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Regression Discontinuity Analysis with Latent Variables

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Title: Regression Discontinuity Analysis with Latent Variables
Language: English
Authors: Monica Morell, Muwon Kwon, Youngjin Han, Youjin Sung, Yang Liu, Ji Seung Yang
Source: Grantee Submission. 2026 61(1):178-191.
Peer Reviewed: Y
Page Count: 15
Publication Date: 2026
Sponsoring Agency: Institute of Education Sciences (ED)
Contract Number: R305D220030
Document Type: Journal Articles
Reports - Research
Descriptors: Regression (Statistics), Models, Item Response Theory, Research Design, Research Problems, Scores, Statistical Inference, Sample Size
DOI: 10.1080/00273171.2025.2565591
ISSN: 0027-3171
1532-7906
Abstract: A regression discontinuity (RD) design is often employed to provide causal evidence when the randomization of the treatment assignment is infeasible. When variables of interest are latent constructs measured by observed indicators, the conventional RD analysis using observed variable scores does not allow researchers to examine heterogeneity in the estimated local average treatment effect (ATE) and to generalize the ATE to participants away from the cutoff. We propose a novel methodological augmentation to the conventional RD analysis, which assumes the availability of multiple indicator variables (i.e., raw item responses) that measure the latent construct underlying the running variable. By specifying an explicit measurement model based on those indicator variables, our latent RD framework allows 1) defining the local ATE conditional on the latent construct, 2) disentangling the heterogeneity of the local ATE, and 3) generalizing the local ATE to running variable scores away from the cutoff. In a proof-of-concept simulation we illustrate the proposed augmentation recovers parameters of interest well under practical test length and sample size conditions.
Abstractor: As Provided
IES Funded: Yes
Entry Date: 2026
Accession Number: ED680414
Database: ERIC
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  Value: <anid>AN0192371394;3zw01jan.26;2026Mar23.03:18;v2.2.500</anid> <title id="AN0192371394-1">Regression Discontinuity Analysis with Latent Variables </title> <p>A regression discontinuity (RD) design is often employed to provide causal evidence when the randomization of the treatment assignment is infeasible. When variables of interest are latent constructs measured by observed indicators, the conventional RD analysis using observed variable scores does not allow researchers to examine heterogeneity in the estimated local average treatment effect (ATE) and to generalize the ATE to participants away from the cutoff. We propose a novel methodological augmentation to the conventional RD analysis, which assumes the availability of multiple indicator variables (i.e., raw item responses) that measure the latent construct underlying the running variable. By specifying an explicit measurement model based on those indicator variables, our latent RD framework allows 1) defining the local ATE conditional on the latent construct, 2) disentangling the heterogeneity of the local ATE, and 3) generalizing the local ATE to running variable scores away from the cutoff. In a proof-of-concept simulation we illustrate the proposed augmentation recovers parameters of interest well under practical test length and sample size conditions.</p> <p>Keywords: regression discontinuity; latent variable modeling; item response theory; causal inferences</p> <hd id="AN0192371394-2">Introduction</hd> <p>Regression discontinuity (RD; Thistlethwaite & Campbell, [<reflink idref="bib51" id="ref1">51</reflink>]; Campbell & Cook, [<reflink idref="bib8" id="ref2">8</reflink>]; Campbell & Stanley, [<reflink idref="bib9" id="ref3">9</reflink>]) designs have become increasingly popular in social sciences as they allow for valid causal inferences without the use of random assignment. For example, Jacob and Lefgren ([<reflink idref="bib23" id="ref4">23</reflink>]) examined the effects of attending summer school (i.e., treatment) on academic outcomes (e.g., future test scores after one year) using the Chicago Public School data. Each year, students who scored below the promotional cutoff were required to attend summer school. Therefore, treatment assignment was almost perfectly determined by whether a student scored above or below the cutoff. If all students are compliant to such decision rule, then the RD design is said to be <emph>sharp</emph>, which is what we focus on in the sequel.[<reflink idref="bib1" id="ref5">1</reflink>] The current test score in this example is termed an <emph>observed running variable (ORV)</emph>. Assuming that no unobserved, individual-level covariate varies abruptly at the ORV cutoff, we can causally identify the treatment effect on the outcome locally at the cutoff score, which is referred to as the local average treatment effect (ATE). An illustrative graph of the sharp RD design can be found in Figure 1A using simulated data. The conventional RD analysis has two limitations. First, while we are able to establish causality at the cutoff, ATEs conditional on other ORV levels cannot be identified. This is because no data point falls on the opposite side of the cutoff in both treatment and control groups (see Figure 1A). Second, the local ATE only reflects the average impact of the treatment; however, it lacks a quantification for effect heterogeneity.</p> <p>Graph: Figure 1. Conventional regression discontinuity analysis. Panel A: Outcome scores plotted against the observed running variable (ORV) scores. Individuals whose ORV scores are lower than the cutoff (shown as the vertical dotted line) receive treatment (black crosses), while those whose ORV scores are above the cutoff form the control group (gray circles). Group-specific regression lines (black = treatment, gray = control) trace the average outcomes conditional on the ORV. The difference in group averages at the cutoff (highlighted by a thick vertical line segment) is the local Average Treatment Effect (ATE). Panel B: ORV scores plotted against the latent running variable (LRV) scores. The ORV cutoff is now marked by a horizontal dotted line. The superimposed probability density characterizes the conditional distribution of the LRV given ORV = cutoff, which reflects measurement error. Panel C: Outcome scores plotted against the LRV scores. Regressions of the outcome onto the LRV scores are depicted as black and gray lines within the treatment and control groups, respectively.</p> <p>In many evaluation studies using RD, ORVs are aggregates of observed indicators that reflect latent constructs (e.g., motivation, language proficiency, and patient satisfaction). For example, the ORV in Jacob and Lefgren ([<reflink idref="bib23" id="ref6">23</reflink>]) is a standardized test score, which is computed from responses to test items and serves as a proxy to academic achievement. Under these circumstances, we often operationalize the underlying constructs as latent variables (e.g., De Boeck et al., [<reflink idref="bib15" id="ref7">15</reflink>]; Hoyle et al., [<reflink idref="bib22" id="ref8">22</reflink>]), which, for better alignment with the RD language, are henceforth referred to as <emph>latent running variables</emph> (LRVs). We rely on a measurement model to specify a joint distribution of LRVs and observed indicators (e.g., factor analysis and item response theory [IRT] models; Thissen & Steinberg, [<reflink idref="bib50" id="ref9">50</reflink>]). Because ORVs are functions of observed indicators, the measurement model further induces the joint distribution of LRVs and ORVs; as a direct consequence, the same ORV score could map stochastically onto a range of LRV scores and <emph>vice versa</emph>. In Figure 1B, ORV scores are plotted against the underlying LRV scores, and the superimposed probability density characterizes the spread of LRV scores at the ORV cutoff level.</p> <p>Under additional assumptions on the measurement and structural models, we demonstrate that augmenting a conventional RD analysis with an explicit measurement model for the LRV allows us to (a) identify ATEs conditional on the LRV, (b) identify ATEs at ORV levels away from the cutoff, (c) quantify the heterogeneity of treatment effects at the ORV cutoff. Heuristically, the validity behind ATE extrapolation and heterogeneity quantification can be attributed to the imperfect mapping between ORV and LRV: At each LRV value, there is a positive probability for an individual to be assigned to either treatment or control group (Figure 1B). Therefore, it is possible to regress the outcome variable onto the LRV across its entire domain (Figure 1C). We consider fully parametric measurement and structural models, in which all parameters are estimated in one-stage by maximum likelihood (ML). Recovery of model parameters, extrapolated ATEs, and posterior treatment effect quantiles are evaluated in a Monte Carlo study.</p> <p>The rest of the paper is organized as follows. We start from a brief introduction of the conventional RD model and its limitations. Then we review existing literature about RD extrapolation that motivates our work. We then formulate our RD model with latent variables, which features a two-parameter logistic (2PL) measurement model with a single LRV and a linear structural model that links the outcome to the LRV. We highlight various effect size measures for ATEs conditional on LRV scores, RD extrapolation, and heterogeneity of treatment effects. We present a simulation study to evaluate the performance of ML estimation in recovering selected effects. We conclude the article with a discussion on limitations and future avenues of research.</p> <hd id="AN0192371394-3">Regression discontinuity designs</hd> <p></p> <hd id="AN0192371394-4">Conventional regression discontinuity design</hd> <p>RD can be conceptualized under two frameworks, the continuity-based framework and the local randomization framework (Cattaneo & Titiunik, [<reflink idref="bib12" id="ref10">12</reflink>]). Our presentation is aligned with the continuity-based framework; a brief summary of the alternative, local randomization framework is provided in Discussion. In the potential outcome framework of causal inference (Holland, [<reflink idref="bib21" id="ref11">21</reflink>]; Rubin, [<reflink idref="bib44" id="ref12">44</reflink>]), each individual <emph>i</emph> has two possible outcomes which would result from receiving the treatment condition, denoted</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> and not receiving the treatment condition, denoted</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mtext>.</mtext></math> </ephtml> Under the Stable Unit Treatment Value Assumption (SUTVA; Rubin, [<reflink idref="bib45" id="ref13">45</reflink>]), the causal effect of treatment on an outcome for individual <emph>i</emph> is defined as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><mtext>.</mtext></math> </ephtml> The fundamental problem of causal inference is that the two potential outcomes cannot be simultaneously observed for the same individual.</p> <p>An RD design applies to situations where a participant is assigned to treatment (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mn>1</mn></mrow></math> </ephtml> ) or control (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ) based solely on whether their value on a continuous ORV</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> reaches a pre-determined cutoff <emph>c</emph>, i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mn>1</mn><mo>{</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>≤</mo><mi>c</mi><mo>}</mo></mrow><mtext>.</mtext></math> </ephtml> [<reflink idref="bib2" id="ref14">2</reflink>] Full compliance is assumed throughout the current paper as we only focus on the sharp RD design. The observed outcome is then expressed as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mrow><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mrow><msub><mrow><mi>T</mi></mrow><mi>i</mi></msub></mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mtext>.</mtext></math> </ephtml> Assume that 1) the conditional expectation of the potential outcomes given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>s</mi></mrow><mtext>,</mtext></math> </ephtml> denoted</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>s</mi><mo>}</mo></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mtext>,</mtext></math> </ephtml> is continuous with respect to <emph>s</emph> in the neighborhood of <emph>c</emph>, and 2) the ORV</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> admits a positive density at the cutoff value <emph>c</emph>. Then the local ATE at the cutoff, denoted</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> is identified (Hahn et al., [<reflink idref="bib20" id="ref15">20</reflink>]):</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi><mo>}</mo><mo>−</mo><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi><mo>}</mo><mo>=</mo><mrow><mrow><munder><mrow><mtext mathvariant="normal">lim</mtext></mrow><mrow><mi>s</mi><mo>↑</mo><mi>c</mi></mrow></munder></mrow></mrow><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>s</mi><mo>}</mo><mo>−</mo><mrow><mrow><munder><mrow><mtext mathvariant="normal">lim</mtext></mrow><mrow><mi>s</mi><mo>↓</mo><mi>c</mi></mrow></munder></mrow></mrow><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>s</mi><mo>}</mo><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib1" id="ref16">1</reflink>)</p> <p>in which the conditional expectation of the observable</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>s</mi></mrow></math> </ephtml> can be estimated from the treatment group if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo><</mo><mi>c</mi></mrow></math> </ephtml> and from the control group if</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>></mo><mi>c</mi></mrow><mtext>.</mtext></math> </ephtml> </p> <hd id="AN0192371394-5">Limitations of conventional regression discontinuity analyses</hd> <p>A major limitation of the conventional RD design under the continuity-based framework is that, without further assumptions, the ATE is no longer identified at ORV values away from the cutoff because no observed data point in the treatment group falls on the right side of the cutoff and similarly no observed data point in the control group falls on the left side of the cutoff (see Figure 1A). As a consequence, RD analyses appear less useful when "more global" causal inferences are desired in the evaluation of program effectiveness. Furthermore, stakeholders often want to know whether making a change to the current treatment assignment mechanism would result in better effectiveness. The external validity of an RD analysis is important yet difficult to establish; however, there has been a growing literature on methods to estimate ATEs for participants with ORV values away from the cutoff. For example, Dong and Lewbel ([<reflink idref="bib16" id="ref17">16</reflink>]) considered extrapolating the ATE to ORV values that are marginally above the cutoff by assuming mild smoothness conditions. Bertanha and Imbens ([<reflink idref="bib4" id="ref18">4</reflink>]) proposed an approach for estimating generalized ATEs away from the cutoff when using a fuzzy RD design. Cattaneo et al. ([<reflink idref="bib11" id="ref19">11</reflink>]) exploited the presence of multiple cutoffs and imposed additional assumptions to extrapolate the treatment effect away from the cutoff value. Mealli and Rampichini ([<reflink idref="bib35" id="ref20">35</reflink>]) and Wing and Cook ([<reflink idref="bib52" id="ref21">52</reflink>]) proposed to use a pretreatment measure of the outcome variable to impute treatment effects above the cutoff.</p> <p>Another limitation of the conventional RD analysis is that it alone does not inform about the potential variability of local treatment effects. For example, suppose that individuals differ along an additional variable</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>X</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>.</mtext></math> </ephtml> [<reflink idref="bib3" id="ref22">3</reflink>] Then</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi><mo>}</mo><mo>=</mo><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>X</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi><mo>}</mo><mo>|</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi></mrow><mo stretchy="true">}</mo></mrow></mrow></math> </ephtml> (<reflink idref="bib2" id="ref23">2</reflink>)</p> <p>due to the law of iterated expectations. In the light of Equation 2, the local ATE</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> should be conceived as a weighted average of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="sans-serif">E</mi><mo>{</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>X</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi><mo>}</mo></mrow><mtext>,</mtext></math> </ephtml> in which the weights are equal to the conditional probability density function (pdf) of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>S</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>c</mi></mrow></math> </ephtml> (Lee & Lemieux, [<reflink idref="bib26" id="ref24">26</reflink>]). Heterogeneity in causal effects has profound impact on the practice of program evaluation: It potentially conveys information about intervention mechanism, participation process, defects in program designs and administrations, and so forth (e.g., Smith, [<reflink idref="bib48" id="ref25">48</reflink>]).</p> <hd id="AN0192371394-6">Existing literature that motivates the current work</hd> <p>According to Bloom ([<reflink idref="bib7" id="ref26">7</reflink>]), there are three views on the generalization of RD findings. The "strict-constructionist" view states that the ATE can only be identified for participants locally at the cutoff without additional assumptions. On the other hand, the "old-school" view approves the extrapolation of results beyond the cutoff using parametric or non-parametric models. Finally, the "more expansive" view of, e.g., Lee and Lemieux ([<reflink idref="bib26" id="ref27">26</reflink>]), suggests that measurement error in the running variable (RV) facilitates the generalization of local ATE as the RD estimand can be interpreted as "a weighted [ATE] where the weights are directly proportional to the ex ante likelihood that an individual's realization of [the ORV] will be close to the threshold." They acknowledged that the "ex ante likelihood" at the individual level is not obtainable from a single measure of the RV. Extra information such as "reliability", "a second test measurement", or "other covariates" that can predict the assignment is needed to generalize the local ATE. As an example, Wing and Cook ([<reflink idref="bib52" id="ref28">52</reflink>]) used a counterfactual approach which relies on a baseline outcome variable score and is in the same spirit of using "other covariates" as suggested by Lee and Lemieux ([<reflink idref="bib26" id="ref29">26</reflink>]).</p> <p>Indeed, the item-level data or sub-scores from which the RV scores are calculated resemble a "second test measurement" in the language of Lee and Lemieux ([<reflink idref="bib26" id="ref30">26</reflink>]), which therefore opens up the possibility of generalizing the local ATE. Angrist and Rokkanen ([<reflink idref="bib1" id="ref31">1</reflink>]) proposed a strategy to identify the ATE at RV values away from the cutoff. The key to Angrist and Rokkanen ([<reflink idref="bib1" id="ref32">1</reflink>]<emph>)</emph> proposal is to find suitable exogenous covariates conditional on which the RV and outcome are independent. Additionally, Rokkanen ([<reflink idref="bib43" id="ref33">43</reflink>]) established that if the ORV is governed by a LRV satisfying the conditional independence assumption of Angrist and Rokkanen ([<reflink idref="bib1" id="ref34">1</reflink>]), then the ATE can be identified beyond the cutoff.</p> <p>Recent efforts have focused on integrating latent variables into conventional RD designs. For example, Soland et al. ([<reflink idref="bib49" id="ref35">49</reflink>]) introduced a structural equation modeling (SEM) approach to RD analysis, incorporating latent outcome variables. The main benefits of Soland et al. ([<reflink idref="bib49" id="ref36">49</reflink>]<emph>)</emph> proposal are its ability to circumvent the limitations of sum scores and effectively handle measurement invariance violations.</p> <p>In alignment with the effort to improve the utility of the RD design in evaluation studies, our proposed integration of RD analysis with measurement modeling builds upon the "more expansive" view of Lee and Lemieux ([<reflink idref="bib26" id="ref37">26</reflink>]) and extends the work of Rokkanen ([<reflink idref="bib43" id="ref38">43</reflink>]).</p> <p>When the ORV represents a latent construct (i.e., the LRV), incorporating an explicit measurement model for the LRV permits generalized causal claims, such as ATEs conditional on ORV levels away from the cutoff, without the need to re-conduct an RD study. Furthermore, it is possible to quantify the variability of local treatment effects with the aid of a fully specified measurement model for the LRV. It is also worth pointing out that the local ATE can be identified for both continuous and discrete ORVs under the new framework (cf. assuming a continuous ORV in order to apply the continuity-based argument in Equation 1). We emphasize that latent variable measurement models are routinely assumed when the measurement instruments are validated and scale scores are computed. Therefore, the proposed work not only improves the utility of conventional RD analysis without imposing unrealistically strong assumptions but also bridges psychometrics and causal inference to better serve the practice evaluating program effectiveness.</p> <hd id="AN0192371394-7">Regression discontinuity models with latent variables</hd> <p>We now extend the conventional RD framework to include a measurement model for the ORV. In particular, we assume that the ORV is computed from a collection of observable indicator variables that reflect a single LRV. Meanwhile, the outcome variable remains observed, and no covariate is involved in the analysis. Extensions to allow latent outcome and additional covariates are discussed in the Supplementary Document.</p> <hd id="AN0192371394-8">Measurement model</hd> <p>For each individual <emph>i</emph>, let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> be the LRV, measured by a collection of <emph>J</emph> indicator variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mrow><msup><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow><mo>,</mo><mo>...</mo><mo>,</mo><mrow><msub><mrow><mi>X</mi></mrow><mrow><mtext mathvariant="italic">iJ</mtext></mrow></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>⊤</mo></msup></mrow></mrow><mtext>.</mtext></math> </ephtml> Realizations of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> are denoted by</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>θ</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="bold">x</mi><mo>=</mo><mrow><msup><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>x</mi></mrow><mn>1</mn></msub></mrow><mo>,</mo><mo>...</mo><mo>,</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>J</mi></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>⊤</mo></msup></mrow></mrow><mtext>,</mtext></math> </ephtml> respectively. A <emph>measurement model</emph> determines the joint distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>.</mtext></math> </ephtml> [<reflink idref="bib4" id="ref39">4</reflink>] In particular, we assume that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi><mn>1</mn></mrow></msub></mrow><mo> ⊥⊥</mo><mo>...</mo><mo>⊥⊥ </mo><mrow><msub><mrow><mi>X</mi></mrow><mrow><mtext mathvariant="italic">iJ</mtext></mrow></msub></mrow><mtext /><mo>|</mo><mtext /><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> where the symbol</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>⊥⊥</mo></mrow></math> </ephtml> stands for stochastic independence. This is often referred to as the local independence assumption in the factor analytic literature (McDonald, [<reflink idref="bib34" id="ref40">34</reflink>]). We then obtain the joint pdf at</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi mathvariant="bold">x</mi></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>θ</mi></mrow><mtext>:</mtext></math> </ephtml> [<reflink idref="bib5" id="ref41">5</reflink>]</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>f</mi></mrow><mi>θ</mi></msub></mrow><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mrow><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></munderover></mrow><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>|</mo><mi>θ</mi></mrow></msub></mrow></mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>|</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib3" id="ref42">3</reflink>)</p> <p>in which</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>|</mo><mi>θ</mi></mrow></msub></mrow></mrow></math> </ephtml> is the conditional pdf of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow></msub></mrow></mrow></math> </ephtml> given</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>f</mi></mrow><mi>θ</mi></msub></mrow></mrow></math> </ephtml> is the marginal pdf of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>.</mtext></math> </ephtml> </p> <p>In the current work, we focus on the 2PL model (Birnbaum, [<reflink idref="bib6" id="ref43">6</reflink>]) for dichotomous indicators (i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow></msub></mrow><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi></mrow></math> </ephtml> ), which is characterized by the following conditional pdf:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>|</mo><mi>θ</mi></mrow></msub></mrow><mo>(</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>|</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mi>P</mi><mo>{</mo><mrow><msub><mrow><mi>X</mi></mrow><mtext mathvariant="italic">ij</mtext></msub></mrow><mo>=</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>|</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>θ</mi><mo>}</mo><mo>=</mo><mrow><mfrac><mrow><mo /><mtext>exp</mtext><mo stretchy="true">[</mo><mrow><msub><mrow><mi>x</mi></mrow><mi>j</mi></msub></mrow><mo>(</mo><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub></mrow><mi>θ</mi><mo>+</mo><mrow><msub><mrow><mi>d</mi></mrow><mi>j</mi></msub></mrow><mo>)</mo><mo stretchy="true">]</mo></mrow><mrow><mn>1</mn><mo>+</mo><mo /><mtext>exp</mtext><mo /><mo>(</mo><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub></mrow><mi>θ</mi><mo>+</mo><mrow><msub><mrow><mi>d</mi></mrow><mi>j</mi></msub></mrow><mo>)</mo></mrow></mfrac></mrow><mo>,</mo></math> </ephtml> (<reflink idref="bib4" id="ref44">4</reflink>)</p> <p>in which</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>a</mi></mrow><mi>j</mi></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>d</mi></mrow><mi>j</mi></msub></mrow></mrow></math> </ephtml> are the item slope and intercept parameters for item <emph>j</emph>, respectively. In addition, it is assumed that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> follows a standard normal distribution marginally.</p> <hd id="AN0192371394-9">Structural model</hd> <p>By means of a <emph>structural model</emph>, we specify the conditional distribution of potential outcomes</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow></math> </ephtml> (where</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mn>0</mn></mrow></math> </ephtml> denotes the control group and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow></math> </ephtml> denotes the treatment group), given indicator variables</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and the LRV</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>.</mtext></math> </ephtml> Note that the nature of the intervention does not change across the ORV and the LRV (SUTVA; Rubin, [<reflink idref="bib45" id="ref45">45</reflink>]; also called no hidden variations in treatment). We make an additional assumption that the potential outcomes are independent to the observable indicators conditional on the LRV: that is,</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊥⊥ </mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mtext /><mo>|</mo><mtext /><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib5" id="ref46">5</reflink>)</p> <p>Immediately from Equation 5, we have</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>|</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>|</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib6" id="ref47">6</reflink>)</p> <p>In line with the parametric framework we have considered so far, we assume a linear structural model that relates the potential outcomes to the LRV:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>0</mn></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mi>g</mi><mo>+</mo><mo stretchy="false">(</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>1</mn></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mi>g</mi><mo stretchy="false">)</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>ε</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>,</mo><mtext /><mi>g</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib7" id="ref48">7</reflink>)</p> <p>in which</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>ε</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>ε</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> are independent and normally distributed error terms with mean zero and homoscedastic variance</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>ε</mi><mn>2</mn></msubsup></mrow></mrow><mtext>.</mtext></math> </ephtml> In Equation 7, we assume that the LRV, rather than the ORV, influences the potential outcomes. This assumption, which defines the relationship between the outcome and the latent variable, is a common approach in causal inference for addressing measurement error (e.g., Sengewald et al., [<reflink idref="bib47" id="ref49">47</reflink>]; Lockwood and McCaffrey, [<reflink idref="bib29" id="ref50">29</reflink>]). Additionally, the interaction term between the group membership and the LRV allows for heterogeneous treatment effect across the LRV.[<reflink idref="bib6" id="ref51">6</reflink>] It follows from Equation 7 and the normality of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>ε</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></math> </ephtml> that</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>|</mo><mi>θ</mi></mrow></msub></mrow><mo>(</mo><mi>y</mi><mo>|</mo><mi>θ</mi><mo>)</mo><mo>=</mo><mrow><mfrac><mn>1</mn><msqrt><mn>2</mn><mi>π</mi><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>ε</mi><mn>2</mn></msubsup></mrow></msqrt></mfrac></mrow><mtext>exp</mtext><mo /><mrow><mo stretchy="true">{</mo><mo>−</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="true">[</mo><mrow><mi>y</mi><mo>−</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>0</mn></msub></mrow><mo>−</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mi>g</mi><mo>−</mo><mo>(</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>1</mn></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mi>g</mi><mo>)</mo><mi>θ</mi></mrow><mo stretchy="true">]</mo></mrow><mn>2</mn></msup></mrow><mrow><mn>2</mn><mrow><msubsup><mrow><mi>σ</mi></mrow><mi>ε</mi><mn>2</mn></msubsup></mrow></mrow></mfrac></mrow><mo stretchy="true">}</mo></mrow><mo>,</mo><mtext /><mi>g</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo></math> </ephtml> (<reflink idref="bib8" id="ref52">8</reflink>)</p> <p>which characterizes the structural model (Equation 6).</p> <hd id="AN0192371394-10">Model for observable data</hd> <p>Even though all the component variables are treated as latent, the treatment assignment must be determined based on an ORV score. Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> be the ORV score for individual <emph>i</emph>. In the sequel, we only consider ORVs that are sums of indicator variables: that is,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mrow><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>J</mi></msubsup></mrow><mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mtext mathvariant="italic">ij</mtext></mrow></msub></mrow></mrow></mrow><mtext>.</mtext></math> </ephtml> [<reflink idref="bib7" id="ref53">7</reflink>] Define the treatment indicator</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>{</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>≤</mo><mi>c</mi><mo>}</mo></mrow><mtext>,</mtext></math> </ephtml> in which <emph>c</emph> is a pre-specified cutoff. It is further assumed that</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="sans-serif">P</mi><mo>{</mo><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>|</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>θ</mi><mo>}</mo><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> (<reflink idref="bib9" id="ref54">9</reflink>)</p> <p>for all</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>θ</mi></mrow><mtext>.</mtext></math> </ephtml> In other words, a participant may be assigned to either the treatment or control group at any level of the LRV. We remark that it is possible to use more than one ORV scores or more sophisticated eligibility criteria for treatment assignment (e.g., Wong et al., [<reflink idref="bib53" id="ref55">53</reflink>]); however, those extensions are not further elaborated and left for future work.</p> <p>Note that</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">[</mo><mn>1</mn><mo>−</mo><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></math> </ephtml> under a sharp RD design. The joint pdf of the observable outcome</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> observable indicators</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> and the LRV</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> can then be factorized as follows:</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mrow><mo stretchy="false">(</mo><mrow><mi>T</mi><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>,</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mrow><mo stretchy="false">(</mo><mrow><mi>T</mi><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>·</mo><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></mtd><mtd columnalign="left"><mrow /></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mo>=</mo><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mrow><mo stretchy="false">(</mo><mrow><mi>T</mi><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">|</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>|</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>·</mo><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow /></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib10" id="ref56">10</reflink>)</p> <p>in which the last equality follows from Equation 5. Integrating out</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>θ</mi></mrow></math> </ephtml> from Equation 10 yields the marginal pdf of observable data</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>:</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi></mrow></msub></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mi mathvariant="italic">∫</mi><mrow><mrow><msub><mrow><mi>f</mi></mrow><mrow><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi></mrow></msub></mrow></mrow></mrow><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi mathvariant="bold">x</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>θ</mi><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib11" id="ref57">11</reflink>)</p> <p>With independent and identically distributed sample data</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">(</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo>,</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mtext>,</mtext></math> </ephtml> parameters in the measurement model (i.e., 2PL item parameters in Equation 4) and the structural model (i.e., regression coefficients in Equation 7) can be estimated by ML simultaneously. Due to the involvement of nonlinear IRT models for the indicators, Equation 11 cannot be expressed in closed form. In practice, this intractable integral can be approximated by, for example, numerical quadrature. The parametric version of our model (Equation 11) can be estimated using any SEM software that allows discrete indicators and latent variable interaction. A path diagram that illustrates the SEM specification can be found in Figure 2.</p> <p>DIAGRAM: Figure 2. Path diagram for the regression discontinuity model with a single latent running variable (LRV). The treatment assignment variable T(Xi) is a deterministic function of the indicator variables Xi1,...,XiJ. Similarly, the product term θiT(Xi) is defined deterministically from T(Xi) and the LRV θi. All deterministic dependencies are shown as dashed lines. Yi is the observable outcome variable. β1,β2, and β3 next to the arrows represent structural coefficients following Equation 7.</p> <hd id="AN0192371394-11">Inferences</hd> <p></p> <hd id="AN0192371394-12">Average Treatment Effects Conditional on the Latent Running Variable</hd> <p>One prominent feature of the latent RD framework is the possibility to identify ATEs conditional on not only the ORV but the LRV as well. Specifically, let</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>θ</mi></mrow><mo stretchy="true">}</mo></mrow></mrow></math> </ephtml> (<reflink idref="bib12" id="ref58">12</reflink>)</p> <p>be the ATE conditional on the LRV score</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>θ</mi></mrow><mtext>.</mtext></math> </ephtml> To make the paper self-contained, we provide in the Supplementary Document a proof for the nonparametric identification of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> which is adapted from Rokkanen ([<reflink idref="bib43" id="ref59">43</reflink>]). With the linear structural model (Equation 7), it is straightforward to verify that</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mi>θ</mi><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib13" id="ref60">13</reflink>)</p> <p>Modern test theory considers latent variables better representations for constructs (e.g., proficiency) than observed scores, because the definitions of latent variables remain invariant when different test forms, or different tests designed for the same purpose, are in use (Lord, [<reflink idref="bib30" id="ref61">30</reflink>]). For instance, suppose that it is desired to substitute a newly developed test for the original one whose scores were used to determine treatment assignment in an RD study. Provided that the new test has been calibrated and linked to the original test (see Kolen & Brennan, [<reflink idref="bib25" id="ref62">25</reflink>], for more information about test linking), the ATEs conditional on observed scores of the new test can be immediately obtained, since the ATE conditional on the (same) underlying LRV is readily available.</p> <hd id="AN0192371394-13">Extrapolation of average treatment effects</hd> <p>Stakeholders are often interested in a program's effectiveness across a range of ORV scores, not just at the cutoff, in order to inform overall evaluation of a program and make further decisions, such as making adjustment to the cutoff. ATEs away from the cutoff are not identified from the data in a conventional RD analysis, and re-performing the RD study using a different cutoff is usually not viable due to lack of resources. With the latent RD framework, however, the ATE conditional on an arbitrary RV score becomes identifiable. The ATE conditional on an arbitrary ORV score</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>∈</mo><mo>{</mo><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>J</mi><mo>}</mo></mrow></math> </ephtml> can be expressed as</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtable><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mo stretchy="true">}</mo></mrow></mrow></mtd><mtd columnalign="left"><mtext /></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext /><mo>=</mo><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>|</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>=</mo><mi>θ</mi><mo>,</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mo stretchy="true">}</mo></mrow><mo stretchy="true">|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mo stretchy="true">}</mo></mrow></mrow></mtd><mtd columnalign="left"><mtext /></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><mtext /><mo>=</mo><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mo stretchy="true">}</mo></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mtext /></mtd></mtr></mtable></mrow></math> </ephtml> (<reflink idref="bib14" id="ref63">14</reflink>)</p> <p>in which the last equality is a consequence of Equation 5. With the linear structural model (Equation 7), Equation 14 reduces to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>τ</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mi mathvariant="sans-serif">E</mi><mrow><mo stretchy="true">{</mo><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mo stretchy="true">}</mo></mrow><mo>.</mo></mrow></math> </ephtml> (<reflink idref="bib15" id="ref64">15</reflink>)</p> <p>Equation 15 indicates that the ATE conditional on any ORV scores can be calculated solely based on the posterior distribution</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mtext>,</mtext></math> </ephtml> without additional assumptions.</p> <hd id="AN0192371394-14">Heterogeneity of treatment effects</hd> <p>Assessing the extent to which treatment effects vary among individuals is another research objective in RD analyses. Harking back to Lee and Lemieux ([<reflink idref="bib26" id="ref65">26</reflink>]), the local ATE can be expressed as a weighted average of individual treatment effects across the entire population when the ORV is not precisely measured. In other words, measurement error, or equivalently, the LRV behind the ORV, can be a source of heterogeneity for treatment effects. Within the latent RD framework, the conclusion that the local ATE is an weighted average is an immediate corollary of Equation 14 with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>=</mo><mi>c</mi></mrow><mtext>.</mtext></math> </ephtml> Equation 14 also suggests quantifying the heterogeneity of treatment effects by the dispersion of the posterior distribution</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><mtext>.</mtext></math> </ephtml> For example, we may calculate the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi></mrow></math> </ephtml> th posterior quantile with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow><mtext>.</mtext></math> </ephtml> Under the linear structural model (Equation 7), the posterior quantile is simplified to</p> <p>Graph</p> <p> <ephtml> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="sans-serif">Q</mi></mrow><mi>α</mi></msub></mrow><mrow><mo stretchy="true">{</mo><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><mo stretchy="true">}</mo></mrow><mo>=</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mrow><msub><mrow><mi mathvariant="sans-serif">Q</mi></mrow><mi>α</mi></msub></mrow><mrow><mo stretchy="true">{</mo><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><mo stretchy="true">}</mo></mrow><mo>,</mo></mrow></math> </ephtml> (<reflink idref="bib16" id="ref66">16</reflink>)</p> <p>in which</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="sans-serif">Q</mi></mrow><mi>α</mi></msub></mrow><mo>{</mo><mo>·</mo><mo>}</mo></mrow></math> </ephtml> is a generic notation for the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi></mrow></math> </ephtml> th quantile of a distribution. For example,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="sans-serif">Q</mi></mrow><mi>α</mi></msub></mrow><mo>{</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi><mo>}</mo></mrow></math> </ephtml> indicates the</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi></mrow></math> </ephtml> th percentile of the posterior distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow><mtext>.</mtext></math> </ephtml> We are usually interested in extreme</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi mathvariant="italic">α</mi></mrow></math> </ephtml> levels (e.g.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mn>0.05</mn></mrow></math> </ephtml> and 0.95) to capture the dispersion of the treatment effects.</p> <hd id="AN0192371394-15">Simulation</hd> <p>We conducted a simulation study to examine the parameter recovery of the proposed latent RD model. We also illustrate how the model can be used to generalize the ATE away from the cutoff and disentangle the heterogeneity in the ATE.</p> <hd id="AN0192371394-16">Data generation</hd> <p></p> <hd id="AN0192371394-17">Step 1: Generating item responses</hd> <p>Let</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>...</mo><mo>,</mo><mi>n</mi></mrow><mtext>,</mtext></math> </ephtml> denote the LRV.</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> was generated from a standard normal distribution; two sample size conditions,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>500</mn></mrow></math> </ephtml> and 1,000, were considered. The LRV was measured by 20 items following the 2PL model (Birnbaum, [<reflink idref="bib6" id="ref67">6</reflink>]). The slope parameters for the LRV were drawn from a lognormal distribution with a mean of 0.3 and a standard deviation of 0.2, and difficulty parameters were generated from a standard normal distribution truncated to</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo stretchy="false">[</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></math> </ephtml> (Feinberg & Rubright, [<reflink idref="bib17" id="ref68">17</reflink>]; Mislevy & Stocking, [<reflink idref="bib36" id="ref69">36</reflink>]). The intercept parameters were calculated by negating the product of the slope and difficulty parameters. The true measurement parameters are tabulated in Table S1 in the Supplementary Document. Item parameters and latent variable values were used in Equation 4 to calculate probability of endorsement of each item for each individual. These probabilities were compared to independent draws from a uniform distribution [0, 1] to generate item responses.</p> <hd id="AN0192371394-18">Step 2. Participant classification</hd> <p>The generated item responses for the LRV were summed to create the ORV</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mtext>.</mtext></math> </ephtml> Treatment assignment</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> was determined by comparing the ORV with a pre-specified cutoff value 7. The cutoff value was chosen so that approximately 40% of the sample were assigned to the treatment condition. Participants with ORV scores at or below the cutoff score were assigned to treatment (i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow></math> </ephtml> ), and participants with ORV scores above the cutoff were assigned to the control group (i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>T</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow></math> </ephtml> ).</p> <hd id="AN0192371394-19">Step 3. Generating outcome</hd> <p>The potential outcomes</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>Y</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mtext>,</mtext></math> </ephtml> were computed from Equation 7, in which</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>0</mn></msub></mrow><mo>=</mo><mn>0</mn></mrow><mtext>,</mtext></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>1</mn></msub></mrow><mo>=</mo><mn>0.5</mn></mrow><mtext>.</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow></math> </ephtml> collectively determine the ATE with respect to the LRV (see Equation 13). Three pairs of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow></math> </ephtml> values were considered such that the resulting ATE at LRV =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>0.25</mn></mrow></math> </ephtml> (approximately the 40th percentile under a standard normal distribution) equals to 0, 0.3, and 0.5, representing zero, moderate, and large effect sizes, respectively (see e.g., Lee & Munk, [<reflink idref="bib27" id="ref70">27</reflink>]; Rhoads & Dye, [<reflink idref="bib42" id="ref71">42</reflink>]). In particular,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mo>=</mo><mn>0.025</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mo>=</mo><mn>0.1</mn></mrow></math> </ephtml> gives a zero effect,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mo>=</mo><mn>0.325</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mo>=</mo><mn>0.1</mn></mrow></math> </ephtml> gives a moderate effect, and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow><mo>=</mo><mn>0.525</mn></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow><mo>=</mo><mn>0.1</mn></mrow></math> </ephtml> gives a large effect. The error term</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>ε</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> </p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>g</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></mrow><mtext>,</mtext></math> </ephtml> was generated from a standard normal distribution.</p> <hd id="AN0192371394-20">Parameters of interest and evaluation criteria</hd> <p>We first estimated the measurement and structural parameters (Equations 4 and 7). Using these estimates, we proceeded to compute extrapolated ATEs away from the ORV cutoff (Equation 15) and the posterior quantiles (Equation 16) reflecting effect heterogeneity locally at the ORV cutoff.</p> <p>To estimate the measurement and structural parameters, we fit the correctly specified latent RD model to the generated data. The simulation was conducted using R (R Core Team, [<reflink idref="bib41" id="ref72">41</reflink>]) and M<emph>plus</emph> version 8.5 (Muthén and Muthén, [<reflink idref="bib39" id="ref73">39</reflink>]). The measurement and structural parameters were simultaneously estimated using maximum likelihood with robust standard errors (ESTIMATOR = MLR in M<emph>plus</emph>) with the default configuration. 500 replications were done for each condition. Evaluation criteria for the point estimates of the measurement and structural parameters include bias and root mean squared error (RMSE).</p> <p>We also calculated the ATE and its standard error conditional on each possible ORV value</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>s</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>...</mo><mo>,</mo><mn>20</mn></mrow><mtext>.</mtext></math> </ephtml> First, the posterior distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><mtext>,</mtext></math> </ephtml> was obtained using the Lord-Wingersky algorithm (Lord & Wingersky, [<reflink idref="bib31" id="ref74">31</reflink>]). Then the ATE was calculated following Equation 15 and the standard error was derived using the delta method (e.g., Bickel & Doksum, [<reflink idref="bib5" id="ref75">5</reflink>]). The true ATE at each ORV level was calculated by plugging-in the true parameters into Equation 15. To assess the recovery of extrapolated ATEs, we apply two criteria: RMSE and empirical coverage of the 95% confidence intervals (CIs).</p> <p>To quantify the heterogeneity of treatment effects at the cutoff (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>c</mi><mo>=</mo><mn>7</mn></mrow></math> </ephtml> ), we focused on the middle 90% region under the posterior distribution</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></mrow><mtext>.</mtext></math> </ephtml> The lower and upper bounds of the region were calculated using Equation 16, with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mn>0.05</mn></mrow></math> </ephtml> for the lower bound and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi><mo>=</mo><mn>0.95</mn></mrow></math> </ephtml> for the upper bound. The posterior distribution of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></mrow></math> </ephtml> was evaluated with the aid of the Lord-Wingersky algorithm. The true values of the lower and upper bounds were calculated using the true model parameters. Evaluation criteria for the quantile estimates include bias and RMSE.</p> <hd id="AN0192371394-21">Results</hd> <p>The parameter estimation algorithm always converged within 500 iterations (which is the default setting of M<emph>plus</emph>). For measurement parameters, a graphical comparison between the average measurement parameter estimates and the true generating values can be found in Figure S1 in the Supplementary Document. We conclude that measurement parameters are well recovered across all the conditions. Table 1 shows that structural parameters used to calculate the ATE,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow></mrow></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> are accurately recovered (see Equation 13). The recovery of the other structural parameters not used to calculate the ATE is accurate as well and reported in Table S2 in the Supplementary Document. For all parameters across conditions, the RMSE values are smaller with the larger sample size of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow><mtext>.</mtext></math> </ephtml> </p> <p>Table 1. Bias and root mean squared error (RMSE) for structural parameters used to calculate the average treatment effect (ATE).</p> <p> <ephtml> <table><thead><tr><td><p>Effect size</p></td><td><p>Parameter</p></td><td><p><graphic href="hmbr_a_2565591_ilm0151.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>n</mi><mo>=</mo><mn>500</mn></mrow></math></p></td><td><p><graphic href="hmbr_a_2565591_ilm0152.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow></math></p></td></tr><tr><td><p>Bias</p></td><td><p>RMSE</p></td><td><p>Bias</p></td><td><p>RMSE</p></td></tr></thead><tbody valign="top"><tr><td><p>0.0</p></td><td><p><graphic href="hmbr_a_2565591_ilm0153.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow></mrow></math></p></td><td><p><graphic href="hmbr_a_2565591_ilm0154.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo><mn>0.013</mn></mrow></math></p></td><td char="."><p>0.163</p></td><td><p>0.003</p></td><td char="."><p>0.116</p></td></tr><tr><td /><td><p><graphic href="hmbr_a_2565591_ilm0155.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow></math></p></td><td><p><graphic href="hmbr_a_2565591_ilm0156.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo><mn>0.015</mn></mrow></math></p></td><td char="."><p>0.179</p></td><td><p>0.005</p></td><td char="."><p>0.126</p></td></tr><tr><td><p>0.3</p></td><td><p><graphic href="hmbr_a_2565591_ilm0157.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow></mrow></math></p></td><td><p><graphic href="hmbr_a_2565591_ilm0158.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo><mn>0.014</mn></mrow></math></p></td><td char="."><p>0.177</p></td><td><p>0.002</p></td><td char="."><p>0.124</p></td></tr><tr><td /><td><p><graphic href="hmbr_a_2565591_ilm0159.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow></math></p></td><td><p><graphic href="hmbr_a_2565591_ilm0160.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo><mn>0.021</mn></mrow></math></p></td><td char="."><p>0.181</p></td><td><p>0.005</p></td><td char="."><p>0.123</p></td></tr><tr><td><p>0.5</p></td><td><p><graphic href="hmbr_a_2565591_ilm0161.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><msub><mrow><mi>β</mi></mrow><mn>2</mn></msub></mrow></mrow></math></p></td><td><p>0.003</p></td><td char="."><p>0.178</p></td><td><p><graphic href="hmbr_a_2565591_ilm0162.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo><mn>0.008</mn></mrow></math></p></td><td char="."><p>0.124</p></td></tr><tr><td /><td><p><graphic href="hmbr_a_2565591_ilm0163.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow></math></p></td><td><p>0.000</p></td><td char="."><p>0.169</p></td><td><p><graphic href="hmbr_a_2565591_ilm0164.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo><mn>0.001</mn></mrow></math></p></td><td char="."><p>0.126</p></td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. Effect size refers to the average treatment effect (ATE) at latent running variable (LRV) =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>0.25</mn></mrow><mtext>.</mtext></math> </ephtml> Values between 0 and 0.001 are displayed as 0.000; similarly, values between</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>0.001</mn></mrow></math> </ephtml> and 0 are displayed as</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>0.000</mn></mrow><mtext>.</mtext></math> </ephtml> </p> <p>Figure 3 summarizes RMSE values for ATEs at various ORV levels. In the smaller sample size condition (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>500</mn></mrow></math> </ephtml> ), the minimum RMSE is attained at the cutoff score of 7 for the zero and moderate effect size conditions, while for the large effect size condition, the smallest RMSE is obtained at the ORV score of 6. In the larger sample size condition (</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow></math> </ephtml> ), the minimum RMSE is attained at the cutoff score of 7 for the zero effect size condition. For the moderate and large effect size conditions, the smallest RMSE is obtained at the ORV score of 6. As the ORV moves away from the cutoff, the RMSE increases. The RMSE values are smaller with the larger sample size of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow></math> </ephtml> across all effect size conditions. Figure 4 shows bias values for ATEs across ORV levels. The small bias values across all conditions indicate that their contribution to the RMSE is negligible. Therefore, given the RMSE is comprised of both the bias and variability of the estimates, the pattern of the RMSE in Figure 3 reflects that the estimates are more variable at the extreme ORV scores. Figure 5 shows the empirical coverage of 95% CIs for the ATE at each value of the ORV for sample size of 500 and 1,000. We conclude that the coverage remains accurate across all ORV levels, with no distinct patterns observed across conditions. The accurate coverage of the ATE interval estimates across all ORV levels, combined with the RMSE results in Figure 3, demonstrates that the quantification of sampling uncertainty can still be trusted, although the ATE estimation becomes less efficient as the ORV moves away from the cutoff.</p> <p>Graph: Figure 3. Root mean sqaured error (RMSE) for the average treatment effect (ATE) calculated at each value of the observed running variable (ORV). ES refers to the effect size of the ATE at latent running variable (LRV) = −0.25. The smallest RMSE value for each effect size condition is highlighted in gray. A gray dashed vertical line indicates the cutoff score of 7.</p> <p>PHOTO (COLOR): Figure 4. Bias for the average treatment effect (ATE) calculated at each value of the observed running variable (ORV). ES refers to the effect size of the ATE at latent running variable (LRV) = −0.25. A gray vertical dashed line indicates the cutoff score of 7. A red horizontal solid line represents 0.</p> <p>Graph: Figure 5. Empirical coverage of 95% confidence intervals for the average treatment effect (ATE) calculated at each value of the observed running variable (ORV). ES refers to the effect size of the ATE at latent running variable (LRV) = −0.25. A gray horizontal solid line indicates the 95% level of confidence interval coverage. Two gray horizontal dashed lines represent the Monte Carlo confidence intervals. A gray vertical dashed line indicates the cutoff score of 7.</p> <p>To quantify the heterogeneity in the treatment effects attributable to measurement error, we are interested in estimating the lower and upper quantiles of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mtext>,</mtext></math> </ephtml> the ATE at the LRV score</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> conditional on the event that the ORV falls right on the cutoff (i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></mrow></math> </ephtml> ). These quantiles are reported in Table 2. The third column shows the true values of the lower and upper bounds of the middle 90% region under the posterior distribution</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>|</mo><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></mrow><mtext>.</mtext></math> </ephtml> For example, for the zero effect size condition, the lower and upper bounds of the ATE are</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>0.077</mn></mrow></math> </ephtml> and 0.042, respectively. This range of the ATE indicates that even though some participants have the same ORV cutoff score (i.e.,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>S</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></mrow></math> </ephtml> ), their ATEs can differ due to the LRV behind the ORV (i.e., measurement error). The bias and RMSE for the estimates show that the minimum and maximum of ATEs are generally well recovered across all three effect size conditions. The recovery improves with increasing sample size, reducing both bias and RMSE. Note that the sign of the interaction effect</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow></math> </ephtml> was incorrectly estimated in a portion of replications, especially when</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>n</mi><mo>=</mo><mn>500</mn></mrow><mtext>.</mtext></math> </ephtml> In those problematic replications, the lower quantile of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>ω</mi><mo stretchy="false">(</mo><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow><mo stretchy="false">)</mo></mrow></math> </ephtml> corresponds to the upper quantile of</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and <emph>vice versa</emph>, which typically leads to poor estimates of the true quantiles.</p> <p>Table 2. Bias and root mean squared error (RMSE) for parameters reflecting heterogeneity of treatment effects at the cutoff score of the observed running variable (ORV).</p> <p> <ephtml> <table><thead><tr><td><p>ES</p></td><td><p>Quantile</p></td><td><p>True</p></td><td><p><graphic href="hmbr_a_2565591_ilm0168.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>n</mi><mo>=</mo><mn>500</mn></mrow></math></p></td><td><p><graphic href="hmbr_a_2565591_ilm0169.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>000</mn></mrow></math></p></td></tr><tr><td><p>% of neg.</p></td><td><p>Bias</p></td><td><p>RMSE</p></td><td><p>% of neg.</p></td><td><p>Bias</p></td><td><p>RMSE</p></td></tr></thead><tbody valign="top"><tr><td><p>0.0</p></td><td><p>Lower</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0170.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.077</p></td><td char="."><p>0.33</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0171.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.034</p></td><td char="."><p>0.173</p></td><td char="."><p>0.10</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0172.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.020</p></td><td char="."><p>0.127</p></td></tr><tr><td /><td><p>Upper</p></td><td char="."><p>0.042</p></td><td char="." /><td char="."><p>0.025</p></td><td char="."><p>0.164</p></td><td char="." /><td char="."><p>0.020</p></td><td char="."><p>0.119</p></td></tr><tr><td><p>0.3</p></td><td><p>Lower</p></td><td char="."><p>0.223</p></td><td char="."><p>0.32</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0173.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.040</p></td><td char="."><p>0.182</p></td><td char="."><p>0.10</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0174.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.020</p></td><td char="."><p>0.118</p></td></tr><tr><td /><td><p>Upper</p></td><td char="."><p>0.342</p></td><td char="." /><td char="."><p>0.030</p></td><td char="."><p>0.177</p></td><td char="." /><td char="."><p>0.019</p></td><td char="."><p>0.126</p></td></tr><tr><td><p>0.5</p></td><td><p>Lower</p></td><td char="."><p>0.423</p></td><td char="."><p>0.27</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0175.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.031</p></td><td char="."><p>0.175</p></td><td char="."><p>0.12</p></td><td char="."><p><graphic href="hmbr_a_2565591_ilm0176.gif" content-type="Graph" /><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>−</mo></mrow></math></p><p>0.027</p></td><td char="."><p>0.126</p></td></tr><tr><td /><td><p>Upper</p></td><td char="."><p>0.542</p></td><td char="." /><td char="."><p>0.036</p></td><td char="."><p>0.184</p></td><td char="." /><td char="."><p>0.010</p></td><td char="."><p>0.125</p></td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note</emph>. ES refers to the effect size of the average treatment effect (ATE) at latent running variable (LRV) =</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>−</mo><mn>0.25</mn></mrow><mtext>.</mtext></math> </ephtml> Lower and Upper refer to Equation 16 with</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>α</mi></mrow></math> </ephtml> equal to 0.05 and 0.95, respectively. % of neg. represents the percentage of the interaction term,</p> <p>Graph</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>β</mi></mrow><mn>3</mn></msub></mrow></mrow><mtext>,</mtext></math> </ephtml> that is estimated to be negative.</p> <hd id="AN0192371394-22">Discussion</hd> <p></p> <hd id="AN0192371394-23">Summary</hd> <p>The proposed RD analysis assumes that the ORV is a proxy of an LRV, which is equipped with its own measurement model. This is not only beneficial to researchers who aim to estimate ATEs on the scale of the LRV, but also to those who are mainly interested in treatment effects on the ORV scale. Our model facilitates the quantification of treatment effect heterogeneity at given ORV scores due to measurement error and the estimation of ATEs at ORV scores away from the cutoff. As such, researchers can use the proposed method to supplement results from a conventional RD analysis when raw response data (e.g., item responses) leading to the ORV scores are available. To evaluate the recovery of extrapolated ATE estimates and measures of effect heterogeneity, we conducted a simulation study under various combinations of sample sizes (medium and large) and effect sizes (zero, moderate, and large). Our results indicate that all parameters of interest are well recovered.</p> <p>Often researchers are interested in not only the effect of the treatment with the existing ORV cutoff, but also how the treatment might have performed if the cutoff were changed. The proposed analysis allows estimation of ATEs at any ORV value. As expected, the recovery of true effects is the best at ORV values around the cutoff. As ORV values farther from the cutoff were used, the RMSE increases mainly due to the excessive variability of the estimates. Meanwhile, the empirical coverage of 95% CIs is always on target regardless of the ORV levels, providing trustworthy uncertainty quantification. Thus, generalizing the ATE beyond the cutoff requires careful consideration of both the estimation efficiency and the quantification of sample uncertainty. Estimating ATEs too far from the cutoff is generally not recommended for two reasons. First, the sampling variability of the estimated ATEs tends to be large as one moves away from the cutoff. Second, ATEs far from the cutoff may extend beyond the range of practical utility for program evaluators.</p> <p>Quantifying the heterogeneity allows researchers to examine the effect of measurement error (in the instrument used to measure the ORV) on the local ATE. We calculated the range of the ATEs on the LRV scale for participants with an ORV at the cutoff and LRV within the middle 90% of the posterior distribution (i.e., Equation 16). These values are well recovered with a moderate test length of 20 items, under both medium and large sample size conditions. This range of ATEs is expected to increase with a shorter test and decrease with a longer test, for a given range (e.g., 90%). For example, as the test length increases, the posterior distribution of LRV at the cutoff will narrow, thus the heterogeneity in the local ATE accounted by the measurement error is small. Note that 90% is an arbitrarily set percentage; researchers may choose a different interval of LRV values depending on their interest.</p> <p>In addition to allowing researchers to examine the generalizability and quantify the heterogeneity in the treatment effect, the proposed model has two other potential benefits: It may be used to aid in the selection of an optimal treatment assignment mechanism and it allows for the estimation of a treatment effect with respect to not just the ORV but also the LRV. Because the ATE is generalizable to ORV values around the cutoff, the proposed model allows researchers the ability to search for a treatment assignment mechanism that maximizes utility by allowing for a targeted balance between the effectiveness of the intervention and resource consumption. This would allow policymakers more flexibility in enacting existing interventions in new contexts. Additionally, the proposed model allows the conditional ATE to be identified given the LRV score. As the definitions of an LRV remain the same across different test forms, the estimation of an ATE with respect to the LRV would allow researchers to synthesize results from multiple studies using different ORVs measuring the same LRV. To enhance the accessibility of RD analysis with latent variables for researchers, an R Shiny application is currently under development (https://lrddbeta.shinyapps.io/lrddshiny%5fsingle). The R Shiny application, which is operated using a different computational algorithm, supports not only LRVs but also latent covariates and latent outcome variables, as detailed in the Supplementary Document.</p> <hd id="AN0192371394-24">A local randomization framework</hd> <p>We remark that the continuity-based framework is not the only available statistical theory of RD designs. An alternative formulation, called the local randomization framework, builds upon the idea that a sharp RD design resembles a randomized control trial for ORV values "in the vicinity" of the cutoff (Cattaneo et al., [<reflink idref="bib10" id="ref76">10</reflink>]). More formally, local randomization requires that the potential outcomes are independent to the RV within a window of positive length around the cutoff and that the joint distribution of the outcome and the ORV is known (Cattaneo et al., [<reflink idref="bib13" id="ref77">13</reflink>]). Under these assumptions, extrapolation of RD treatment effect is automatically allowed within the window of local randomization (Cattaneo et al., [<reflink idref="bib11" id="ref78">11</reflink>]); however, it does not support further generalization outside the window. In practice, it is possible to apply both continuity and local randomization views of RD for the purpose of program evaluation. For instance, Cattaneo et al. ([<reflink idref="bib14" id="ref79">14</reflink>]) recommend using continuity-based local polynomial methods for analysis and supplementing with local randomization methods as a robustness check. For a more detailed review of the two frameworks refer to Cattaneo and Titiunik ([<reflink idref="bib12" id="ref80">12</reflink>]).</p> <hd id="AN0192371394-25">Limitations</hd> <p>The proposed augmentation to conventional sharp RD analyses has several limitations.</p> <p>First, compared to the conventional RD analysis, the new RD framework with an LRV relies on more assumptions that must be examined closely. Nonparametric identification of ATEs on the LRV scale requires three assumptions (see the Supplementary Document): 1) the measurement model for the LRV (Equation 3) is correctly specified and ensures assignment to both treatment and control with positive probability across the entire range of LRV values, 2) the potential outcomes are conditionally independent to the indicators (and thus the ORV) given the LRV, and 3) distributional assumptions on the observable outcome. In the present study, we also restrict ourselves to fitting a fully parametric version of the latent RD model, which involves extra assumptions about distributional and functional forms. In practice, it is necessary to assess the tenability of various parametric assumptions and, if necessary, extend the proposed latent RD framework to incorporate more complex structural relations. Future research is encouraged to develop goodness-of-fit tests and indices to evaluate the key assumptions of the proposed model, which can be based on the recent development in model-fit assessment for categorical data (e.g., Joe & Maydeu-Olivares, [<reflink idref="bib24" id="ref81">24</reflink>]; Liu et al., [<reflink idref="bib28" id="ref82">28</reflink>]; Maydeu-Olivares, [<reflink idref="bib32" id="ref83">32</reflink>]; Maydeu-Olivares & Liu, [<reflink idref="bib33" id="ref84">33</reflink>]). It is also of interest to evaluate the robustness of ATE estimation in the presence of model misspecification.</p> <p>Second, we only considered summed scores in the present study, while response-pattern scores, such as IRT scale scores, are also commonly used in practice. Our analytical framework can be generalized to accommodate response-pattern scores since the ORV in our model can be any function of the observed indicators for the LRV. The evaluations of posterior means and quantiles are even simpler as no aggregation of response patterns producing the same summed score is needed. However, unique challenges arise as IRT scale scores depend on item parameters. To map each response pattern to the corresponding ORV score, we need to acquire measurement model parameters used for IRT scoring prior to treatment assignment, instead of estimating measurement parameters together with structural parameters in one stage. Statistical inference based on two-stage estimation is more involved: Valid standard errors for structural parameters can be obtained using pseudo ML estimation (Gong & Samaniego, [<reflink idref="bib19" id="ref85">19</reflink>]; Parke, [<reflink idref="bib40" id="ref86">40</reflink>]), which is left for future work.</p> <p>Third, we only examined the performance of the proposed model when using the full sample for estimation. In practice, researchers tend to fit RD models to a subsample of the data that contains individuals with RV scores within a certain distance of the cutoff, called a bandwidth. This is largely done to avoid gross misspecification of the functional form. However, fitting the latent RD model using a subsample within a bandwidth would lead to poor estimation of measurement model parameters. To rectify this, pseudo ML estimation (Gong & Samaniego, [<reflink idref="bib19" id="ref87">19</reflink>]; Parke, [<reflink idref="bib40" id="ref88">40</reflink>]) can be used to first estimate the measurement model using the full data set. Those parameters can then be fixed and the structural parameters can be estimated using only the subsample within the designated bandwidth. Alternatively, the first stage fitting of the measurement model can be omitted if measurement model parameters are available to the analyst (e.g., for computing IRT scale scores). Standard error adjustment by pseudo ML ensures that the sampling variability from the previous measurement model fitting is properly taken into account.</p> <p>Fourth, response data behind the calculation of ORVs may not be routinely accessible in RD studies. It is particularly difficult to retrieve raw item responses if the test is proprietary. Obtaining item-level data requires a close collaboration between the program evaluation team and the test-maintaining company. However, some national panel studies provide item-level data, such as the German National Education Panel Study (NEPS; Aßmann et al., [<reflink idref="bib2" id="ref89">2</reflink>]), the Longitudinal Internet Studies for the Social sciences in the Netherlands (LISS panel; Scherpenzeel, [<reflink idref="bib46" id="ref90">46</reflink>]), and the UK Household Longitudinal Study (UKHLS; Fisher et al., [<reflink idref="bib18" id="ref91">18</reflink>]). The repeated measurement designs and their associated interventions may provide promising opportunities for applying the proposed approach in future research. Given the dividends our method can pay, we hope to bridge the worlds of psychometrics and causal inference to better serve the practice of program evaluation. If item-level data are not available but more than one subscores of the test are, then we can still take advantage of the proposed model for RD extrapolation and heterogeneity quantification. In this case, we can build a measurement model for subscores rather than raw item responses.</p> <p>Fifth, this study primarily focused on a unidimensional measurement model and a single-level structural model. However, our approach can be extended in multiple ways. Given that RD designs can involve multiple assignments variables (Wong et al., [<reflink idref="bib53" id="ref92">53</reflink>]), incorporating multidimensional measurement models would allow for the integration of latent variables within a multivariate RD framework. Additionally, data in educational settings are often collected through cluster sampling. To account for this, our approach can be extended to multilevel structural models that explicitly model hierarchical data structures. We are currently pursuing both extensions. For the multilevel RD design, building on our previous works (Morell, [<reflink idref="bib37" id="ref93">37</reflink>]; Morell et al., [<reflink idref="bib38" id="ref94">38</reflink>]), a full manuscript is in progress.</p> <hd id="AN0192371394-26">Article Information</hd> <p> <bold>Conflict of Interest Disclosures:</bold> Each author signed a form for disclosure of potential conflicts of interest. No authors reported any financial or other conflicts of interest in relation to the work described.</p> <p> <bold>Ethical Principles:</bold> The authors affirm having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.</p> <p> <bold>Funding:</bold> The research reported here was supported, in whole or in part, by the Institute of Education Sciences, U.S. Department of Education, through grant R305D220030 to Universtiy of Maryland. The opinions expressed are those of the authors and do not represent the views of the Institute or the U.S. Department of Education.</p> <p> <bold>Role of the Funders/Sponsors:</bold> None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.</p> <p> <bold>Acknowledgments:</bold> The authors would like to thank Drs. Peter M. Steiner, Alberto Maydeu-Olivares and anonymous Associate Editor and Reviewers for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the authors alone, and endorsement by the authors' institution or the Institute of Educational Sciences is not intended and should not be inferred.</p> <ref id="AN0192371394-27"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref5" type="bt">1</bibl> <bibtext> In the analysis of Jacob and Lefgren ([23]), however, compliance is not perfect, leading to a fuzzy RD design, which we do not further discuss in the current paper.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref14" type="bt">2</bibl> <bibtext> Throughout the paper, we keep the direction of inequality in the treatment indicator consistent with the illustrative example due to Jacob and Lefgren ([23]). In practice, the direction can be flipped in cases when the cutoff represents a lower bound of eligible ORV values for treatment assignment.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref22" type="bt">3</bibl> <bibtext> It is assumed that the presence of</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> does not affect the identification of local ATE (e.g., Equation 1).</bibtext> </blist> <blist> <bibl id="bib4" idref="ref18" type="bt">4</bibl> <bibtext> Some authors (e.g., Bartholomew et al., [3]) term the conditional distribution of</bibtext> </blist> <blist> <bibtext>Graph</bibtext> </blist> <blist> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> given</bibtext> </blist> <blist> <bibl id="bib5" idref="ref41" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib6" idref="ref43" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> the "measurement model" and the marginal distribution of</bibtext> </blist> <blist> <bibl id="bib7" idref="ref26" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib8" idref="ref2" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> the "structural model." We, however, reserve the term "structural model" for the conditional distribution of the potential outcomes given</bibtext> </blist> <blist> <bibl id="bib9" idref="ref3" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib10" idref="ref56" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi mathvariant="bold">X</mi></mrow><mi>i</mi></msub></mrow></mrow></math> </ephtml> and</bibtext> </blist> <blist> <bibl id="bib11" idref="ref19" type="bt"></bibl> <bibtext>Graph</bibtext> </blist> <blist> <bibl id="bib12" idref="ref10" type="bt"></bibl> <bibtext> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><msub><mrow><mi>θ</mi></mrow><mi>i</mi></msub></mrow></mrow><mtext>.</mtext></math> </ephtml> </bibtext> </blist> <blist> <bibtext> For simplicity, we use the term pdf for both discrete and continuous random variables. It should be understood as the Radon-Nikodym derivative with respect to the Lebesgue measure in the continuous case and to the counting measure in the discrete case.</bibtext> </blist> <blist> <bibtext> Since the LRV is connected to the ORV by the joint pdf in Equation 3, we can also conclude that the treatment effect is different across the ORV.</bibtext> </blist> <blist> <bibtext> See Discussion for using IRT scale scores as the ORV.</bibtext> </blist> <blist> <bibtext> Monica Morell and Muwon Kwon contributed equally to this paper.</bibtext> </blist> <blist> <bibtext> Supplemental data for this article can be accessed online at https://doi.org/10.1080/00273171.2025.2565591.</bibtext> </blist> </ref> <ref id="AN0192371394-28"> <title> References </title> <blist> <bibtext> Angrist, J. D., & Rokkanen, M. (2015). Wanna get away? Regression discontinuity estimation of exam school effects away from the cutoff. 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Items – Name: Title
  Label: Title
  Group: Ti
  Data: Regression Discontinuity Analysis with Latent Variables
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Monica+Morell%22">Monica Morell</searchLink><br /><searchLink fieldCode="AR" term="%22Muwon+Kwon%22">Muwon Kwon</searchLink><br /><searchLink fieldCode="AR" term="%22Youngjin+Han%22">Youngjin Han</searchLink><br /><searchLink fieldCode="AR" term="%22Youjin+Sung%22">Youjin Sung</searchLink><br /><searchLink fieldCode="AR" term="%22Yang+Liu%22">Yang Liu</searchLink><br /><searchLink fieldCode="AR" term="%22Ji+Seung+Yang%22">Ji Seung Yang</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22Grantee+Submission%22"><i>Grantee Submission</i></searchLink>. 2026 61(1):178-191.
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  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 15
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2026
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  Label: Sponsoring Agency
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  Data: Institute of Education Sciences (ED)
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  Data: R305D220030
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  Data: Journal Articles<br />Reports - Research
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  Label: Descriptors
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  Data: <searchLink fieldCode="DE" term="%22Regression+%28Statistics%29%22">Regression (Statistics)</searchLink><br /><searchLink fieldCode="DE" term="%22Models%22">Models</searchLink><br /><searchLink fieldCode="DE" term="%22Item+Response+Theory%22">Item Response Theory</searchLink><br /><searchLink fieldCode="DE" term="%22Research+Design%22">Research Design</searchLink><br /><searchLink fieldCode="DE" term="%22Research+Problems%22">Research Problems</searchLink><br /><searchLink fieldCode="DE" term="%22Scores%22">Scores</searchLink><br /><searchLink fieldCode="DE" term="%22Statistical+Inference%22">Statistical Inference</searchLink><br /><searchLink fieldCode="DE" term="%22Sample+Size%22">Sample Size</searchLink>
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  Label: DOI
  Group: ID
  Data: 10.1080/00273171.2025.2565591
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0027-3171<br />1532-7906
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: A regression discontinuity (RD) design is often employed to provide causal evidence when the randomization of the treatment assignment is infeasible. When variables of interest are latent constructs measured by observed indicators, the conventional RD analysis using observed variable scores does not allow researchers to examine heterogeneity in the estimated local average treatment effect (ATE) and to generalize the ATE to participants away from the cutoff. We propose a novel methodological augmentation to the conventional RD analysis, which assumes the availability of multiple indicator variables (i.e., raw item responses) that measure the latent construct underlying the running variable. By specifying an explicit measurement model based on those indicator variables, our latent RD framework allows 1) defining the local ATE conditional on the latent construct, 2) disentangling the heterogeneity of the local ATE, and 3) generalizing the local ATE to running variable scores away from the cutoff. In a proof-of-concept simulation we illustrate the proposed augmentation recovers parameters of interest well under practical test length and sample size conditions.
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  Data: 2026
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  Data: ED680414
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      – Type: doi
        Value: 10.1080/00273171.2025.2565591
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 15
        StartPage: 178
    Subjects:
      – SubjectFull: Regression (Statistics)
        Type: general
      – SubjectFull: Models
        Type: general
      – SubjectFull: Item Response Theory
        Type: general
      – SubjectFull: Research Design
        Type: general
      – SubjectFull: Research Problems
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      – SubjectFull: Scores
        Type: general
      – SubjectFull: Statistical Inference
        Type: general
      – SubjectFull: Sample Size
        Type: general
    Titles:
      – TitleFull: Regression Discontinuity Analysis with Latent Variables
        Type: main
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      – PersonEntity:
          Name:
            NameFull: Monica Morell
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            NameFull: Muwon Kwon
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            NameFull: Youngjin Han
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            NameFull: Youjin Sung
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            NameFull: Yang Liu
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            NameFull: Ji Seung Yang
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              Value: 61
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            – TitleFull: Grantee Submission
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