the first stage <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi><mo>=</mo><mn>1</mn></math> </ephtml> , a sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of chosen size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> is selected from the first-stage sampling frame <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> (eligible persons) using the sampling fraction <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p>Recall that eligible persons have been non-randomly chosen from the entire source population. They may all be selected at this stage [i.e., when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> ], otherwise <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> leads to simple random sampling of eligible persons in each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . Under either choice, all selective mechanisms, including that of non-generalizability and collider-stratification, contribute to the disparity estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> .</p> <p>In the second stage <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>ℓ</mi><mo>=</mo><mn>2</mn></math> </ephtml> , a sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of chosen size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> is selected from the second-stage sampling frame <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> (selected in stage one) of size <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> using the sampling fraction <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>The final sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup></math> </ephtml> [collapsed over <emph>R</emph> ] is where disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> is measured. If no allowable covariates are specified, stage 2 enrolls all persons [i.e., when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> ] or selects by simple random sampling for each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . Otherwise, the sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> similarly situates the social groups on the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> according to their distribution in the standard population, denoted <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> .</p> <hd id="AN0192656324-18">Time Zero</hd> <p>Time zero indicates the temporal anchor during calendar time for the start of follow-up for the outcomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>j</mi></mrow></msub></math> </ephtml> , where the time-scale of follow-up (i.e., time on study) is denoted as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi></math> </ephtml> (with <emph>J</emph> the end of follow-up). Time-zero determines when outcomes are counted towards disparity. We propose to anchor time zero at calendar time <emph>k</emph>, for two reasons: (<reflink idref="bib1" id="ref60">1</reflink>) to avoid differential alignment of outcomes from the point of eligibility; (<reflink idref="bib2" id="ref61">2</reflink>) to avoid underestimating disparity from a relevant point in time. For example, disparities (e.g., in appropriate treatment) may occur early after enrollment (e.g., after hospital discharge for myocardial infarction). If early treatment is critical for preventing adverse outcomes (e.g., a second myocardial infarction), we want to characterize that disparity by setting time zero right after enrollment.</p> <hd id="AN0192656324-19">Follow-up and Outcome Ascertainment</hd> <p>We specify how outcomes are defined (e.g., incident or prevalent), what constructs are considered, how they are measured, and for how long they will be assessed. These details add precision that can aid future interventional work or policy actions to reduce disparity. For example, if our enrollment window is indexed around an incident diagnosis of hypertension, resolving disparities early on may require a focus on addressing patient knowledge, awareness, and structures that prevent adherence to a healthy diet and regular physical activity. Resolving disparities five years post-onset also involves supports to improve medication adherence, enable home-based blood-pressure monitoring, and resources and protocols to facilitate timely and appropriate treatment intensification by clinicians for patients with uncontrolled hypertension.</p> <hd id="AN0192656324-20">Statistical Analysis</hd> <p>Last, we need to specify how the data will be analyzed. We choose the scale (e.g., additive or ratio) and coding of the outcome (shortfall [e.g., uncontrolled hypertension] or gain [controlled hypertension]) for reporting disparity. For repeatedly measured outcomes or time-to-event outcomes, we also choose whether to present measures indexed at the end of follow-up (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>J</mi></math> </ephtml> ) or to present a graphical summary of the disparity or of group-specific measures indexed at each time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>j</mi></math> </ephtml> during follow-up (i.e., at time on study <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi></math> </ephtml> ) such as group-specific growth curves, cumulative incidence curves, or survival curves.</p> <p>If there are multiple target studies across calendar times <emph>k</emph>, we can always present trends in disparity or group-specific outcomes across calendar time <emph>k</emph>. We may also provide a summary measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Ψ</mi></mrow></math> </ephtml> , which is a weighted average of calendar-specific disparity estimates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> for outcomes indexed at any point during follow-up <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>j</mi></math> </ephtml> (or the end of follow-up <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>J</mi></math> </ephtml> ), as in (<reflink idref="bib2" id="ref62">2</reflink>). To summarize the additive disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup></math> </ephtml> using (<reflink idref="bib2" id="ref63">2</reflink>), we suggest the weight <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> which is the probability that an instance of enrollment among the standard population has calendar time <emph>k</emph>. To summarize the relative disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup></math> </ephtml> , the weight <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub></math> </ephtml> is multiplied by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , the mean outcome among the enrolled privileged group at time <emph>k</emph> ([<reflink idref="bib55" id="ref64">55</reflink>]). This approach standardizes the additive disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup></math> </ephtml> (or the relative disparity <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup></math> </ephtml> ) to the distribution of enrollment timing among the standard population (see the subsection "Standard Distribution"), removing impacts of differential enrollment timing (see the subsection "Temporal Framing").[<reflink idref="bib19" id="ref65">19</reflink>] The summary measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Ψ</mi></mrow></math> </ephtml> is interpretable as a difference in standardized mean outcomes (for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup></math> </ephtml> ) or a ratio of standardized mean outcomes (for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup></math> </ephtml> ). Thus, one may alternatively pool instances of enrollment and weight each instance in the statistical analysis by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></math> </ephtml></p> <p>Graph</p> <p>Such a pooled analysis permits aggregation of trends in the outcome over the target study timescale <emph>j</emph>.</p> <p>A person may be eligible many times (e.g., they may have prior hypertension at multiple visits). Of course, under certain eligibility criteria (e.g., recent onset of hypertension) a person may only be eligible at one point in calendar time. When persons enroll in multiple studies over calendar time, this leads to correlated outcomes which can be addressed by using a stratified cluster bootstrap ([<reflink idref="bib17" id="ref66">17</reflink>]; [<reflink idref="bib24" id="ref67">24</reflink>]; [<reflink idref="bib65" id="ref68">65</reflink>]; [<reflink idref="bib34" id="ref69">34</reflink>]) to obtain confidence intervals.</p> <hd id="AN0192656324-21">Extension of the Target Study to Address Nonrandom Sample Selection</hd> <p></p> <hd id="AN0192656324-22">Overview</hd> <p>When investigators wish to include all contributions of non-random sample selection to disparity, the target study described in the section "A Target Study Conceptual Model for Measuring Disparity" is sufficient. To address non-random sample selection, we introduce sampling strategies that allow data from the eligible population described in the previous section, denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> , to infer about disparity that may exist in a broader population (Design 2), a different population (Design 3), or a counterfactual population (Design 4) in which collider stratification from selecting on full or partial eligibility does not occur. Designs 2 and 3 allow inference to studies where eligibility criteria are changed whereas Design 4 allows inference to target studies that, through intervention, change who is eligible. Design 4, if chosen, adds a causal element to the model in that the eligibility-related variables are intervened on, but Design 4 remains descriptive with respect to social group membership and allowables. We will see that, unlike Designs 2 and 3, Design 4 may be used when eligibility-related variables affect the outcome.</p> <p>In addition to the innocuous sampling assumption (<reflink idref="bib4" id="ref70">4</reflink>) and variants of the overlap assumption (<reflink idref="bib3" id="ref71">3</reflink>), each modified sampling design relies on independence (or exchangeability) assumptions and positivity assumptions. In each design, these additional assumptions may partly depend on a set of non-allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., socioeconomic status) that are measured by the time of enrollment <emph>k</emph> and, when specified, are factored into the sampling design. Specifically, the first-stage and second-stage sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi mathvariant="script">D</mi></mrow></msubsup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mi mathvariant="script">D</mi></mrow></msubsup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> may depend on the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> . When the target study design uses non-allowalbes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , ultimately, it does not balance them across social groups <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> .[<reflink idref="bib20" id="ref72">20</reflink>]</p> <p>Designs 2 and 3 operate under the same minimal structural constraints as Design 1.[<reflink idref="bib21" id="ref73">21</reflink>] The sampling strategy for Design 4, invoking counterfactuals, has more constraints which we discuss later. Aside from the sampling plan and aggregation over calendar time, the other elements are unchanged from Design 1.</p> <hd id="AN0192656324-23">Design 2: Sampling as if from a Broader Population (Generalizability)</hd> <p>As in Figure 1b, suppose that the indicator of full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> (1: yes, 0: no) is based on partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> (1: yes if has prior hypertension, established care in the health system, with a visit within <emph>k</emph>, 0: no otherwise) and partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (1: yes if enrolled in the EPPP, 0: otherwise), i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>×</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> . Suppose we must study persons with prior hypertension, established care, a current visit who are enrolled in the EPPP (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> as in Figure 2a) but want to assess disparity regardless of EPPP enrollment (i.e., a broader version of eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>*</mo></math> </ephtml> =1 based on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> alone, as in Figure 2b). For this we use first-stage sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> that act as if we sample the broader population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> alone: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>Graph: Figure 2. Venn diagrams depicting populations eligible and inferred to under Designs 2, 3, and 4 using partial eligibility indicators (labeled Qk‡, Qk†, and Qk≀ [1:yes, 0: no]) based on eligibility-related variables Wk‡, Wk†, and Wk≀ (where Wk† affects Wk≀). The first row pertains to a target study (Design 2) that (a) enrolls a fully eligible population defined by Qk=1, i.e., Qk‡=1 and Qk†=1 but (b) infers to a broader population defined by Qk‡=1. The second row pertains to a target study (Design 3) that (c) enrolls a fully eligible population defined by Qk=1, i.e., Qk‡=1 and Qk†=1 but (d) infers to a different population defined by Qk‡=1 and Qk†=0. The third row pertains a target study (Design 4) that (e) enrolls a fully eligible population defined by Qk=1, i.e., Qk‡=1, Qk†=1, and Qk≀=1 but (d) infers to a fully eligible counterfactual population defined by QkGk=1, i.e., Qk‡=1, Qk†Gk=1, and Qk≀Gk=1 after an intervention Gk that eliminates collider stratification through Wk†. A target study (Design 1) may enroll and infer to the same fully eligible population defined by Qk=1.</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> is the size of the first-stage sampling frame <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , the source population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">P</mi></mrow><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> . We use second-stage sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> to create a final sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> where the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are balanced across groups to follow their distribution in the standard population, defined among <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup></math> </ephtml> [collapsed over R]: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msup><mi>Q</mi><mtext>‡</mtext></msup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msup><mi>Q</mi><mtext>‡</mtext></msup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>When the marginalized group in the broader population is the standard population, the marginalized group undergoes simple random sampling in stage 2, so that its expected outcome in the target study and in the broader population are the same (i.e., the inference for the marginalized group is purely descriptive of the marginalized group in the broader population).</p> <p>The design permits inference to the broader population under an independence assumption:</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>∐</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">a</mi><mo mathvariant="bold">,</mo></mrow><mi>k</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">for</mi><mspace width=".1em" /><mi mathvariant="normal">all</mi></mrow><mspace width=".1em" /><mi>k</mi></math> </ephtml> </p> <p>Graph</p> <p>In words, the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) must be independent of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> (e.g., based on EPPP enrollment) among the broader population (denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> ) given social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., age and sex), and non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., SES). This assumption would hold in Figure 1b if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment) did not affect the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (i.e., the arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo stretchy="false">→</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> is absent). A positivity assumption is also required: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml></p> <p>Graph</p> <p>for all <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml> and all <emph>k</emph>.</p> <p>For each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , we note at each time <emph>k</emph> the pattern of allowable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and non-allowable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> covariate values among the broader population (e.g., denoted only by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> ). For each pattern, we must observe persons who belong to the population that our target study enrolls (i.e., persons who meet our narrower version of full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> based on partial eligibility indicators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> ). The overlap assumption (<reflink idref="bib3" id="ref74">3</reflink>) needs to hold among the broader population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> .[<reflink idref="bib22" id="ref75">22</reflink>]</p> <hd id="AN0192656324-24">Design 3: Sampling as if from a Different Population (Transportability)</hd> <p>Suppose again that full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> is based on partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (prior hypertension, established care, current visit) and partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (enrolled in the EPPP) as in Figure 1b. The target study again enrolls those who are fully eligible (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> as in Figure 2c) but we want to assess disparity for those not enrolled in the EPPP (i.e., a different version of full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mrow><mo>*</mo><mo>*</mo></mrow></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> based on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn></math> </ephtml> , as in Figure 2d). For this we use modified first-stage sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> that act as if we sample the different population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> is the size of the first-stage sampling frame <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , i.e., the source population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">P</mi></mrow><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> . We use second-stage sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> to create a final sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> where allowables are balanced across groups to their distribution in the standard population, defined among <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup></math> </ephtml> [collapsed over R ]: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>When the marginalized group in the different population is the standard population, the marginalized group undergoes simple random sampling in stage 2, so that its expected outcome in the target study and in the different population is the same (i.e., the inference for the marginalized group is purely descriptive of the marginalized group in the different population).</p> <p>The design permits inference to the different population under the independence assumption (<reflink idref="bib11" id="ref76">11</reflink>) which, again, would hold in Figure 1b if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment) did not affect the outcome (e.g., hypertension control <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (i.e., the arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo stretchy="false">→</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> ) is absent. A positivity assumption is also required: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml></p> <p>Graph</p> <p>for all <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo><mn>0</mn></math> </ephtml> and all <emph>k</emph>.</p> <p>For each social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , we note at each time <emph>k</emph> the pattern of allowable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and nonallowable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> covariate values among the different population (e.g., denoted by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn></math> </ephtml> ). For each pattern, we must observe persons who belong to the population our target study enrolls (i.e., who meet our narrower version of full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> based on partial eligibility indicators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> ). The overlap assumption (<reflink idref="bib3" id="ref77">3</reflink>) needs to hold among the different population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> .[<reflink idref="bib23" id="ref78">23</reflink>]</p> <hd id="AN0192656324-25">Design 4: Sampling as if from a Counterfactual Population (Inference in a Selected Population...</hd> <p>As in Figure 1c, now we express full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> (1: yes, 0: no) with finer partial eligibility indicators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> (prior hypertension, established care), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment), and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup></math> </ephtml> (e.g., current visit), i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><msup><mi>Q</mi><mo>≀</mo></msup><mo>×</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>×</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> , where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.25em" /><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mtext>‡</mtext></msubsup></math> </ephtml> may affect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> which may affect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>≀</mo></msubsup></math> </ephtml> .[<reflink idref="bib24" id="ref79">24</reflink>] Suppose we want to infer to those with full eligibility (Figure 2e) but worry that selecting persons enrolled in the EPPP (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> ) induces collider-stratification that masks disparity. Suppose that we accept collider stratification through conditioning on other partial eligibility indicators <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> (e.g., prior hypertension, established care) and conditioning on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup></math> </ephtml> (e.g., current visit) as part of disparity. To avoid collider stratification through <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> , we infer to a counterfactual population where such collider-stratification is absent (Figure 2f). Denote <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>k</mi></msub></math> </ephtml> as an intervention to allocate[<reflink idref="bib25" id="ref80">25</reflink>] the partial eligibility variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment) according to a distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>g</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> that does not simultaneously depend on (i) social group <emph>R</emph> and (ii) non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., risk factors <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> ) (Table 1). Let <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>V</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup></math> </ephtml> be the potential outcome of a variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>k</mi></msub></math> </ephtml> under intervention <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>k</mi></msub></math> </ephtml> . Our use of a superscript to denote the intervention <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>k</mi></msub></math> </ephtml> differs from our use of a superscript to denote a sampling design <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">D</mi></mrow></math> </ephtml> . We use sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> that act as if we sample the counterfactual population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> : <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>Table 1. Example Interventions to Eliminate Forms of Collider-Stratification Under Design 4.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="center" /></colgroup><thead><tr><th align="left">Sub-design</th><th align="left">Definition of the intervention <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>G</mi><mi>k</mi></msub></math></p> to allocate <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> according to the distribution <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>g</mi><mi>k</mi></msub><mo stretchy="false" xmlns="">(</mo><mo xmlns="">⋅</mo><mo stretchy="false" xmlns="">)</mo></math></p></th><th align="left">The allocation strategy <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>g</mi><mi>k</mi></msub><mo stretchy="false" xmlns="">(</mo><mo xmlns="">⋅</mo><mo stretchy="false" xmlns="">)</mo></math></p> for <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> under <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>G</mi><mi>k</mi></msub></math></p></th><th align="left">Distribution <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>q</mi><mi>k</mi></msub><mo stretchy="false" xmlns="">(</mo><mo xmlns="">⋅</mo><mo stretchy="false" xmlns="">)</mo></math></p> of partial eligibility <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn></math></p> under <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>G</mi><mi>k</mi></msub></math></p></th></tr></thead><tbody><tr><td>4a</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo xmlns="">∼</mo><msub xmlns=""><mi>g</mi><mi>k</mi></msub><mo stretchy="false" xmlns="">(</mo><mo xmlns="">⋅</mo><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mi xmlns="">P</mi><mo stretchy="false" xmlns="">(</mo><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo fence="false" stretchy="false" xmlns="">|</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><mi xmlns="">k</mi><mo stretchy="false" xmlns="">)</mo></math></p></td><td>Randomly assign <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> (e.g., EPPP enrollment) by the observed probability of <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo xmlns="">=</mo><msup xmlns=""><mrow><mi mathvariant="bold-italic">w</mi></mrow><mo>†</mo></msup></math></p> among those with <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn></math></p> (e.g., prior hypertension) at calendar time <italic>k</italic>. This removes associations between <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> and each of the allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> (e.g., age <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>X</mi><mi>k</mi></msub></math></p>)<bold>,</bold> non-allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> (e.g., SES <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>L</mi><mi>k</mi></msub></math></p>), and social group <italic>R</italic>.</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">P</mi><mo stretchy="false" xmlns="">(</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo fence="false" stretchy="false" xmlns="">|</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><mi xmlns="">k</mi><mo stretchy="false" xmlns="">)</mo></math></p></td></tr><tr><td>4b</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo xmlns="">∼</mo><msub xmlns=""><mi>g</mi><mi>k</mi></msub><mo stretchy="false" xmlns="">(</mo><mo xmlns="">⋅</mo><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mi xmlns="">P</mi><mo stretchy="false" xmlns="">(</mo><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo fence="false" stretchy="false" xmlns="">|</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><mi xmlns="">R</mi><mo xmlns="">=</mo><mi xmlns="">r</mi><mo xmlns="">,</mo><msub xmlns=""><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo xmlns="">,</mo><mi xmlns="">k</mi><mo stretchy="false" xmlns="">)</mo></math></p></td><td>Randomly assign <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> (e.g., EPPP enrollment) by the observed probability of <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo xmlns="">=</mo><msup xmlns=""><mrow><mi mathvariant="bold-italic">w</mi></mrow><mo>†</mo></msup></math></p> among those with <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn></math></p> (e.g., prior hypertension) given social group <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">R</mi><mo xmlns="">=</mo><mi xmlns="">r</mi></math></p>, the allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> (e.g., age <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>X</mi><mi>k</mi></msub></math></p>) at calendar time <italic>k</italic>. This removes direct associations (with respect to social group <italic>R</italic> and allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo stretchy="false" xmlns="">)</mo><mspace width="0.25em" xmlns="" /></math></p>between <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> and non-allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> (e.g., SES <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>L</mi><mi>k</mi></msub></math></p>).</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">P</mi><mo stretchy="false" xmlns="">(</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo fence="false" stretchy="false" xmlns="">|</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><mi xmlns="">R</mi><mo xmlns="">=</mo><mi xmlns="">r</mi><mo xmlns="">,</mo><msub xmlns=""><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo xmlns="">,</mo><mi xmlns="">k</mi><mo stretchy="false" xmlns="">)</mo></math></p></td></tr><tr><td>4c</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo xmlns="">∼</mo><msub xmlns=""><mi>g</mi><mi>k</mi></msub><mo stretchy="false" xmlns="">(</mo><mo xmlns="">⋅</mo><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mi xmlns="">P</mi><mo stretchy="false" xmlns="">(</mo><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">w</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo fence="false" stretchy="false" xmlns="">|</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><mi xmlns="">T</mi><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><msub xmlns=""><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo xmlns="">,</mo><msub xmlns=""><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo xmlns="">,</mo><mi xmlns="">k</mi><mo stretchy="false" xmlns="">)</mo></math></p></td><td>Randomly assign <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> (e.g., EPPP enrollment) by the observed probability of <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo xmlns="">=</mo><msup xmlns=""><mrow><mi mathvariant="bold-italic">w</mi></mrow><mo>†</mo></msup></math></p> among those with <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn></math></p> (e.g., prior hypertension) in the standard population <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">T</mi><mo xmlns="">=</mo><mn xmlns="">1</mn></math></p> given the allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> (e.g., age <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>X</mi><mi>k</mi></msub></math></p>) and a set of non-allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> (e.g., SES <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mi>L</mi><mi>k</mi></msub></math></p>) that satisfy exchangeability (18) or (19) at calendar time <italic>k</italic>. This removes direct associations (with respect to allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p> and non-allowables <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub xmlns=""><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math></p>) between <p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup xmlns=""><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math></p> and social group <italic>R</italic>.a</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">P</mi><mo stretchy="false" xmlns="">(</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo fence="false" stretchy="false" xmlns="">|</mo><msubsup xmlns=""><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><mi xmlns="">T</mi><mo xmlns="">=</mo><mn xmlns="">1</mn><mo xmlns="">,</mo><msub xmlns=""><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo xmlns="">,</mo><msub xmlns=""><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo xmlns="">,</mo><mi xmlns="">k</mi><mo stretchy="false" xmlns="">)</mo></math></p></td></tr></tbody></table> </ephtml> </p> <p>1 As explained at the end of the subsection "Design 4: Sampling as if from a Counterfactual Population (Inference in a Selected Population)" and in Footnote 22, when the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are multivariate and follow certain causal structures, Design 4c may leave residual contributions from collider stratification that would be eliminated under Designs 4a and 4b.</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> is the size of the first-stage sampling frame <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>0</mn></mrow><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , i.e., the counterfactual source population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="script">P</mi></mrow><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> . We use second-stage sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> to create a final sample <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> where the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are balanced according to the standard distribution defined among <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mrow><mrow><mi mathvariant="double-struck">S</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup></math> </ephtml> [collapsed over R]: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mrow><mspace width="0.25em" /></mrow><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>This design permits inference to the counterfactual population via an exchangeability assumption:[<reflink idref="bib26" id="ref81">26</reflink>] <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msubsup><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>,</mo><mi>Q</mi><msup><mrow><msubsup><mrow /><mi>k</mi><mo>≀</mo></msubsup></mrow><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msup></mrow><mo stretchy="false">)</mo><mo>∐</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">for</mi><mspace width=".1em" /><mi mathvariant="normal">all</mi></mrow><mspace width=".1em" /><mi>k</mi></math> </ephtml></p> <p>Graph</p> <p>In words, the potential outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup></math> </ephtml> (e.g., hypertension control) and the potential value of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mrow><mo>≀</mo><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></mrow></msubsup></math> </ephtml> (e.g., current visit) must be jointly independent of observed partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment) given partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> , social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , and non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , i.e., no unmeasured selection-bias. Positivity (<reflink idref="bib12" id="ref82">12</reflink>) is required as well as consistency: the intervention <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>k</mi></msub></math> </ephtml> returns observed values for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup></math> </ephtml> when it assigns a person's observed values for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> . The overlap assumption (<reflink idref="bib3" id="ref83">3</reflink>) needs to hold among the counterfactual population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> .[<reflink idref="bib27" id="ref84">27</reflink>] Now, if no partial eligibility variables occur after <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> (i.e., if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>≀</mo></msubsup><mo>=</mo><mo>⊘</mo></math> </ephtml> the empty set) exchangeability (<reflink idref="bib18" id="ref85">18</reflink>) simplifies to: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>∐</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi><mspace width=".1em" /><mrow><mi mathvariant="normal">for</mi><mspace width=".1em" /><mi mathvariant="normal">all</mi></mrow><mspace width=".1em" /><mi>k</mi></math> </ephtml></p> <p>Graph</p> <p>The exchangeability assumptions (<reflink idref="bib18" id="ref86">18</reflink>) or (<reflink idref="bib19" id="ref87">19</reflink>) of Design 4 holds where the independence assumption (<reflink idref="bib11" id="ref88">11</reflink>) of Designs 2 and 3 fails: when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> affects the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> , i.e., the arrow <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo stretchy="false">→</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mspace width="0.25em" /></mrow></math> </ephtml> in Figure 1c (see Figure 4). Design 4 operates under additional structural constraints compared to Designs 1, 2, and 3.[<reflink idref="bib28" id="ref89">28</reflink>]</p> <p>Table 1 specifies interventions for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>k</mi></msub></math> </ephtml> (Designs 4a, 4b, and 4c) that eliminate collider stratification through partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> . Design 4a makes partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> random given partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and calendar time <emph>k</emph>. Design 4b makes partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> random with respect to non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> given partially eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> , social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and calendar time <emph>k</emph>. Design 4c makes partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> random with respect to social group <emph>R</emph> given partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> , the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , a set of non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> that satisfy exchangeability (<reflink idref="bib18" id="ref90">18</reflink>) or (<reflink idref="bib19" id="ref91">19</reflink>), and calendar time <emph>k</emph>. These designs target different counterfactual populations and may return different estimates of disparity. To choose, one may consider the design's feasibility (in how <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> is allocated) or the design's inferential utility. When there are no downstream partial eligibility-related variables (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>≀</mo></msubsup><mo>=</mo><mo>⊘</mo></math> </ephtml> ), Design 4a generalizes to the broader population defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> even when eligibility variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> affect the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (unlike Design 2). Under the same conditions, when the marginalized group is the standard population, Design 4c reduces to simple random sampling among the marginalized group across both stages of sampling. Then, the model fully describes the fully eligible marginalized group defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> . One may also consider meaningfulness. Designs 4b and 4c do not remove all social group differences in partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> , which may be a more 'realistic' setting for characterizing disparity in outcomes. Finally, one may also consider effectiveness. When <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> is multivariate, under certain causal structures Design 4c, unlike Designs 4a and 4b, may leave some residual collider-stratification through conditioning on partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> .[<reflink idref="bib29" id="ref92">29</reflink>]</p> <hd id="AN0192656324-26">Modified Statistical Analysis</hd> <p>In the subsections "Overview" and "Statistical Analysis", we discussed procedures to aggregate results over calendar time use the distribution in the standard population implied by the design. This is the broader population under Design 2, the different population under Design 3, and the counterfactual population under Design 4.[<reflink idref="bib30" id="ref93">30</reflink>]</p> <hd id="AN0192656324-27">Emulation of the Target Study with Secondary Data</hd> <p></p> <hd id="AN0192656324-28">Overview</hd> <p>In theory, the target study protocol could be implemented in real life to measure disparity. Often, a target study will have to be emulated through the design and analysis of secondary data. We outline data structures and estimators to emulate the target study under our motivating example of assessing racial disparity in hypertension control in a healthcare system among those with prior hypertension, established care and a current visit who are (a) enrolled in the EPPP (Design 1); (b) may or may not be enrolled in EPPP (Design 2); not enrolled in the EPPP (Design 3); enrolled in the EPPP under a hypothetical allocation of EPPP (Design 4). These applications are plausible when we only have outcomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> measured in the EPPP. The example target study protocols and emulation steps for each design are shown in Table 2. In this example, we assume multiple target studies across calendar time whose results are to be aggregated. Recall that with multiple target studies, a person may possibly be eligible for and enroll in multiple studies. We show how the emulation simplifies with one target study. For each design <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">D</mi></mrow></math> </ephtml> , we present estimators for each social group's mean outcome aggregated over calendar time <emph>k</emph>, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mi mathvariant="script">D</mi></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> (see the subsections "Overview", "Statistical Analysis", and "Modified Statistical Analysis"). The aggregated additive disparity is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="normal">Ψ</mi></mrow><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msup><mo>=</mo><msup><mi>τ</mi><mrow><mi mathvariant="script">D</mi></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><msup><mi>τ</mi><mrow><mi mathvariant="script">D</mi></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> and the relative disparity is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="normal">Ψ</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msup><mo>=</mo><msup><mi>τ</mi><mrow><mi mathvariant="script">D</mi></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>/</mo><msup><mi>τ</mi><mrow><mi mathvariant="script">D</mi></mrow></msup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>Table 2. Example Target Study Protocol Specification and its Emulation With Secondary Data.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /></colgroup><thead><tr><th align="left" /><th align="left">Target Study</th><th align="left">Emulation with EMR data</th></tr></thead><tbody><tr><td>Enrollment windows</td><td>Weekly over 2015</td><td>Same</td></tr><tr><td>Enrollment groups</td><td>Self-reported Non-Hispanic Black persons and Non-Hispanic White Persons</td><td>Same, implemented as self-reported race/ethnicity as recorded in EMR</td></tr><tr><td>Eligibility criteria</td><td>Established care in health system, prior diagnosis of hypertension, not pregnant, not diagnosed with ESKD, enrolled in EPPP, and current visit for primary care</td><td>Same, implemented as 2+ primary care visit in past 2 years, diagnosis of hypertension, in past 2 years, not pregnant, not diagnosed with ESKD, enrolled in EPPP before current visit</td></tr><tr><td>Allowable covariates</td><td>Age and sex assigned at birth</td><td>Same, implemented using age and sexa in EMR at current visit</td></tr><tr><td>Standard population</td><td>Black population</td><td>Same</td></tr><tr><td>Enrollment Process</td><td>Stratified sampling...</td><td>With pooled data (Figure 4), apply...</td></tr><tr><td><italic>...Design 1 for inference in the fully eligible population</italic></td><td>...to balance age and sex (allowable), sampling from the eligible population</td><td>...G-Computation (20) or weighting (21) using age and sex (allowable)</td></tr><tr><td><italic>...Design 2: for inference in the broader population (e.g., regardless of EPPP enrollment)</italic></td><td>... to balance age and age (allowable), and account for comorbidity and SES (non-allowable), sampling as if from the population regardless of EPPP enrollment</td><td>...G-Computation (22) or weighting (23) using age, sexa (allowable), and comorbidity, and SESb (non-allowable)</td></tr><tr><td><italic>...Design 3: for inference in a different population (e.g., not enrolled in EPPP)</italic></td><td>...to balance age and sex (allowable), and account for comorbidity and SES (non-allowable), sampling as if from the population not enrolled in EPPP</td><td>...G-Computation (24) or weighting (25) using age, sexa (allowable), and comorbidity, and SESb (non-allowable)</td></tr><tr><td><italic>...Design 4: for inference in a counterfactual population after intervention to allocate EPPP enrollment (e.g., to remove the impact of collider-stratification)</italic></td><td>...to balance age and sex (allowable), and account for comorbidity and SES (non-allowable), sampling as if from a counterfactual eligible population after intervening to allocate EPPP enrollment</td><td>...G-Computation (28) or weighting (29) using age, sexa (allowable), and comorbidity, and SESb (non-allowable); or simplified versions of G-computation and weighting, i.e., (22) and (23) under Design 4a, (30) and (31) under Design 4b, or (32) and (33) under Design 4c</td></tr><tr><td>Time zero</td><td>Time of current visit</td><td>Same</td></tr><tr><td>Outcome assessment</td><td>Uncontrolled hypertension at current visit (i.e., systolic blood pressure ≥140 mm Hg or diastolic blood pressure ≥90 mm Hg)</td><td>Same</td></tr><tr><td>Statistical analysis</td><td>Mean difference in uncontrolled hypertension</td><td>Same</td></tr><tr><td><italic>...Aggregation of results</italic></td><td>Weighted average of results according to the distribution of calendar time enrollment in the Black population</td><td>Same</td></tr></tbody></table> </ephtml> </p> <ulist> <item>2 Abbreviations: EMR = Electronic Medical Records; EPPP = Electronic Patient Portal Program; ESKD = End Stage Kidney Disease; SES = Socioeconomic Status; Hg = Mercury.</item> <item>3 Sex as recorded in the EMR.</item> <item>4 In the EMR, SES is approximated by health insurance type and categorized CDC Social Vulnerability Index.</item> </ulist> <p>We present two types of estimators that, given the appropriate data structure, are used to emulate the sampling-based enrollment and aggregation. G-computation ([<reflink idref="bib73" id="ref94">73</reflink>]), akin to model-based standardization, sequentially regresses the outcome and predicted values. Weighting ([<reflink idref="bib31" id="ref95">31</reflink>]), which takes a weighted average of the outcome, models membership in the social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , the standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> and, for Designs 2 through 4, indicators of partial eligibility. G-computation estimators are usually more efficient ([<reflink idref="bib64" id="ref96">64</reflink>]) but weighting is more objective as the weights are constructed without outcome data. To construct confidence intervals that account for clustering by individual, we use a cluster bootstrap that samples each individual with replacement ([<reflink idref="bib17" id="ref97">17</reflink>]; [<reflink idref="bib24" id="ref98">24</reflink>]; [<reflink idref="bib65" id="ref99">65</reflink>]; [<reflink idref="bib34" id="ref100">34</reflink>]). We abbreviate a weighted mean, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo movablelimits="false">∑</mo></mrow><mi>i</mi></munder><mspace width="0.2em" /><msub><mi>Y</mi><mi>i</mi></msub><msub><mi>ω</mi><mi>i</mi></msub><mo>/</mo><munder><mrow><mo movablelimits="false">∑</mo></mrow><mi>i</mi></munder><mspace width="0.2em" /><msub><mi>ω</mi><mi>i</mi></msub></math> </ephtml> where <emph>i</emph> represents the unit of observation, as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>Y</mi><mo>×</mo><mi>ω</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> .</p> <hd id="AN0192656324-29">Data Structure</hd> <p>To emulate a target study, we specify a unit of calendar time for enrollment windows (e.g., months), enrollment groups <emph>R</emph> (e.g., Black persons [marginalized <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> ] and White persons [privileged <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>0</mn></math> </ephtml> ]), and full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care, and enrolled in EPPP before <emph>k</emph>, and current visit within <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> ). For Design 1, we do not disaggregate full eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml> into partial eligibility. For Designs 2 and 3, we distinguish partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> that we generalize or transport over (e.g., EPPP enrollment) from partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> that we do not (e.g., prior hypertension, established care, current visit). For Design 4, we distinguish partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> allocated by intervention (e.g., EPPP enrollment), partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup></math> </ephtml> affected by the allocation (e.g., current visit), and partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup></math> </ephtml> not affected by the allocation (e.g., prior hypertension, established care). We choose allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> to similarly situate social groups (e.g., age <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>k</mi></msub></math> </ephtml> ) and the within-sample standard population, coded <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> , determining their within sample distribution (e.g., the Black group). We choose non-allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> to meet assumptions of independence (<reflink idref="bib11" id="ref101">11</reflink>) for Designs 2 and 3 or exchangeability (<reflink idref="bib18" id="ref102">18</reflink>) or (<reflink idref="bib19" id="ref103">19</reflink>) for Design 4 (e.g., SES <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> ).</p> <p>We form a 'long' dataset where every row is the vector <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="script">O</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mrow><mi>i</mi><mo>,</mo><mi>k</mi><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>,</mo><mi>R</mi><mo>,</mo><mi>T</mi><mo>,</mo><msub><mi>X</mi><mi>k</mi></msub><mo>,</mo><msub><mi>L</mi><mi>k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> for an individual <emph>i</emph> at month <emph>k</emph> (Figure 4). Each person <emph>i</emph> contributes one record per calendar time <emph>k</emph>. The data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">O</mi></mrow></math> </ephtml> only include calendar times <emph>k</emph> where all social groups of interest are represented. The indicator <emph>T</emph> of membership in the standard population is constructed (e.g., if the marginalized group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> is the standard population, we set <emph>T</emph> as equal to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> </ephtml> ). For Design 1, we subset the data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">O</mi></mrow></math> </ephtml> to those fully eligible at time <emph>k</emph>, i.e., by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> . For Designs 2, 3, and 4, we subset the data <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="script">O</mi></mrow></math> </ephtml> to those partially eligible by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> . We attach outcomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> at follow-up time <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>+</mo><mi>J</mi></math> </ephtml> to records indexed at time <emph>k</emph>.</p> <hd id="AN0192656324-30">Identification and Estimation for Design 1 (Default Model)</hd> <p>Under overlap (<reflink idref="bib3" id="ref104">3</reflink>) and innocuous sampling (<reflink idref="bib4" id="ref105">4</reflink>), we can identify the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">]</mo><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>To estimate (<reflink idref="bib20" id="ref106">20</reflink>) by G-computation, in step 1 we fit a model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> for the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) given the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., age) and calendar time <emph>k</emph> (e.g., months) among those fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care, current visit, enrolled in EPPP) in the social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . In step 2, we obtain predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup></math> </ephtml> from the model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> on those fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> . In step 3, we average <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup></math> </ephtml> in the pooled, fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> to estimate the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> (conditionally on calendar time <emph>k</emph> for time-specific means, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>μ</mi><mi>k</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> ).</p> <p>We may also estimate the aggregated mean outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> by the weighting estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>×</mo><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo>=</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml></p> <p>The first term of the weight is the ratio of (i) the probability of belonging to the standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> among those fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> , conditional on the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and calendar time <emph>k</emph> to (ii) the corresponding probability of belonging to the person's observed social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . The second term of the weight is inverse of this ratio but with unconditional probabilities (with respect to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></math> </ephtml> ). The probabilities in the first term may be estimated by predictions from models for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , and the probabilities in the second term are estimated directly. A weighted average of the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) in the fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> estimates the aggregate mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> . For time-specific means, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>μ</mi><mi>k</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>1</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , all terms condition on calendar time <emph>k</emph>.</p> <hd id="AN0192656324-31">Identification and Estimation for Design 2 (Generalizability)</hd> <p>Under a version of overlap (<reflink idref="bib3" id="ref107">3</reflink>) (see footnote 16), innocuous sampling (<reflink idref="bib4" id="ref108">4</reflink>), independence (<reflink idref="bib11" id="ref109">11</reflink>), and positivity (<reflink idref="bib12" id="ref110">12</reflink>), we can identify the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of the social group in the broader population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care in the health system, regardless of EPPP enrollment) as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mrow><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi>E</mi><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p></p> <ulist> <item> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mspace width="0.25em" /><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml></item> <p></p> <item> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml></item> </ulist> <p>To estimate (<reflink idref="bib22" id="ref111">22</reflink>) by G-computation, in step 1 we fit a model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> for the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) given the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., age), non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., SES), and calendar time <emph>k</emph> (e.g., months) among the fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care, EPPP enrollment, current visit) social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . In step 2, we obtain predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup></math> </ephtml> from the model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> in broader population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., regardless of EPPP enrollment). In step 3, we fit a model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> for the predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup></math> </ephtml> given the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and calendar time <emph>k</emph> on the broader <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . In step 4, we obtain predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup></math> </ephtml> from the model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> on the broader population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> . In step 5, we average <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup></math> </ephtml> in the pooled broader <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> to estimate the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> (conditionally on calendar time <emph>k</emph> for time-specific means, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>μ</mi><mi>k</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> ).</p> <p>We may also estimate the aggregated mean outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> by the weighting estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>×</mo><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo>=</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml></p> <p>The second and third terms are similar to (<reflink idref="bib21" id="ref112">21</reflink>) but are defined by the broader <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> population (e.g., prior hypertension, established care, current visit) rather than by those fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., also enrolled in the EPPP). The first term is the ratio of (i) the unconditional (with respect to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> probability of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., enrolled in the EPPP) in the broader <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> population (e.g., prior hypertension, established care, current visit) in the social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , to (ii) the corresponding conditional probability given the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , and calendar time <emph>k</emph>. The denominator is estimated by predictions from a model for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> and the numerator is estimated directly. A weighted average of the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) in the fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> estimates the aggregate mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> . For time-specific means, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>μ</mi><mi>k</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , all terms condition on calendar time <emph>k</emph>.</p> <hd id="AN0192656324-32">Identification and Estimation for Design 3 (Transportability)</hd> <p>Under a version of overlap (<reflink idref="bib3" id="ref113">3</reflink>) (see footnote 17), innocuous sampling (<reflink idref="bib4" id="ref114">4</reflink>), independence (<reflink idref="bib11" id="ref115">11</reflink>), and positivity (<reflink idref="bib15" id="ref116">15</reflink>), we identify the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of the social group in the different population ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> ) (e.g., prior hypertension, established care, current visit, but <emph>not</emph> enrolled in the EPPP) as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd /><mtd><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mrow><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi>E</mi><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo></mrow></mrow></mtd></mtr><mtr><mtd /><mtd><mrow><mrow><mrow><mspace width="0.25em" /><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></mrow></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p></p> <ulist> <item> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mspace width="0.25em" /><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml></item> <p></p> <item> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml></item> </ulist> <p>To estimate (<reflink idref="bib24" id="ref117">24</reflink>) by G-computation, in step 1 we fit a model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> for the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) given the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., age), non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., SES), and calendar time <emph>k</emph> (e.g., months) among the fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care, current visit, enrolled in EPPP) social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . In step 2, we obtain predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup></math> </ephtml> from the model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> in the different population ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> (e.g., not enrolled in EPPP). In step 3, we fit a model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> for the predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>1</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup></math> </ephtml> given the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and calendar time <emph>k</emph> among the different ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> . In step 4, we obtain predicted values <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup></math> </ephtml> from the model <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>η</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> on the different population ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> . In step 5, we average <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>p</mi><mn>2</mn><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup></math> </ephtml> in the pooled different ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> to estimate the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> (conditionally on calendar time <emph>k</emph> for time-specific means, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>μ</mi><mi>k</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> ).</p> <p>We may also estimate the aggregated mean outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> by the weighting estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>×</mo><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo>=</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml> .</p> <p>The fourth and fifth terms of the weight are similar to those used in (<reflink idref="bib21" id="ref118">21</reflink>), except that they are among the different population ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> (e.g., prior hypertension, established care, current visit, but not enrolled in EPPP) rather than those fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., also enrolled in the EPPP). The first term of the weight is equivalent to the inverse odds of being fully eligible ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> versus in the different population ( <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>0</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> , conditional on social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , and calendar time <emph>k</emph>. It is estimated by fitting a model for partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> , making predictions, obtaining its complement, and taking the ratio of the compliment to the prediction. The second term is the odds but is unconditional (with respect to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> and is estimated directly. A weighted average of the outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub></math> </ephtml> (e.g., hypertension control) among the fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> who are estimates the aggregate mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> . For calendar-time specific means, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>μ</mi><mi>k</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>3</mn></mrow></msubsup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> , the probabilities in the second term and fourth terms of the weight condition on calendar time <emph>k</emph>.</p> <hd id="AN0192656324-33">Identification and Estimation for Design 4 (Inference in a Counterfactual Selected Population...</hd> <p>Identification and estimation under Design 4 uses special weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> that generally are: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>ϕ</mi><mi>k</mi></msub><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">[</mo><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></mfrac></mrow></mstyle></mrow></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml><ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd /><mtd><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></mtd></mtr><mtr><mtd /><mtd><mo>=</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">[</mo><msub><mi>E</mi><mi>r</mi></msub><mo fence="false" stretchy="false">{</mo><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msup><mi>Q</mi><mtext>‡</mtext></msup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo fence="false" stretchy="false">}</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>Q</mi><mo>≀</mo></msup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msup><mi>Q</mi><mo>†</mo></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mi>Q</mi><mtext>‡</mtext></msup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><mrow><msub><mi>E</mi><mrow><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">k</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><msub><mi>E</mi><mi>r</mi></msub><mo fence="false" stretchy="false">{</mo><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msup><mi>Q</mi><mtext>‡</mtext></msup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo fence="false" stretchy="false">}</mo><mi>P</mi><mo stretchy="false">(</mo><msup><mi>Q</mi><mo>≀</mo></msup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msup><mi>Q</mi><mo>†</mo></msup><mo>=</mo><mn>1</mn><mo>,</mo><msup><mi>Q</mi><mtext>‡</mtext></msup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></mfrac></mrow></mstyle></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p></p> <ulist> <item> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>r</mi></msub><mo fence="false" stretchy="false">{</mo><mo>⋅</mo><mo fence="false" stretchy="false">}</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> ,</item> <p></p> <item> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> ,</item> <p></p> <item> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml></item> </ulist> <p>In (<reflink idref="bib26" id="ref119">26</reflink>) and (<reflink idref="bib27" id="ref120">27</reflink>) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is the probability of being partially eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., enrolled in EPPP) in the counterfactual source population given partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care), social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and calendar time <emph>k</emph> which, under the interventional distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> for the Designs 4a, 4b, and 4c, is shown in the third column of Table 1. The term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> can thus be estimated as the predicted value from an appropriate model for partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> .[<reflink idref="bib31" id="ref121">31</reflink>]</p> <p>The weights <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> account for how non-random selection on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><msup><mrow><msubsup><mrow /><mi>k</mi><mo>≀</mo></msubsup></mrow><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msup></math> </ephtml> affects the distribution of non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>≀</mo></msubsup></math> </ephtml> (e.g., current visit) may be affected by both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment) the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> (e.g., as in Figure 1c and Figure 3). They incorporate a product <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>k</mi></msub></math> </ephtml> of two parts (i) and (ii). The first part (i) is the conditional probability of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., current visit) given other indicators of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> =1 (e.g., EPPP enrollment) and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care), non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , calendar time <emph>k</emph>, and membership in the social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> (in the case of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> ) or the standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> (in the case of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> ). The second part (ii) is either (ii-a) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> as described above (in the case of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> ) or (ii-b) <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> standardized over the conditional distribution of social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> within the standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> (in the case of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> ).[<reflink idref="bib32" id="ref122">32</reflink>] For <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> the numerator is this product <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>k</mi></msub></math> </ephtml> and the denominator standardizes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>k</mi></msub></math> </ephtml> over the conditional distribution of the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> . For <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> the numerator standardizes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>k</mi></msub></math> </ephtml> over the conditional distribution of the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> while the denominator further standardizes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>k</mi></msub></math> </ephtml> over the conditional joint distribution of the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and calendar time <emph>k</emph>. Both <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> may be estimated by G-computation (see sample code in the Supplementary Material).</p> <p>Graph: Figure 3. Single World Intervention Graph ([<reflink idref="bib67" id="ref123">67</reflink>]) depicting Design 4a (a) without intervention (b) with intervention Gk to set the partial eligibility variable Wk† [e.g., enrollment in an electronic patient portal program (EPPP) ] according to a random draw. Yk+J is the outcome (e.g., hypertension control) at time k+J, R represents social group membership (e.g., race), Xk (e.g., age) and Lk (e.g., SES) are covariates that may be deemed allowable Ak or non-allowable Nk. Qk† is the partial eligibility indicator (1: yes, 0: no) for the partial eligibility variable Wk† [e.g., EPPP enrollment) ]. Wk≀ represents another partial eligibility variable (e.g., current visit) affected by Wk†, with its partial eligibility indicator Qk≀ (1: yes, 0: no). For simplicity, the historical process variable H and the partial eligibility variables Wk‡ (e.g., prior hypertension, established care in health system) and their indicator Qk‡ that appear on Figure 1c are omitted but if included would inherit their causal relationships from Figure 1c. Note that on (a) independence (<reflink idref="bib11" id="ref124">11</reflink>) Yk+J∐Qk†|Xk,Lk,R does not hold because Wk† affects Yk+J. However, on (b) exchangeability (<reflink idref="bib18" id="ref125">18</reflink>) (Yk+JGk,Qk≀Gk)∐Qk†|Xk,Lk,R does hold. Note also that (i) though Wk†Gk is randomly assigned, (Xk,Lk) are not independent of Qk†Gk given Qk≀Gk (ii) there is no collider stratification between R and Yk+JGk from conditioning on Qk†Gk.</p> <p>Graph: Figure 4. Example data structure to emulate the target study (i,k,Qk‡,Qk†,Qk≀,R,T,Xk,Lk,Yk+J) where i is a person identifier, k representes the coarsened moment of calendar time, R represents social group membership, T is an indicator of membership in the standard population, and the covariates Xk,Lk are chosen from to select allowable covariates Ak (for all designs) and, if needed, non-allowable covariates Nk (for Designs 2, 3, and 4 that address non-random sample selection), Qk represents full eligibility and Qk‡ and Qk† are partial eligibility indicators for Designs 2 and 3, and Qk≀ is an additional partial eligibility indicator for Design 4, and Yk+J is the outcome. Assuming Design 4, Person 1 is evaluated for seven studies and eligible (i.e., Qk=1) for two (at k=4,10). Person 2 is evaluated for three studies and eligible for one (at k=7). For Design 1, we subset the data to those fully eligible Qk=1. For Designs 2, 3, and 4, we subset the data to those partially eligible by Qk‡=1.</p> <p>Under a version of overlap (<reflink idref="bib3" id="ref126">3</reflink>) (see footnote 21), innocuous sampling (<reflink idref="bib4" id="ref127">4</reflink>), exchangeability (<reflink idref="bib18" id="ref128">18</reflink>), positivity (<reflink idref="bib12" id="ref129">12</reflink>), and consistency, we identify the aggregated mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> of the social group in the fully eligible counterfactual population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care, enrolled in EPPP, current visit) after an intervention <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>G</mi><mi>k</mi></msub></math> </ephtml> on partial eligibility-related variables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> (e.g., EPPP enrollment) as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em"><mtr><mtd /><mtd><mrow><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mrow><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msub><mi>ϕ</mi><mi>k</mi></msub><mi>E</mi><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo></mrow></mrow></mrow></mtd></mtr><mtr><mtd /><mtd><mrow><mrow><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></mtd></mtr></mtable></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , identified by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> , identified by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>To estimate (<reflink idref="bib28" id="ref130">28</reflink>) by G-computation, it suffices to follow the same procedure as for Design 2 (see the subsection "Identification and Estimation for Design 1 (Default Model)") with a slight change, to weight the model in step 3 by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> and use <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> as a weight for a weighted average for step 5. For calendar-time specific means, we condition the last expectation and all terms in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> on <emph>k</emph>.</p> <p>We may also estimate the aggregated mean outcome <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> by the weighting estimator: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>×</mo><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn></mrow></msubsup><mo>=</mo><msub><mi>ϕ</mi><mi>k</mi></msub><mo>×</mo><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mstyle displaystyle="true" scriptlevel="0"><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></mstyle></mstyle></mstyle></mstyle></math> </ephtml></p> <p>Graph</p> <p>The first two terms are <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> . The third term is the ratio of the unconditional probability (with respect to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> of being partially eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., current visit) given other indicators of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml> =1 (e.g., EPPP enrollment) and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> (e.g., prior hypertension, established care), and social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> to the corresponding conditional probability given the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , and calendar time <emph>k</emph>. The remaining terms are identical to the expressions for the weight (<reflink idref="bib23" id="ref131">23</reflink>) used in Design 2. For calendar-time specific means, we condition all terms in the weight and all terms in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> on <emph>k</emph>.</p> <hd id="AN0192656324-34">Identification and Estimation of Design 4 Under Simplifying Conditions</hd> <p>Emulation of Design 4 simplifies greatly with no indicators of partial eligibility <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup></math> </ephtml> affected by an intervention on the partial eligibility-related variable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> . All terms for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mo>≀</mo></msubsup></math> </ephtml> in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ϕ</mi><mi>k</mi></msub></math> </ephtml> (<reflink idref="bib26" id="ref132">26</reflink>), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> (<reflink idref="bib27" id="ref133">27</reflink>), in the estimators (<reflink idref="bib28" id="ref134">28</reflink>) and (<reflink idref="bib29" id="ref135">29</reflink>) disappear. Under Design 4a (i.e., randomly assign <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> ; Table 1), the term <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> cancels and the estimators (<reflink idref="bib28" id="ref136">28</reflink>) and (<reflink idref="bib29" id="ref137">29</reflink>) reduce to those of Design 2, i.e., (<reflink idref="bib22" id="ref138">22</reflink>) and (<reflink idref="bib23" id="ref139">23</reflink>). Then, Design 4a generalizes the results <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Ψ</mi></mrow></math> </ephtml> to those partially eligible by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn></math> </ephtml> even when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> affects the outcome.</p> <p>Under Design 4b, (i.e., randomly assign <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> ; Table 1), the identifying expression for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mi>b</mi></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> behind the G-computation estimator reduces to: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mrow><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi>E</mi><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mspace width="0.25em" /><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>The weighting estimator reduces to: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>×</mo><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mi>b</mi><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mi>b</mi><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo>=</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml> .</p> <p>Then, Design 4b addresses non-random selection while retaining social group differences in eligibility. Note that if the chosen allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are sufficient to satisfy independence (<reflink idref="bib11" id="ref140">11</reflink>) or exchangeability (<reflink idref="bib18" id="ref141">18</reflink>), so that no non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are needed, the estimators (<reflink idref="bib30" id="ref142">30</reflink>) and (<reflink idref="bib31" id="ref143">31</reflink>) reduce to (<reflink idref="bib20" id="ref144">20</reflink>) and (<reflink idref="bib21" id="ref145">21</reflink>) of Design 1.</p> <p>Under Design 4c (i.e., randomly assign <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> given <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> as in <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>1</mn></math> </ephtml> ; Table 1), <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>θ</mi><mi>k</mi><mrow><mspace width=".1em" /><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup></math> </ephtml> reduces to one, and the identifying expression for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>τ</mi><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mi>c</mi></mrow></msup><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> behind the G-computation estimator reduces to: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mrow><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msubsup><mi>ϕ</mi><mi>k</mi><mi>c</mi></msubsup><mi>E</mi><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , identified by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ϕ</mi><mi>k</mi><mi>c</mi></msubsup><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>k</mi></mrow></msub><mo stretchy="false">[</mo><mspace width="-0.15em" /><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo><mspace width="-0.15em" /><mo stretchy="false">]</mo></math> </ephtml> is over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ϕ</mi><mi>k</mi><mi>c</mi></msubsup><mo>=</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><msub><mrow><mi mathvariant="bold-italic">E</mi></mrow><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">[</mo><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">]</mo></mrow></mfrac></mrow></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mi mathvariant="bold-italic">n</mi></mrow></msub><mo stretchy="false">[</mo><mo>⋅</mo><mo stretchy="false">]</mo></math> </ephtml> over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>The weighting estimator reduces to: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo>×</mo><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mi>c</mi><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">]</mo></math> </ephtml></p> <p>Graph</p> <p>where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ω</mi><mrow><mi>r</mi><mo>,</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><mi>k</mi></mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow><mn>4</mn><mi>c</mi><mo>−</mo><mi>p</mi><mi>o</mi><mi>o</mi><mi>l</mi></mrow></msubsup><mo>=</mo><msubsup><mi>ϕ</mi><mi>k</mi><mi>c</mi></msubsup><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>×</mo><mrow><mfrac><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>P</mi><mo stretchy="false">(</mo><mi>T</mi><mo>=</mo><mn>1</mn><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow></math> </ephtml> .</p> <p>Estimation under Design 4c is thus similar to that of Design 4b except with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ϕ</mi><mi>k</mi><mi>c</mi></msubsup></math> </ephtml> factored in. If the marginalized group is the standard population, its aggregated mean outcome (<reflink idref="bib32" id="ref146">32</reflink>) reduces to its pooled mean <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msub><mi>Q</mi><mi>k</mi></msub></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>,</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> and its weight (<reflink idref="bib33" id="ref147">33</reflink>) reduces to one. Then, the target study model is purely descriptive of the fully eligible <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Q</mi><mi>k</mi></msub><mo>=</mo><mn>1</mn></math> </ephtml> marginalized group (i.e., its expected outcome in the target study and in the fully eligible population are the same), even when addressing non-random sample selection. It does so by making the selection process of the privileged group match that of the marginalized group.</p> <hd id="AN0192656324-35">Contributions and Comparison to Existing Literature</hd> <p></p> <hd id="AN0192656324-36">Target Trial Emulation</hd> <p>Our target study model is inspired by the trial emulation framework ([<reflink idref="bib32" id="ref148">32</reflink>]) used to evaluate causal effects of treatment strategies. Our work differs in that (i) there are no treatment groups, only observed social groups; (ii) we balance allowable covariates by design (sampling) rather than balance confounders by intervention (randomization); (iii) under multiple studies across calendar time, we enforce one person per unit of time, all social groups compared must be represented at each unit of time, and our aggregated estimators adjust for calendar time enrollment by balancing it across groups, without any assumption that disparity is homogeneous over time. Our summary estimate is interpretable as a weighted average of disparity measures for populations indexed at different points in calendar time, and it avoids potential bias due to differential timing of enrollment by social groups. Our model can incorporate treatment strategies and within-group randomization post sampling to evaluate causal effects of treatment strategies on disparity ([<reflink idref="bib39" id="ref149">39</reflink>]).</p> <hd id="AN0192656324-37">Sampling as a Conceptual Model</hd> <p>[<reflink idref="bib77" id="ref150">77</reflink>], [<reflink idref="bib48" id="ref151">48</reflink>], and [<reflink idref="bib14" id="ref152">14</reflink>], use sampling designs of target populations to generalize or transport results from randomized trials. [<reflink idref="bib78" id="ref153">78</reflink>] proposes simple random sampling to define causal effects of observing social groups to measure inequality. [<reflink idref="bib52" id="ref154">52</reflink>] uses simple random sampling to critique inference about causal effects on inequality in a superpopulation. [<reflink idref="bib56" id="ref155">56</reflink>], in a commentary on [<reflink idref="bib37" id="ref156">37</reflink>], mentions enrollment in a target trial with balanced allowables to motivate causal inference with somewhat vague interventions. Our model uses stratified sampling of an eligible population to construct a sample with desirable properties (i.e., distributions of covariates that reflect populations of interest to address non-random sample selection, balance of allowable covariates across social groups) to measure disparity. Our formal presentation discusses the sampling designs and emulation procedures in considerable detail.</p> <hd id="AN0192656324-38">Alternative Conceptual Models of Disparity</hd> <p>Other conceptual models that employ allowability are widely used to define disparity in healthcare or used to audit or improve algorithmic fairness. We outline these models and then compare them to our model. We ignore issues of non-random sample selection to focus on core differences between the models. Each of these models, including our own, presumes that the constructs measured by allowables have the same meaning for each social group (i.e., that there is no differential interpretation or measurement error).</p> <p>[<reflink idref="bib12" id="ref157">12</reflink>] frame disparity as the difference in healthcare utilization outcomes unexplained by differences in the allowable covariates, a generalization of the Oaxaca-Blinder Decomposition ([<reflink idref="bib3" id="ref158">3</reflink>]; [<reflink idref="bib61" id="ref159">61</reflink>]) originally used to measure labor discrimination.[<reflink idref="bib33" id="ref160">33</reflink>] Under this interpretation, they define an IOM concordant disparity that compares outcomes between an observed marginalized group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> and a counterfactual privileged group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>0</mn></math> </ephtml> with certain distributional properties. Its joint distribution of the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> and non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> , must: (a) return <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mi mathvariant="bold-italic">A</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> the factual marginal distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> among the marginalized group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> when integrated over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> ; (b) return <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mi mathvariant="bold-italic">N</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> the factual marginal distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> among the privileged group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>0</mn></math> </ephtml> when integrated over <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> . Criterion (a) ensures balance of the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> . Criterion (b) picks up the mediating role of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> leading to differences in healthcare utilization in the overall population. They avoid strict causal assumptions by not placing further constraints on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></math> </ephtml> . They adapt a 'Rank and Replace' procedure ([<reflink idref="bib54" id="ref161">54</reflink>]) to implement this model by producing a counterfactual privileged population satisfying criteria (a) and (b).</p> <p>[<reflink idref="bib21" id="ref162">21</reflink>] agree with the decomposition perspective but argue that [<reflink idref="bib12" id="ref163">12</reflink>] criteria and 'Rank and Replace' procedure lead to implausible populations not relevant for policymaking. With a factual marginalized group, a decomposition involves an intervention to assign the privileged group the allowable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> distribution of the marginalized group, and such intervention will impact the non-allowable <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> distribution when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> causes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> . This yields a causal decomposition, one that intervenes on the allowables among the privileged group, and compares the observed marginalized group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>1</mn></math> </ephtml> to a counterfactual privileged group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mn>0</mn></math> </ephtml> where the joint distribution of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> depend on the causal relationship between them. When <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> causes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> (Figure 5a), the joint distribution is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mi mathvariant="bold-italic">N</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn></mrow><mo stretchy="false">)</mo><mo>×</mo><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo fence="false" stretchy="false">|</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo>,</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> , their 'marginal framework', where they assign <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> within levels of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> . Whereas when <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> causes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> (Figure 5b), the joint distribution is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo fence="false" stretchy="false">|</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow></mrow><mo stretchy="false">)</mo><mo>×</mo><msub><mi>f</mi><mrow><mi mathvariant="bold-italic">A</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> , their 'conditional framework', where they assign <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> irrespective of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> . To implement, they propose a density ratio weighting procedure. The model relies on 'nature preserving assumptions' that allow the counterfactuals to be mapped to data, wherein the intervention on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> does not change how the outcome is conditionally distributed. The model does not extend to settings where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> cause each other over time (Figure 5c).</p> <p>Graph: Figure 5. Directed acyclic graphs depicting causal relations between historical processes H , social group R , a set of allowable covariates A, a set of nonallowable covariates N, and a decision-based outcome D. In (a), the nonallowables affect the allowables. In (b) the allowables affect the nonallowables. In (c) there is causal feedback between allowables A and non-allowables N over time.</p> <p>Many authors ([<reflink idref="bib62" id="ref164">62</reflink>]; [<reflink idref="bib85" id="ref165">85</reflink>]; [<reflink idref="bib42" id="ref166">42</reflink>]; [<reflink idref="bib59" id="ref167">59</reflink>]; [<reflink idref="bib84" id="ref168">84</reflink>]; [<reflink idref="bib7" id="ref169">7</reflink>]; [<reflink idref="bib81" id="ref170">81</reflink>]) define discrimination as a direct effect of assigning social group membership <emph>R</emph> (or its perception) on an outcome <emph>D</emph> made by a decider.[<reflink idref="bib34" id="ref171">34</reflink>] The direct effect does not occur through allowable covariates <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> appropriate for decision-making, capturing inappropriate causal paths.[<reflink idref="bib35" id="ref172">35</reflink>] In Figure 5a and b, discrimination is reflected by the direct path set: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mi>D</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo stretchy="false">→</mo><mi>D</mi></math> </ephtml> (again, we say 'direct' as the path set avoids <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> ). In Figure 5c, discrimination is reflected by the path set: <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mi>D</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mn>1</mn></msub><mo stretchy="false">→</mo><mi>D</mi></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mn>2</mn></msub><mo stretchy="false">→</mo><mi>D</mi></math> </ephtml> , and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mn>1</mn></msub><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mn>2</mn></msub><mo stretchy="false">→</mo><mi>D</mi></math> </ephtml> . A direct effect can be defined by potential outcomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>D</mi><mrow><mi>r</mi><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mi>r</mi></msup></mrow></msup></math> </ephtml> under interventions that jointly assign social group <emph>R</emph> and allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> . For exposition, consider the direct effect defined among the marginalized group, where persons' social group <emph>R</emph> is set to marginalized <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn></math> </ephtml> versus privileged <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn></math> </ephtml> but the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> are held to what they would be under marginalized status, i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></msup></math> </ephtml> (to measure discrimination): <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mrow><mo>[</mo><msup><mi>D</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></msup></mrow></msup><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo>]</mo></mrow><mrow><mo>−</mo><mi>E</mi></mrow><mrow><mo>[</mo><msup><mi>D</mi><mrow><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></msup></mrow></msup><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo>]</mo></mrow></math> </ephtml></p> <p>Graph</p> <p>Under consistency and composition assumptions, ([<reflink idref="bib80" id="ref173">80</reflink>]) the expression <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msup><mi>D</mi><mrow><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></msup></mrow></msup><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> is identified as the marginalized group's factual mean, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> . Under Figure 5b, if the graph includes all confounders of <emph>R</emph> 's effects on <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> and <emph>D</emph> (e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> </ephtml> ), all confounders of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> 's effect on <emph>D</emph>, and no confounder is affected by <emph>R</emph> (i.e., the recanting witness criterion holds; [<reflink idref="bib6" id="ref174">6</reflink>]; [<reflink idref="bib71" id="ref175">71</reflink>]), the 'cross-world' expression <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><msup><mi>D</mi><mrow><mi>r</mi><mo>=</mo><mn>0</mn><mo>,</mo><msup><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo><mn>1</mn></mrow></msup></mrow></msup><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">]</mo></math> </ephtml> is identified as: <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo movablelimits="false">∑</mo></mrow><mrow><mi>h</mi><mo>,</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">n</mi></mrow></mrow></munder><mspace width="0.2em" /><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>D</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo>,</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>h</mi></mrow><mo stretchy="false">]</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">n</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">a</mi></mrow><mo>,</mo><mi>h</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></math> </ephtml></p> <p>Graph</p> <p>The expression shows a joint distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo fence="false" stretchy="false">|</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mi>H</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mi>H</mi></mrow><mo stretchy="false">)</mo><mo>×</mo><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mi>H</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>,</mo><mi>H</mi><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> where the confounder <emph>H</emph> is treated as allowable.[<reflink idref="bib36" id="ref176">36</reflink>] Under Figure 5a and c, the direct effect is not identified because the non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> confound the effect of the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> on the outcome <emph>D</emph> but are affected by social group <emph>R</emph>.</p> <p>Our model defines disparity by comparing groups who are (distributionally) similarly situated (i.e., balanced) on the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> by design. We argued in the subsection "Is a Causal Framing of Disparity Necessary?" that this design is IOM concordant.[<reflink idref="bib37" id="ref177">37</reflink>] Our model's estimand is also interpretable as a standardized measure, where a disparity metric is calculated for each level of the allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo>=</mo><mrow><mi mathvariant="bold-italic">a</mi></mrow></math> </ephtml> and these metrics are standardized to a common distribution. In contrast, the [<reflink idref="bib12" id="ref178">12</reflink>], [<reflink idref="bib21" id="ref179">21</reflink>], and direct effect models decompose either a crude difference or a total effect into a portion that captures unjust differences (e.g., disparity, discrimination) and another that does not.[<reflink idref="bib38" id="ref180">38</reflink>] Direct effects are defined by nuanced interventions that either act in a 'cross-world' sense ([<reflink idref="bib1" id="ref181">1</reflink>]), act on distinct mechanisms that stem from assigning social group <emph>R</emph> ([<reflink idref="bib68" id="ref182">68</reflink>]), or change how information flows from assigning <emph>R</emph> ([<reflink idref="bib19" id="ref183">19</reflink>]).</p> <p>Our default model, Design 1, assumes overlap (<reflink idref="bib3" id="ref184">3</reflink>) of allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> [<reflink idref="bib39" id="ref185">39</reflink>] and does not specify <emph>H</emph> (i.e., determinants of <emph>R</emph> as in Figure 5) or non-allowables <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> . That is, unlike the [<reflink idref="bib12" id="ref186">12</reflink>], [<reflink idref="bib21" id="ref187">21</reflink>], and direct effect models, our model does not require investigators to understand all the non-allowable factors that lead to the outcome. If we do express <emph>H</emph> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow><mspace width="0.25em" /></math> </ephtml> and choose the marginalized group as the standard population, our model would compare the observed outcomes of the marginalized group to a privileged group with a joint distribution <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo>,</mo><mi>H</mi><mo fence="false" stretchy="false">|</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow></mrow></msub><mo stretchy="false">(</mo><mrow><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo>,</mo><mi>H</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo><mn>0</mn><mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow></mrow><mo stretchy="false">)</mo><mo>×</mo><msub><mi>f</mi><mrow><mi mathvariant="bold-italic">A</mi></mrow></msub><mo stretchy="false">(</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">A</mi></mrow></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold-italic">N</mi></mrow></math> </ephtml> , and <emph>H</emph> (i.e., <emph>H</emph> is treated as if it were non-allowable). Our default model is agnostic about causal structure. Our model is identified under Figure 5a to c, whereas the [<reflink idref="bib21" id="ref188">21</reflink>] model does not cover Figure 5c, and the direct effect model is not identified in Figure 5a or c. Our model also assumes that the sampling process does not impact the data-generating process (<reflink idref="bib4" id="ref189">4</reflink>). The [<reflink idref="bib21" id="ref190">21</reflink>] and direct effect models assume that their interventions leave aspects (e.g., the conditional outcome distribution) of the data generation process intact. This assumption is very demanding given how society and social groups are currently structured ([<reflink idref="bib47" id="ref191">47</reflink>]; [<reflink idref="bib38" id="ref192">38</reflink>]).</p> <hd id="AN0192656324-39">Selection Bias (Including Generalizability and Transportability)</hd> <p>Nonrandom sample selection can bias the causal effect of assigning perceived group membership (e.g., a measure of discrimination) ([<reflink idref="bib28" id="ref193">28</reflink>]; [<reflink idref="bib53" id="ref194">53</reflink>]; [<reflink idref="bib46" id="ref195">46</reflink>]; [<reflink idref="bib26" id="ref196">26</reflink>]; [<reflink idref="bib74" id="ref197">74</reflink>]). It also impacts a descriptive measure of inequality ([<reflink idref="bib79" id="ref198">79</reflink>]). We underscored that nonrandomly selected populations can be inherently meaningful and that collider stratification may affect baseline covariates to further disadvantage a marginalized group on outcomes, aligning with the Healthy People 2020 and NIMHD definitions of disparity (see the subsections "Defining Disparity" and "Non-Random Sample Selection"). We provided Design 1 for use in these settings.</p> <p>We argued that non-random sample selection may be addressed whenever (a) it limits the ability to infer to a population of interest or (b) masks disparity through collider-stratification. We provided Designs 2 and 3 for use in setting (a), and Design 4 for use in setting (b). Design 4 may be contrasted with Design 1 to understand how much non-random sample selection impacts disparity. Designs 2 and 3 use similar assumptions used to generalize or transport descriptive measures and the results of randomized trials ([<reflink idref="bib18" id="ref199">18</reflink>]). When there are no allowables specified and the combined sample [collapsed over social group <emph>R</emph> ] is the standard population, and there is only one study (i.e., indexed at a single moment of calendar time), then the identifying expressions of Designs 2 and 3 reduce to those of [<reflink idref="bib2" id="ref200">2</reflink>], the weighting estimators for Design 2 reduce to inverse selection weights of [<reflink idref="bib10" id="ref201">10</reflink>], the weighting estimators for Design 3 reduce to the inverse odds sampling weights of [<reflink idref="bib82" id="ref202">82</reflink>], and the G-computation estimators for Design 2 reduce to those of [<reflink idref="bib48" id="ref203">48</reflink>] and also [<reflink idref="bib15" id="ref204">15</reflink>].</p> <p>Design 4 is a novel approach to address non-random sample selection. It envisions an intervention to allocate <emph>certain</emph> (i.e., not necessarily all) eligibility-related variables, even when other eligibility-related variables (not intervened upon) are affected by the intervention. [<reflink idref="bib16" id="ref205">16</reflink>] envision an intervention to scale up trial participation, the last step in study enrollment. The exchangeability assumption of Design 4, like [<reflink idref="bib16" id="ref206">16</reflink>], holds when eligibility-related variables or study participation affect the outcome. The independence assumption of Designs 2 and 3 used to generalize or transport does not. Estimation under Design 4 simplifies greatly when the intervention point is the last step in enrollment (e.g., an intervention to allocate study participation).</p> <p>We focused on non-random sample selection. Missing data, loss to follow-up, and competing risks during follow-up may bias a disparity estimate ([<reflink idref="bib33" id="ref207">33</reflink>]). These may be addressed in the statistical analysis of the target study and its emulation. Events (e.g., death, hospital discharge for study in inpatient disparity in prognosis) before enrollment may affect who (non-randomly) selects into the study sample ([<reflink idref="bib69" id="ref208">69</reflink>]). If survivors at enrollment (i.e., those without the event) are not considered to be a meaningful population, an extension of Design 4 with a sustained intervention may help if the event is manipulable, but such an extension is not developed here and is saved for future work.</p> <hd id="AN0192656324-40">Discussion</hd> <p>We have proposed a conceptual model for measuring disparity and a framework to emulate it with secondary data. Through a sampling plan, the model similarly situates social groups on allowable covariates at baseline to map to meaningful definitions of disparity (Design 1). The model extends to address non-random sample selection in various ways to permit generalizability (Design 2), transportability (Design 3), or inference in a selected population without inducing undesirable forms of collider-stratification (Design 4). We motivated Design 4 for when collider stratification due to non-random sample selection attenuates disparity, but it is difficult to know when this will occur ([<reflink idref="bib60" id="ref209">60</reflink>]). Investigators may emulate both Designs 1 and 4 to report how selective mechanisms may contribute to or attenuate disparity. Unlike existing models used to measure disparity, our model involves no intervention to assign social group membership and no intervention to manipulate the allowable covariates. Only under Design 4 does the model intervene on variables used to establish eligibility. Under Designs 1‒3, the model can recover the crude expected outcomes of a social group (e.g., the marginalized group) by using it to determine the standard distribution of the sampling plan. This is also true of Designs 4a and 4c under simplifying conditions (see the subsection "Identification and Estimation of Design 4 Under Simplifying Conditions"). We have described data structures and provided weighting and G-computation estimators to emulate the model in complex data. Our model and emulation procedures avoid bias due to differential enrollment over calendar time without invoking any assumption that disparity is homogeneous over time. The summary estimates are weighted averages of estimates for populations at points in calendar time. In the Supplementary Material, we provide sample code, a data application to electronic medical records, and proof of all results.</p> <p>Our model has translational value for advancing public health and clinical medicine. First, it relies on minimal assumptions and is therefore a practical measure for advancing social justice. Second, its features accommodate specific populations during critical life stages: eligibility, time zero, follow-up, and outcome definition, aspects which actual interventions must consider in practice. Third, it maps to definitions of disparity that have strong moral foundations and have long guided public health action. Fourth, it can be extended to evaluate causal effects of (i) hypothetical interventions to inform future interventions ([<reflink idref="bib37" id="ref210">37</reflink>]) and (ii) actual interventions in (non)-randomized trials ([<reflink idref="bib39" id="ref211">39</reflink>]).</p> <p>Our model also has conceptual value. Our Designs 1‒3 are grounded in the observed world. They pick up the realized effects of unjust mechanisms as they operate in this world. Mechanisms of injustice are exquisitely complex, inter-dependent, mutually constituted, and dynamically reinforcing ([<reflink idref="bib66" id="ref212">66</reflink>]). Causal approaches that leverage observational data assume that the way outcomes are conditionally distributed in the factual world will be unchanged in the counterfactual world created by hypothetical interventions, ignoring this complexity ([<reflink idref="bib38" id="ref213">38</reflink>]). Our Designs 1‒3 capture the impact of this complexity as observed without specifying how this complexity works or assuming it away. Our Design 4 invokes a consistency assumption, though, and is subject to this limitation.</p> <hd id="AN0192656324-41">Supplemental Material</hd> <p>Graph: Supplemental material, sj-pdf-1-smr-10.1177_00491241251314037 for The Target Study: A Conceptual Model and Framework for Measuring Disparity by John W. Jackson, Yea-Jen Hsu, Raquel C. Greer, Romsai T. Boonyasai and Chanelle J. Howe in Sociological Methods & Research</p> <ref id="AN0192656324-42"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref60" type="bt">1</bibl> <bibtext> Dr. Jackson conceived of the work, developed the formal results, carried out the data application, and drafted the initial and revised manuscripts. Dr. Hsu constructed the analytic cohort for the data application. Drs. Jackson, Hsu, Greer, and Boonyasai oversaw the construction of the analytic cohort and data application. Drs. Hsu, Greer, Boonyasai, and Howe critically edited the initial and revised manuscripts for scientific content.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref61" type="bt">2</bibl> <bibtext> The data application code and sample code to implement all estimators are available at: https://osf.io/ta7vw/ (Open Science Framework) and https://github.com/jwjackson/targetstudy (GitHub).</bibtext> </blist> <blist> <bibl id="bib3" idref="ref55" type="bt">3</bibl> <bibtext> The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: This work does not necessarily represent the views or opinions of the Agency for Healthcare Research and Quality. Dr. Howe has received funding via a grant from Sanofi Pasteur administered directly to Brown University (unrelated to the current work).</bibtext> </blist> <blist> <bibl id="bib4" idref="ref3" type="bt">4</bibl> <bibtext> The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Dr. Jackson was supported by a grant from the National Heart, Lung, and Blood Institute (K01HL145320).</bibtext> </blist> <blist> <bibl id="bib5" idref="ref10" type="bt">5</bibl> <bibtext> John W. Jackson https://orcid.org/0000-0002-1528-7003 Chanelle J. Howe https://orcid.org/0000-0001-5379-472X</bibtext> </blist> <blist> <bibl id="bib6" idref="ref174" type="bt">6</bibl> <bibtext> This manuscript used data from electronic patient medical records of patients seen within a large healthcare system. To protect patient privacy and to comply with HIPAA, we are unable to share or post the data with third parties for re-analysis.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref169" type="bt">7</bibl> <bibtext> Supplemental material (i.e., data application and proofs) for this article is available online.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref16" type="bt">8</bibl> <bibtext> Disparity compares across groups. It is broader than the legal notions of discrimination defined as disparate treatment (<emph>Civil Rights Act of 1964</emph>) and disparate impact (<emph>Civil Rights Act of 1991</emph>). It captures contemporary and intergenerational marginalization at personal, institutional, and societal levels that the restrict conditions and opportunities for a reasonably good life, whatever else one may desire in life ([20]; [4]; [63]).</bibtext> </blist> <blist> <bibl id="bib9" idref="ref22" type="bt">9</bibl> <bibtext> An alternative causal model is to balance the allowables by hypothetically assigning at random which social group a person is perceived to belong to, to identify discrimination for a decision outcome <emph>D</emph>. ([28]) However, this hypothetical intervention will also balance non-allowables implicated in disparity, leading to an over-adjusted measure of disparity.</bibtext> </blist> <blist> <bibtext> Similar arguments have been made about the underlying structure behind lack of generalizability for causal effects ([27]; [29]; [51]). We focus on a descriptive estimand where social group is not intervened on.</bibtext> </blist> <blist> <bibtext> We posit that a meaningful population is a set of persons who, within a short span of time, are bound together by a shared set of circumstances, conditions, experiences, or purpose making them distinct from others. See, e.g., [44].</bibtext> </blist> <blist> <bibtext> Many interventions address disparity by altering how SES and comorbidities impact outcomes. ([57]).</bibtext> </blist> <blist> <bibtext> If we interpret the definitions in the subsection "Defining Disparity" strictly, disparity is null when the marginalized group is not disadvantaged on the outcome. On the additive scale, disparity is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mo>+</mo><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup><mo>=</mo><mo stretchy="false">(</mo><mrow><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup><mo>−</mo><mrow><mi mathvariant="normal">Δ</mi></mrow></mrow><mo stretchy="false">)</mo><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup><mo>></mo><mrow><mi mathvariant="normal">Δ</mi></mrow><mo stretchy="false">)</mo></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> is the indicator function (1: true, 0: false) and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Δ</mi></mrow><mo>=</mo>0</math> </ephtml> . On the ratio scale, disparity is <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mo>+</mo><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>exp</mi><mspace width="0.2em" /><mo stretchy="false">(</mo><mrow><mi>log</mi><mspace width="0.2em" /><mo stretchy="false">(</mo><mrow><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup><mo>/</mo><mrow><mi mathvariant="normal">Δ</mi></mrow></mrow><mo stretchy="false">)</mo><mo>×</mo><mi>I</mi><mo stretchy="false">(</mo><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup><mo>></mo><mrow><mi mathvariant="normal">Δ</mi></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Δ</mi></mrow><mo>=</mo>1</math> </ephtml> . Then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mo>+</mo></msubsup></math> </ephtml> replaces <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> everywhere. Disparity may be defined with a threshold <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Δ</mi></mrow></math> </ephtml> , choosing <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Δ</mi></mrow><mo>></mo>0</math> </ephtml> on the additive scale or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="normal">Δ</mi></mrow><mo>></mo>1</math> </ephtml> on the ratio scale.</bibtext> </blist> <blist> <bibtext> For a target study indexed at calendar time <emph>k</emph>, social group may affect eligibility-related variables or allowables (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></math> </ephtml> may exist) and allowables may affect eligibility-related variables or vice versa (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> may exist). The outcome must not affect eligibility-related variables, allowables, or social group (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mrow><mi mathvariant="bold-italic">R</mi></mrow></mrow><mo stretchy="false">)</mo></math> </ephtml> may not exist).</bibtext> </blist> <blist> <bibtext> If <emph>R</emph> is categorical, including multiple marginalized groups, at time <emph>k</emph> we may enroll all groups into the same target study or separate studies, each with one marginalized group and a random partition of the most privileged group. For the latter, the allowables (see subsection "Allowable Covariates"), standard population (see subsection "Standard Distribution"), and sampling design (see subsection "Enrollment Process (for Design 1)" and section "Extension of the Target Study to Address Non-Random Sample Selection") may differ across studies.</bibtext> </blist> <blist> <bibtext> For a social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mi>k</mi></msub></math> </ephtml> that varies over time (e.g., mental illness), a person is classified to their social group as of time <emph>k</emph>.</bibtext> </blist> <blist> <bibtext> If <emph>k</emph> represents a wide span of calendar time (e.g., a month), individuals may be eligible at multiple instances during <emph>k</emph> (e.g., many visits with a primary care provider in a month). The first instance during the unit of time <emph>k</emph> would be used for enrollment.</bibtext> </blist> <blist> <bibtext> The sampling fraction <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> in Sections 4 and 5 may be larger than one. To sample without replacement, we rescale it, i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow><mrow><mo>*</mo><mo>,</mo><mrow><mi mathvariant="script">D</mi></mrow></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>−</mo>1</mrow></msub><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> where <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>×</mo><munder><mo form="prefix" movablelimits="true">min</mo><mi>v</mi></munder><mo stretchy="false">(</mo><mrow><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>−</mo>1</mrow></msub><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>/</mo><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></math> </ephtml> . Then <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML">0<mo>≤</mo><msub><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>≤</mo>1</math> </ephtml> for all strata defined by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mi mathvariant="bold-italic">V</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo>=</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow></math> </ephtml> . <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow><mrow><mo>*</mo><mo>,</mo><mrow><mi mathvariant="script">D</mi></mrow></mrow></msubsup><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> is a stage-specific probability of inclusion at time <emph>k</emph>. This constrains the sample size to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><msub><mo>∑</mo><mrow><mi mathvariant="bold-italic">v</mi></mrow></msub><mspace width="0.2em" /><msub><mrow><mrow><mrow><mi mathvariant="double-struck">N</mi></mrow></mrow></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>*</mo><mo stretchy="false">(</mo><mrow><msub><mi mathvariant="bold-italic">v</mi><mi mathvariant="bold-italic">k</mi></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> The standardization of the disparity measure <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ψ</mi><mi>k</mi></msub></math> </ephtml> to the distribution of enrollment time among the standard population can be informally viewed as treating enrollment time as "allowable," though this is achieved through aggregation rather than sampling.</bibtext> </blist> <blist> <bibtext> Each sampling strategy differs from the typical approach of adding "inappropriate" variables as allowable covariates to address non-random selection, which can lead to an "overadjustment" by making social groups similar on the very factors that are implicated in disparity.</bibtext> </blist> <blist> <bibtext> It is okay if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></math> </ephtml> exists and it is okay if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> exists, and any causal relations between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are permitted. However, it is not okay if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> exists.</bibtext> </blist> <blist> <bibtext> (3) becomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo>0</math> </ephtml> for all <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo>0</math> </ephtml> and all <emph>k</emph>.</bibtext> </blist> <blist> <bibtext> (3) becomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo>0<mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo>0</math> </ephtml> for all <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>,</mo><mo>=</mo>0<mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo>0</math> </ephtml> and all <emph>k</emph>.</bibtext> </blist> <blist> <bibtext> Designs 2 and 3 do not distinguish eligibility-related variables that may be affected by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup></math> </ephtml> , and thus subsume <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>≀</mo></msubsup></math> </ephtml> into <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mtext>‡</mtext></msubsup></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> [58] define causal effects of treatment allocation policies. We allocate eligibility-related variables.</bibtext> </blist> <blist> <bibtext> Exchangeability (18), positivity (12), and consistency (above) help identify the distributions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">n</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> from the observed data. The sampling fractions <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>α</mi><mrow><mi>k</mi><mo>,</mo>2</mrow><mrow><mrow><mi mathvariant="script">D</mi></mrow>4</mrow></msubsup><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> do not balance <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> across <emph>R</emph>.</bibtext> </blist> <blist> <bibtext> (3) becomes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo>0</math> </ephtml> for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> with <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">a</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo>,</mo><mi>k</mi></mrow><mo stretchy="false">)</mo><mo>></mo>0</math> </ephtml> and all <emph>k</emph>.</bibtext> </blist> <blist> <bibtext> It is okay if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mspace width="0.25em" /></math> </ephtml> exists and it is okay if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo><mo stretchy="false">→</mo><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mtext>‡</mtext></msubsup></math> </ephtml> or <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mtext>‡</mtext></msubsup></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> exists, and any causal relations between <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></math> </ephtml> are permitted. It is not okay if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> exists or if <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mrow><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>†</mo></msubsup><mo>,</mo><msubsup><mrow><mi mathvariant="bold-italic">W</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow><mo>≀</mo></msubsup></mrow><mo stretchy="false">)</mo><mo stretchy="false">→</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold-italic">N</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi mathvariant="bold-italic">k</mi></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> exists.</bibtext> </blist> <blist> <bibtext> Under Design 4c, residual contributions from collider-stratification would occur on Figure 2C if, rather than the single node <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> , we had two nodes <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mrow><mi>k</mi><mo stretchy="false">(</mo><mrow><mi>i</mi><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></msub></math> </ephtml> that each inherited all edges from <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>k</mi></msub></math> </ephtml> . The residual contribution would operate through the pathways <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><msub><mi>L</mi><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><msubsup><mi>W</mi><mi>k</mi><mo>†</mo></msubsup><mo stretchy="false">←</mo><msub><mi>L</mi><mrow><mi>k</mi><mo stretchy="false">(</mo><mrow><mi>i</mi><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">→</mo><msub><mi>L</mi><mrow><mi>k</mi><mo stretchy="false">(</mo><mrow><mi>i</mi><mi>i</mi></mrow><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><msubsup><mi>W</mi><mi>k</mi><mo>†</mo></msubsup><mo stretchy="false">←</mo><msub><mi>L</mi><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> . This would not occur under Designs 4a or 4b.</bibtext> </blist> <blist> <bibtext> To aggregate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>a</mi><mi>d</mi><mi>d</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><mi>Q</mi><mtext>‡</mtext><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo stretchy="false">)</mo></math> </ephtml> for Design 2, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo>0<mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo stretchy="false">)</mo></math> </ephtml> for Design 3, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1<mo stretchy="false">)</mo></math> </ephtml> for Design 4. To aggregate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>ψ</mi><mi>k</mi><mrow><mi>r</mi><mi>e</mi><mi>l</mi></mrow></msubsup></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>γ</mi><mi>k</mi></msub></math> </ephtml> is multiplied by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo>0<mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo>0<mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> for Design 2, by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo>0<mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo>0<mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo>0<mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> for Design 3, and by <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>k</mi></msub><mo stretchy="false">(</mo>0<mo stretchy="false">)</mo><mo>=</mo><msub><mi>E</mi><mrow><mi mathvariant="normal">Ω</mi></mrow></msub><mo stretchy="false">[</mo><mrow><msub><mi>Y</mi><mrow><mi>k</mi><mo>+</mo><mi>J</mi></mrow></msub><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo>0<mo>,</mo><mi>k</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> for Design 4. For pooled analyses, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>Q</mi><mtext>‡</mtext><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1</mrow><mo stretchy="false">)</mo><mo>/</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><mi>Q</mi><mtext>‡</mtext><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> for Design 2, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo>0<mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1</mrow><mo stretchy="false">)</mo><mo>/</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo>0<mo>,</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> for Design 3, and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mi>k</mi></msub><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mrow><mi>k</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>T</mi><mo>=</mo>1</mrow><mo stretchy="false">)</mo><mo>/</mo><mi>P</mi><mo stretchy="false">(</mo><mi>k</mi><mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mrow><msub><mi>G</mi><mi>k</mi></msub></mrow></msubsup><mo>=</mo>1<mo>,</mo><mi>R</mi><mo>=</mo><mi>r</mi><mo stretchy="false">)</mo></math> </ephtml> for Design 4.</bibtext> </blist> <blist> <bibtext> For example, under design 4a, we would estimate <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> as <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>P</mi><mo stretchy="false">^</mo></mrow><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo>1<mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> from a model for <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo stretchy="false">(</mo><msubsup><mi>Q</mi><mi>k</mi><mo>†</mo></msubsup><mo>=</mo>1<mo fence="false" stretchy="false">|</mo><msubsup><mi>Q</mi><mi>k</mi><mtext>‡</mtext></msubsup><mo>=</mo>1<mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> If, as in our example, a social group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo><mi>r</mi><mrow><mi mathvariant="normal">′</mi></mrow></math> </ephtml> represents the standard population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo>1</math> </ephtml> , this calculation reduces to <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>q</mi><mi>k</mi></msub><mo stretchy="false">(</mo><mo>⋅</mo><mo stretchy="false">)</mo></math> </ephtml> .</bibtext> </blist> <blist> <bibtext> The decompositions of [12] and [21] use a counterfactual privileged population <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo>0</mrow><msub><mo stretchy="false">]</mo><mrow><mi>c</mi><mi>f</mi></mrow></msub></math> </ephtml> to separate the observed difference <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>1<mo stretchy="false">]</mo><mrow><mo>−</mo><mi>E</mi></mrow><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>0<mo stretchy="false">]</mo></math> </ephtml> into a portion <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo>0</mrow><msub><mo stretchy="false">]</mo><mrow><mi>c</mi><mi>f</mi></mrow></msub><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>0<mo stretchy="false">]</mo></math> </ephtml> due to, and a portion <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>Y</mi><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>1<mo stretchy="false">]</mo><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>Y</mi><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo>0</mrow><msub><mo stretchy="false">]</mo><mrow><mi>c</mi><mi>f</mi></mrow></msub></math> </ephtml> not due to, differences in the distribution of allowables across social groups <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo>1</math> </ephtml> and <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo>=</mo>0</math> </ephtml> .</bibtext> </blist> <blist> <bibtext> In the direct effect model considered here for exposition, the decomposition uses a counterfactual <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mi>D</mi><mrow><mi>r</mi><mo>=</mo>0<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mrow><mi mathvariant="bold-italic">r</mi></mrow><mo>=</mo>1</mrow></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>1<mo stretchy="false">]</mo></math> </ephtml> to separate the total effect of assigning social group (or its perception) among the marginalized group <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>D</mi><mrow><mi>r</mi><mo>=</mo>1<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo>1</mrow></mrow><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo>1</mrow><mo stretchy="false">]</mo><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi>D</mi><mrow><mi>r</mi><mo>=</mo>0<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo>0</mrow></mrow><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>1<mo stretchy="false">]</mo></math> </ephtml> into an indirect effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mrow><msup><mi>D</mi><mrow><mi>r</mi><mo>=</mo>0<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo>1</mrow></mrow></msup><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo>1</mrow><mo stretchy="false">]</mo><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><mi>D</mi><mrow><mi>r</mi><mo>=</mo>0<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo>0</mrow></mrow></math> </ephtml><ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>1<mo stretchy="false">]</mo></math> </ephtml> due to social group's effect on the allowables and a direct effect <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo stretchy="false">[</mo><mrow><mi>D</mi><mrow><mi>r</mi><mo>=</mo>1<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo>1</mrow></mrow><mrow><mo fence="false" stretchy="false">|</mo></mrow><mi>R</mi><mo>=</mo>1</mrow><mo stretchy="false">]</mo><mo>−</mo><mi>E</mi><mo stretchy="false">[</mo><msup><mi>D</mi><mrow><mi>r</mi><mo>=</mo>0<mo>,</mo><mrow><mi mathvariant="bold-italic">A</mi></mrow><mrow><mi>r</mi><mo>=</mo>1</mrow></mrow></msup><mo fence="false" stretchy="false">|</mo><mi>R</mi><mo>=</mo>1<mo stretchy="false">]</mo></math> </ephtml> not due to social group's effect on the allowables.</bibtext> </blist> <blist> <bibtext> A causal path is a directed sequence of edges out a parent node and into a child node in a directed acyclic graph.</bibtext> </blist> <blist> <bibtext> If the path <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mo stretchy="false">←</mo><mi>H</mi><mo stretchy="false">→</mo><mrow><mi mathvariant="bold-italic">N</mi></mrow><mo stretchy="false">→</mo><mi>Y</mi></math> </ephtml> in Figure 5 disadvantages the marginalized group, making <emph>H</emph> allowable underestimates disparity.</bibtext> </blist> <blist> <bibtext> [12] and [49] describe propensity score weighting based on allowables alone as not IOM concordant. 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Jackson; Yea-Jen Hsu; Raquel C. Greer; Romsai T. Boonyasai and Chanelle J. Howe</p> <p>Reported by Author; Author; Author; Author; Author</p> <p></p> <p>Dr. John W. Jackson, Sc.D. is an Associate Professor in the Departments of Epidemiology, Biostatistics, and Mental Health at the Johns Hopkins Bloomberg School of Public Health, and core faculty in the Johns Hopkins Center for Health Equity, Center for Health Disparities Solutions, and Center for Drug Safety and Effectiveness. His research focuses on developing methods for translational health equity research, including methods to define and measure health disparities, to identify high-leverage targets and strategies for interventions that address health disparities, and to evaluate effects of interventions. His work is funded by the National Heart, Lung, and Blood Institute, the Robert E. Meyerhoff Foundation, as well as by pilot funding from Johns Hopkins University.</p> <p>Dr. Yea-Jen Hsu, Ph.D., is an Associate Research Scientist in the Department of Health Policy and Management at the Johns Hopkins Bloomberg School of Public Health. She specializes in health services research, program evaluation, and the application of implementation science theories to healthcare improvement, with a current focus on patient safety and quality improvement with innovative care and evaluation models.</p> <p>Dr. Raquel C. Greer, M.D., M.H.S. is an Adjunct Associate Professor in the Department of Medicine at the Johns Hopkins School of Medicine and her research is focused on identifying and testing strategies to reduce and eliminate health disparities and advance health equity for people with kidney disease.</p> <p>Dr. Romsai Tony Boonyasai, M.D., M.P.H. is a Physician Advisor in the Division of Quality Measurement and Improvement at the Agency for Health Care Quality, a branch of the U.S. Department of Health and Human Services, and an Associate Professor (part-time) in the Department of Medicine at the Johns Hopkins School of Medicine. His research interests include the development of quality-of-care measures and the implementation of systems-based interventions to improve population health and advance health equity.</p> <p>Dr. Chanelle J. Howe, Ph.D. is an Associate Professor in the Department of Epidemiology within the Brown University School of Public Health. She has an appointment with Brown's Center for Epidemiologic Research, is a member of the Providence/Boston Center for AIDS Research, and is a faculty associate with Brown's Population Studies and Training Center. Her research interests include methods, infectious diseases, and health disparities. She also serves as an Editor for the American Journal of Epidemiology.</p> </aug> <nolink nlid="nl1" bibid="bib13" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib36" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib22" firstref="ref4"></nolink> <nolink nlid="nl4" bibid="bib32" firstref="ref5"></nolink> <nolink nlid="nl5" bibid="bib41" firstref="ref6"></nolink> <nolink nlid="nl6" bibid="bib21" firstref="ref7"></nolink> <nolink nlid="nl7" bibid="bib46" firstref="ref8"></nolink> <nolink nlid="nl8" bibid="bib26" firstref="ref9"></nolink> <nolink nlid="nl9" bibid="bib83" firstref="ref11"></nolink> <nolink nlid="nl10" bibid="bib37" firstref="ref17"></nolink> <nolink nlid="nl11" bibid="bib54" firstref="ref18"></nolink> <nolink nlid="nl12" bibid="bib12" firstref="ref19"></nolink> <nolink nlid="nl13" bibid="bib57" firstref="ref21"></nolink> <nolink nlid="nl14" bibid="bib25" firstref="ref27"></nolink> <nolink nlid="nl15" bibid="bib35" firstref="ref31"></nolink> <nolink nlid="nl16" bibid="bib43" firstref="ref32"></nolink> <nolink nlid="nl17" bibid="bib72" firstref="ref33"></nolink> <nolink nlid="nl18" bibid="bib10" firstref="ref34"></nolink> <nolink nlid="nl19" bibid="bib79" firstref="ref35"></nolink> <nolink nlid="nl20" bibid="bib30" firstref="ref36"></nolink> <nolink nlid="nl21" bibid="bib23" firstref="ref37"></nolink> <nolink nlid="nl22" bibid="bib70" firstref="ref38"></nolink> <nolink nlid="nl23" bibid="bib60" firstref="ref39"></nolink> <nolink nlid="nl24" bibid="bib69" firstref="ref42"></nolink> <nolink nlid="nl25" bibid="bib11" firstref="ref43"></nolink> <nolink nlid="nl26" bibid="bib14" firstref="ref47"></nolink> <nolink nlid="nl27" bibid="bib40" firstref="ref51"></nolink> <nolink nlid="nl28" bibid="bib15" firstref="ref52"></nolink> <nolink nlid="nl29" bibid="bib16" firstref="ref53"></nolink> <nolink nlid="nl30" bibid="bib76" firstref="ref54"></nolink> <nolink nlid="nl31" bibid="bib17" firstref="ref56"></nolink> <nolink nlid="nl32" bibid="bib50" firstref="ref57"></nolink> <nolink nlid="nl33" bibid="bib18" firstref="ref58"></nolink> <nolink nlid="nl34" bibid="bib75" firstref="ref59"></nolink> <nolink nlid="nl35" bibid="bib55" firstref="ref64"></nolink> <nolink nlid="nl36" bibid="bib19" firstref="ref65"></nolink> <nolink nlid="nl37" bibid="bib24" firstref="ref67"></nolink> <nolink nlid="nl38" bibid="bib65" firstref="ref68"></nolink> <nolink nlid="nl39" bibid="bib34" firstref="ref69"></nolink> <nolink nlid="nl40" bibid="bib20" firstref="ref72"></nolink> <nolink nlid="nl41" bibid="bib27" firstref="ref84"></nolink> <nolink nlid="nl42" bibid="bib28" firstref="ref89"></nolink> <nolink nlid="nl43" bibid="bib29" firstref="ref92"></nolink> <nolink nlid="nl44" bibid="bib73" firstref="ref94"></nolink> <nolink nlid="nl45" bibid="bib31" firstref="ref95"></nolink> <nolink nlid="nl46" bibid="bib64" firstref="ref96"></nolink> <nolink nlid="nl47" bibid="bib67" firstref="ref123"></nolink> <nolink nlid="nl48" bibid="bib33" firstref="ref147"></nolink> <nolink nlid="nl49" bibid="bib39" firstref="ref149"></nolink> <nolink nlid="nl50" bibid="bib77" firstref="ref150"></nolink> <nolink nlid="nl51" bibid="bib48" firstref="ref151"></nolink> <nolink nlid="nl52" bibid="bib78" firstref="ref153"></nolink> <nolink nlid="nl53" bibid="bib52" firstref="ref154"></nolink> <nolink nlid="nl54" bibid="bib56" firstref="ref155"></nolink> <nolink nlid="nl55" bibid="bib61" firstref="ref159"></nolink> <nolink nlid="nl56" bibid="bib62" firstref="ref164"></nolink> <nolink nlid="nl57" bibid="bib85" firstref="ref165"></nolink> <nolink nlid="nl58" bibid="bib42" firstref="ref166"></nolink> <nolink nlid="nl59" bibid="bib59" firstref="ref167"></nolink> <nolink nlid="nl60" bibid="bib84" firstref="ref168"></nolink> <nolink nlid="nl61" bibid="bib81" firstref="ref170"></nolink> <nolink nlid="nl62" bibid="bib80" firstref="ref173"></nolink> <nolink nlid="nl63" bibid="bib71" firstref="ref175"></nolink> <nolink nlid="nl64" bibid="bib38" firstref="ref180"></nolink> <nolink nlid="nl65" bibid="bib68" firstref="ref182"></nolink> <nolink nlid="nl66" bibid="bib47" firstref="ref191"></nolink> <nolink nlid="nl67" bibid="bib53" firstref="ref194"></nolink> <nolink nlid="nl68" bibid="bib74" firstref="ref197"></nolink> <nolink nlid="nl69" bibid="bib82" firstref="ref202"></nolink> <nolink nlid="nl70" bibid="bib66" firstref="ref212"></nolink>
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  Data: The Target Study: A Conceptual Model and Framework for Measuring Disparity
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  Data: <searchLink fieldCode="AR" term="%22John+W%2E+Jackson%22">John W. Jackson</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-1528-7003">0000-0002-1528-7003</externalLink>)<br /><searchLink fieldCode="AR" term="%22Yea-Jen+Hsu%22">Yea-Jen Hsu</searchLink><br /><searchLink fieldCode="AR" term="%22Raquel+C%2E+Greer%22">Raquel C. Greer</searchLink><br /><searchLink fieldCode="AR" term="%22Romsai+T%2E+Boonyasai%22">Romsai T. Boonyasai</searchLink><br /><searchLink fieldCode="AR" term="%22Chanelle+J%2E+Howe%22">Chanelle J. Howe</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0001-5379-472X">0000-0001-5379-472X</externalLink>)
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  Data: <searchLink fieldCode="SO" term="%22Sociological+Methods+%26+Research%22"><i>Sociological Methods & Research</i></searchLink>. 2026 55(2):405-458.
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  Data: We present a conceptual model to measure disparity--the target study--where social groups may be similarly situated (i.e., balanced) on allowable covariates. Our model, based on a sampling design, does not intervene to assign social group membership or alter allowable covariates. To address nonrandom sample selection, we extend our model to generalize or transport disparity or to assess disparity after an intervention on eligibility-related variables that eliminates forms of collider-stratification. To avoid bias from differential timing of enrollment, we aggregate time-specific study results by balancing calendar time of enrollment across social groups. To provide a framework for emulating our model, we discuss study designs, data structures, and G-computation and weighting estimators. We compare our sampling-based model to prominent decomposition-based models used in healthcare and algorithmic fairness. We provide R code for all estimators and apply our methods to measure health system disparities in hypertension control using electronic medical records.
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