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Predictive Strength of Math Screening across English Language Proficiency Scores in Third to Eighth Grade

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Title: Predictive Strength of Math Screening across English Language Proficiency Scores in Third to Eighth Grade
Language: English
Authors: Garret J. Hall (ORCID 0000-0002-8285-3239), Emma Doyle, Edgardo Mejias Vazquez (ORCID 0009-0006-4339-4531)
Source: Learning Disabilities Research & Practice. 2025 40(2):71-85.
Availability: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
Peer Reviewed: Y
Page Count: 15
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Junior High Schools
Middle Schools
Secondary Education
Early Childhood Education
Grade 3
Primary Education
Grade 4
Intermediate Grades
Grade 5
Grade 6
Grade 7
Grade 8
Descriptors: Screening Tests, Mathematics Tests, Achievement Tests, Mathematics Achievement, Language Proficiency, Predictive Validity, English Learners, Elementary School Students, Middle School Students, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Spanish, Summative Evaluation
Assessment and Survey Identifiers: Measures of Academic Progress
DOI: 10.1177/09388982241309118
ISSN: 0938-8982
1540-5826
Abstract: Using data from students in Grades 3-8 in school years 2018-2019 (N = 1,871) and 2021-2022 (N = 1,740), we examined the strength of fall math screening using Measures of Academic Progress (MAP) for predicting end-of-year state math assessment performance levels and whether this prediction varied across English language proficiency (ELP). In addition, we examined whether differential predictive strength varied across screenings using English or Spanish MAP math (2018-2019 only). MAP math was similarly predictive across the ELP continuum (odds ratios between 6 and 15; log-odds between 1.5 and 2.5). However, in 2018-2019, there was more variation in screening prediction across ELP for students assessed using the English MAP math compared to Spanish MAP math. For 2021-2022, there was little evidence of differential screening prediction across ELP. Our findings suggest that MAP math is moderately predictive across ELP and school years. We discuss limitations, future directions, and implications for research and practice.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1469552
Database: ERIC
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  Value: <anid>AN0184489646;z2z01may.25;2025Apr17.01:46;v2.2.500</anid> <title id="AN0184489646-1">Predictive Strength of Math Screening Across English Language Proficiency Scores in Third to Eighth Grade </title> <p>Using data from students in Grades 3–8 in school years 2018–2019 (N = 1,871) and 2021–2022 (N = 1,740), we examined the strength of fall math screening using Measures of Academic Progress (MAP) for predicting end-of-year state math assessment performance levels and whether this prediction varied across English language proficiency (ELP). In addition, we examined whether differential predictive strength varied across screenings using English or Spanish MAP math (2018–2019 only). MAP math was similarly predictive across the ELP continuum (odds ratios between 6 and 15; log-odds between 1.5 and 2.5). However, in 2018–2019, there was more variation in screening prediction across ELP for students assessed using the English MAP math compared to Spanish MAP math. For 2021–2022, there was little evidence of differential screening prediction across ELP. Our findings suggest that MAP math is moderately predictive across ELP and school years. We discuss limitations, future directions, and implications for research and practice.</p> <p>Keywords: English learner; English language proficiency; mathematics; universal screening; Bayesian</p> <p>Academic universal screening among English learners (ELs)[<reflink idref="bib5" id="ref1">5</reflink>] within multi-tiered systems of support (MTSS) requires special attention to mitigate over- and under-identification of students in need of academic supports ([<reflink idref="bib1" id="ref2">1</reflink>]). Thus, inequities in this screening process have potential to inaccurately inform decision-makers about the strengths and needs of ELs, particularly with respect to academic risk identification. This is particularly important given the potential patterns of over-identification of ELs for learning disabilities (SLD) ([<reflink idref="bib51" id="ref3">51</reflink>]), exacerbated by ambiguities in the implementation of exclusionary criteria for SLD ([<reflink idref="bib21" id="ref4">21</reflink>]).</p> <p>The impacts of COVID-19 have compounded many students' learning challenges, as demonstrated not only by the general population trends of lower achievement ([<reflink idref="bib29" id="ref5">29</reflink>]) but also by those demonstrating existing learning difficulties ([<reflink idref="bib14" id="ref6">14</reflink>]). COVID-19-related learning disruptions may have uniquely impacted the interacting barriers and facilitators to ELs' math learning, including the confluence of both language and content-area learning. This increases the imperative to understand how to disentangle ELs' language proficiencies from their math performance to arrive at equitable decisions regarding their math learning needs.</p> <p>Math screening has not received an adequate degree of attention among ELs, despite math skill development having a strong linguistic basis ([<reflink idref="bib48" id="ref7">48</reflink>]; [<reflink idref="bib47" id="ref8">47</reflink>]; [<reflink idref="bib49" id="ref9">49</reflink>]). This gap is important to address to highlight how math screening data can be used to help identify academic risk (and thereby contribute to the potential identification of disabilities) within a comprehensive MTSS that equitably addresses students' strengths and needs in a preventive manner.</p> <p>Screening ELs' math skills in alternative languages adds another layer to consider in implementing universal screening ([<reflink idref="bib49" id="ref10">49</reflink>]). Recent work has brought attention to math screening among ELs (e.g., [<reflink idref="bib4" id="ref11">4</reflink>]), with some research focusing on how accounting for students' English language proficiency (ELP) may assist screening predictions ([<reflink idref="bib19" id="ref12">19</reflink>]). Other recent studies have also investigated language-specific screening practices ([<reflink idref="bib9" id="ref13">9</reflink>]; [<reflink idref="bib10" id="ref14">10</reflink>]; [<reflink idref="bib20" id="ref15">20</reflink>]), highlighting the importance of attending to linguistic diversity and language proficiencies in math screening for predicting academic risk. In addition, recent work using latent class analysis has shown that lower performance in both reading and math in English and Spanish is characteristic of math learning difficulties among ELs. However, some students demonstrate more consistent challenges across English and Spanish reading and math, whereas others show performance challenges more specific to reading comprehension and math problem solving ([<reflink idref="bib49" id="ref16">49</reflink>]). Early screening and identification of learning difficulties can help practitioners attend to these emerging difficulties before they worsen.</p> <p>A critical concern in universal screening when considering linguistic diversity is screener performance across different language backgrounds ([<reflink idref="bib1" id="ref17">1</reflink>]). The goal of a screener should be that its predictive strength is similar across demographic factors to avoid inadvertently biased decision-making ([<reflink idref="bib2" id="ref18">2</reflink>]). Such screening data offer an essential point of reference for other sources of quantitative and qualitative data (e.g., teacher input) to be integrated into the academic risk (and learning disability) identification process. The impacts of COVID-related school closures and variability in instructional modality prompt questions about how the distribution of academic performance changed and for whom, particularly when considering the added barriers to instruction and service eligibility ELs may face ([<reflink idref="bib21" id="ref19">21</reflink>]; [<reflink idref="bib42" id="ref20">42</reflink>]).</p> <p>A key remaining question to inform math screening among ELs is whether math screener prediction varies across the ELP spectrum among ELs. Since attaining ELP is the primary federally mandated benchmark among ELs ([<reflink idref="bib13" id="ref21">13</reflink>]), in addition to other less formal programmatic benchmarks through their language instructional programs, a closer examination of math screening prediction heterogeneity across ELP is necessary to gain additional evidence on whether math screening is equally likely to predict students' academic risk regardless of their ELP. Evidence to the contrary, such as screeners more strongly predicting criterion outcomes at lower or higher ELP, would suggest that the screener may be more advantageous for predicting academic risk at particular ELP levels. Such a conclusion would draw attention to the need to consider screener prediction not only between ELs and their English-proficient peers (e.g., [<reflink idref="bib4" id="ref22">4</reflink>]) but also within the subpopulation of ELs. The often unclear guidance for determining the SLD exclusionary criteria regarding language and cultural background necessitates a closer examination of how screening data can provide meaningful data points to facilitate equitable identification of learners' needs ([<reflink idref="bib21" id="ref23">21</reflink>]).</p> <p>In sum, the confluence of the lack of research on math screening accuracy among ELs, the role of within-group variability of ELP in screening prediction, and the potential unique impacts of COVID-19 on ELs' math learning warrant an investigation of the predictive strength of math screening across ELP and under different learning environments pre- and post-COVID school closures. The administration of screening in multiple languages (primarily Spanish and English), which is increasingly more available to schools, further compounds these contextual and linguistic issues in screening prediction. Thus, this study had three key aims.</p> <p>First, we examined variability in math screening predictive strength across ELP among ELs in Grades 3–8. Second, we examined how this predictive strength across ELP changes depending on whether students were administered the math assessment in Spanish or English. Finally, we examined whether this screening prediction changed between school years 2018–2019 and 2021–2022. Changes to eligibility for screening in Spanish changed across these years as well, further complicating the determination of whether screening accuracy varies across ELP levels and assessment languages. The confluence of these measurement and contextual factors offers an ecologically valid examination of how ELP interacts with math screening under different environmental and measurement circumstances.</p> <hd id="AN0184489646-2">EL Academic Performance Trends and Screening</hd> <p>Much of the literature on ELs' academic success has focused on the gap between this student population and their English-proficient peers. The gap is largely attributable to the definition of the EL category itself: It is based on a variable, ELP, that directly impacts academic performance in English. As a result, such a gap is not unexpected, and alternative classification schemes for linguistically diverse learners often lead to different academic performance trends (e.g., "multilingual learners" vs only ELs; [<reflink idref="bib25" id="ref24">25</reflink>]). The trend of much of the screening literature on ELs, in particular, focuses on this achievement gap and how screening can assist in accurately identifying ELs' academic risk ([<reflink idref="bib4" id="ref25">4</reflink>]; [<reflink idref="bib24" id="ref26">24</reflink>]). The performance discrepancy between ELs and non-ELs, although often reinforced by inequitable access to instruction ([<reflink idref="bib42" id="ref27">42</reflink>]), should not be the primary focus of screening because it overlooks the substantial variability within the EL group. Recent work has focused on predictive accuracy of both reading ([<reflink idref="bib19" id="ref28">19</reflink>]; [<reflink idref="bib26" id="ref29">26</reflink>]; [<reflink idref="bib32" id="ref30">32</reflink>]) and math ([<reflink idref="bib19" id="ref31">19</reflink>]) <emph>within</emph> language groups. However, no studies to date have focused on how math screening prediction strength may vary across the spectrum of ELP <emph>among</emph> ELs.</p> <p>In considering the potential for differential screening accuracy across ELP, it is also essential to attend to the differences in base rates of academic risk across the range of ELP. "Base rates" refer to the number of individuals exhibiting a target condition. In the case of academic screening, the base rate refers to the number of individuals who demonstrate academic risk or perform in a particular performance category on the criterion outcome (e.g., state tests). Lower base rates (i.e., lower prevalence of a condition) often make it more difficult to detect the condition ([<reflink idref="bib12" id="ref32">12</reflink>]). This creates a unique methodological challenge for assessing screener accuracy among ELs because, regardless of the properties of the screener itself, the accuracy of a screener is sensitive to the base rate(s) of the category (or categories) the screener is predicting ([<reflink idref="bib12" id="ref33">12</reflink>]). Because of the positive correlation between ELP and math performance, the distribution of base rates is not constant across ELP. That is, students with higher ELP are more likely to be performing higher on criterion outcomes and screeners, so a screener's predictive ability may depend on its properties, the base rates of the outcome being predicted, and the variation of base rates across ELP levels.</p> <p>This area of research is further challenged by the increasing use of alternative languages in math assessment. The use of alternative languages when available is current recommended practice ([<reflink idref="bib51" id="ref34">51</reflink>]). This raises additional questions about whether screener predictive strength depends on both the language of assessment and students' ELP, and there are additional limitations on the availability of alternative screening languages (which are often only Spanish; e.g., [<reflink idref="bib41" id="ref35">41</reflink>]). Greater ELP would presumably aid students in accessing the academic and content-specific language on English math screeners, but the relation of ELP to Spanish math screener prediction would be less clear since lower ELP would not necessarily translate to greater Spanish proficiency. In all cases, there is uncertainty around how ELP would interact with the qualities of the math screener to moderate the screener's predictive performance considering the variety of social, contextual, and linguistic factors that interact in the assessment of ELs ([<reflink idref="bib46" id="ref36">46</reflink>]; [<reflink idref="bib48" id="ref37">48</reflink>]). To date, no research has examined these intersections of ELP, math screening prediction, and assessment language, despite the relevance of this intersection for establishing data-based decision processes across language groups, language proficiencies, and assessment languages ([<reflink idref="bib51" id="ref38">51</reflink>]).</p> <hd id="AN0184489646-3">The Current Study</hd> <p>No studies to date have examined the differential predictive value of math screening as a function of ELP. Moreover, there is currently no research on how this may differ across languages of assessment, though some recent work has focused specifically on Spanish reading screening accuracy ([<reflink idref="bib26" id="ref39">26</reflink>]) as well as the predictive validity of dynamic math assessment ([<reflink idref="bib9" id="ref40">9</reflink>]). Other studies have demonstrated the impacts of math test translation ([<reflink idref="bib43" id="ref41">43</reflink>]), differences in screening performance by language across ELP ([<reflink idref="bib20" id="ref42">20</reflink>]), and differential item functioning in math across EL or non-EL status ([<reflink idref="bib5" id="ref43">5</reflink>]). Yet, a substantial research gap remains regarding heterogeneity in math screener predictive strength across ELP.</p> <p>As noted, compounding the problem of addressing these screening issues is the prolonged influence of COVID-19-related school closures and changes to instruction on students' achievement, which resulted in substantial changes to the distribution of academic performance ([<reflink idref="bib29" id="ref44">29</reflink>]). ELs may have faced unique additional challenges in terms of both their academic skills and their language development, necessitating an examination of how sensitive the interactions of ELP and math skills in predicting academic risk were to the pandemic-related changes to instruction. This is most relevant in terms of the substantial changes to the distribution of academic performance following COVID-related disruptions because these performance trends change the base rates of academic risk (if the definition is held constant pre- and post-COVID). However, there is conflicting evidence pertaining to changes in ELs' math performance during the pandemic. Some research suggests that EL's math performance declines were less pronounced compared to non-EL peers during the pandemic ([<reflink idref="bib39" id="ref45">39</reflink>]), though this may not hold across the school year ([<reflink idref="bib40" id="ref46">40</reflink>]). Other studies have demonstrated that learning loss from COVID impacted students demonstrating academic difficulties more than their higher-performing peers ([<reflink idref="bib14" id="ref47">14</reflink>]). ELs already demonstrating mathematics learning difficulties likely faced additional barriers to their math learning during this time given the compounding issue of concurrent language development, though there is currently no research addressing how ELP and math screening interact prior to and after COVID-19-related school closures. This is an essential component in examining how educational contexts shape the function of screeners for making equitable data-based decisions.</p> <p>To address these research gaps, this study addressed three research questions (RQs).</p> <p></p> <ulist> <item> To what degree does universal math screening predictive strength vary as a function of ELP?</item> <p></p> <item> To what extent is the interaction between ELP and math skill nonlinear in predicting state test performance levels?</item> <p></p> <item> Does this predictive strength differ between English and Spanish versions of the same universal screener?</item> </ulist> <p>We expected that, on average, fall math screening would be moderately to strongly predictive of spring state test performance given the consistent normative patterns of math screening's predictive strength for state test performance. However, we expected that math screening would decrease in predictive strength as ELP decreases, given the idiosyncrasies of content-specific academic skill and multi-language development ([<reflink idref="bib47" id="ref48">47</reflink>]), particularly in math, which has a more inconsistent language load than reading. Due to this nonlinear development of math and language skills (and likely a range of individual differences in that), we did not assume that increases in ELP would relate to a linear change in the predictive strength of math screening. Consequently, we used generalized additive mixed models (GAMMs; [<reflink idref="bib55" id="ref49">55</reflink>]) to examine the nonlinearity of the interaction between ELP and math screening in predicting end-of-year state test performance. Last, we examined whether this predictive strength of the ELP × math screening interaction differed depending on whether students took the Spanish version of the math screener of the English version. Due to the lack of research on this, we did not have a priori hypotheses for RQ3. We examined RQs 1 and 2 with data from pre-COVID screening (2018–2019) as well as data from a school year following COVID-related virtual instruction (2021–2022; hereafter "post-COVID"). RQ3 was feasible only in the 2018–2019 school year.</p> <hd id="AN0184489646-4">Method</hd> <p></p> <hd id="AN0184489646-5">Participants and Procedure</hd> <p>The study used data from students in Grades 3–8 in 19 schools (4 middle schools of Grades 7–8) from a large suburban school district in the Midwestern United States from the years 2018–2019 (N = 1,871) and 2021–2022 (N = 1,740). Students were included in the sample if they had ELP scores from the prior year (used to inform instructional decisions in the subsequent year) and fall screening scores (demographic characteristics of the sample are provided in Table 1).</p> <p>Table 1. Sample Demographics.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="char" char="(" /><col align="char" char="(" /></colgroup><thead><tr><th align="left" rowspan="2" /><th align="left" colspan="2">Count (proportion)</th></tr><tr><th align="left">2018–2019 (<italic>N </italic>= 1871)</th><th align="left">2021–2022 (<italic>N </italic>= 1740)</th></tr></thead><tbody><tr><td colspan="3">Demographics</td></tr><tr><td> Black, Asian, NHPI, AI/AN, or two or more races</td><td>319 (.17)</td><td> 330 (.19)</td></tr><tr><td> Hispanic/Latino</td><td>1369 (.73)</td><td>1220 (.70)</td></tr><tr><td> White</td><td>183 (.10)</td><td>190 (.11)</td></tr><tr><td> EL</td><td>1599 (.85)</td><td>1520 (.87)</td></tr><tr><td> Spanish home language</td><td>1272 (.68)</td><td>1207 (.69)</td></tr><tr><td> Female</td><td>872 (.47)</td><td>758 (.44)</td></tr><tr><td> Free/reduced-price lunch eligible</td><td>1368 (.73)</td><td>1224 (.70)</td></tr><tr><td> Has IEP</td><td>221 (.12)</td><td>247 (.14)</td></tr><tr><td colspan="3">Grade</td></tr><tr><td> 3</td><td>432 (.23)</td><td>374 (.21)</td></tr><tr><td> 4</td><td>450 (.24)</td><td>320 (.18)</td></tr><tr><td> 5</td><td>435 (.23)</td><td>328 (.19)</td></tr><tr><td> 6</td><td>266 (.14)</td><td>273 (.16)</td></tr><tr><td> 7</td><td>156 (.08)</td><td>225 (.13)</td></tr><tr><td> 8</td><td>132 (.07)</td><td>220 (.13)</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>. NHPI = Native Hawaiian and other Pacific Islander; AI/AN = American Indian/Alaska Native; IEP = individualized education plan; EL = English learner. Race/ethnicity category names reported as provided in the district data. Polish and Japanese were the next most frequently reported home languages (3%–5%); for 9% of students, home language data were missing in 2018–2019. More than 40 other languages were reported.</p> <p>The participating district adheres to a bilingual literacy model for language instructional programs that includes Spanish one-way (classes composed of only Spanish-speaking students learning both English and Spanish) and two-way (composed of both Spanish and non-Spanish-speaking students) dual-language programs in addition to transitional English programming for less commonly spoken languages (which focuses on promoting only English proficiency).</p> <p>Data on specific language instructional program enrollment are unavailable. As Table 1 shows, not all students in the years of analysis (2018–2019, 2021–2022) were classified as ELs due to many reaching proficiency criteria, though all students had ELP scores from the prior year. The school district shared de-identified student-level data with the first author. All study procedures were determined "not human subjects research" by the authors' Institutional Review Board.</p> <hd id="AN0184489646-6">Measures</hd> <p></p> <hd id="AN0184489646-7">Predictors</hd> <p></p> <hd id="AN0184489646-8">ACCESS for ELLs</hd> <p>ACCESS for ELLs ([<reflink idref="bib52" id="ref50">52</reflink>]) is a summative measure that captures several dimensions of ELP, including comprehension, literacy, oral language, and written language. ACCESS demonstrated strong validity and reliability in the normative sample for measuring students' ELP performance and growth, including internal consistency (coefficient α =.6–.99 across subtests and grades) and classification accuracy of ELP levels ([<reflink idref="bib8" id="ref51">8</reflink>]). ACCESS is administered in the spring of each year to eligible students, so the scores from the previous year inform EL status in the subsequent year. In the current study, we used the ACCESS overall composite scale score from the previous year (i.e., 2017–2018 for 2018–2019) as a measure of ELP. ACCESS scores are also binned into ordinal ELP levels within each grade ranging from 1.0 to 6.0. These levels are typically used for decision-making in schools. However, given the advantageous statistical properties of the overall scale score (e.g., no truncation of scores within grade and more granularity in ELP measurement), we used the overall scale score in all analyses.</p> <hd id="AN0184489646-9">Measures of Academic Progress Growth Math</hd> <p>Measures of Academic Progress (MAP) Growth Mathematics is a computer-adaptive interim assessment used as an initial stage of screening in the participating district ([<reflink idref="bib35" id="ref52">35</reflink>]). It was administered between August and September of each year (2018–2019 and 2021–2022). Based on criteria set by the school district, students with lower ELP levels took the equivalent Spanish MAP Growth Math. In 2018, this criterion was approximately an overall ELP level of 5 (of 6), and 50% of students took Spanish MAP Math. However, in 2021, this criterion was changed, and few students per grade took Spanish MAP (except Grade 3, in which approximately 50% still took Spanish MAP math). Generally, third graders with ELP < 5 took Spanish MAP in 2021, whereas Spanish MAP was limited to those with ELP < 3 in other grades (as well as other information determined on an as-needed basis).</p> <p>The Spanish and English versions of MAP Math are equivalent and share the same norms ([<reflink idref="bib33" id="ref53">33</reflink>]); the assessment demonstrates validity and reliability for assessing students' mathematics performance and growth in fall, winter, and spring of each year as well as across years [[<reflink idref="bib31" id="ref54">31</reflink>]; [<reflink idref="bib34" id="ref55">34</reflink>]]. The measure shows strong correlations with criterion measures (state tests) between fall and spring (<emph>r </emph>=.78–.82 in Grades 3–8) and classification accuracy (area under the curve =.91–.93 in Grades 3–8; [<reflink idref="bib31" id="ref56">31</reflink>]).</p> <hd id="AN0184489646-10">Outcome</hd> <p></p> <hd id="AN0184489646-11">Math State Summative Assessment</hd> <p>The State Summative Assessment (SSA) in math was administered in April of each year. Scale scores are transformed into ordinal performance levels ranging from 1 (<emph>did not yet meet expectations</emph>) to 5 (<emph>exceeded expectations</emph>). We used the full range of performance levels (1–5) to better differentiate screening predictions, as opposed to dichotomizing "at risk" or "not at risk" ([<reflink idref="bib17" id="ref57">17</reflink>]). Language accommodations on the assessment for ELs are available as appropriate for their ELP (e.g., translation of directions; a paper-based Spanish version of the assessment). Data on accommodation types are unavailable.</p> <hd id="AN0184489646-12">Analytic Plan</hd> <p>We used Bayesian multilevel sequential ordinal models to examine the relation between fall MAP performance and spring SSA performance levels. We adopted the terminology of [<reflink idref="bib15" id="ref58">15</reflink>] and referred to fixed effects as <emph>constant</emph> effects and random effects (i.e., intercepts and slopes) as <emph>varying</emph> effects. To examine whether MAP was similarly predictive of SSA across ELP,[<reflink idref="bib6" id="ref59">6</reflink>] and to compare different models, we estimated six separate models in 2018–2019, and we estimated four of those in 2021–2022 (models that include Spanish MAP as a moderator are not included in 2021–2022). Equations for each of the models are presented in Table 2. In all models, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></math> </ephtml> represents the latent distributions corresponding to each of SSA's ordinal performance levels, <emph>c</emph>, for student <emph>i</emph> in partially crossed school <emph>j</emph> and grade <emph>k</emph>. The sequential ordinal model ([<reflink idref="bib50" id="ref60">50</reflink>]) represents the ordered outcome categories as occurring in conditional sequences (i.e., to reach performance Level 2, one must have reached Level 1). We used the logit link function in all models.</p> <p>Table 2. Models Estimated in Each Year.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="left" /></colgroup><thead><tr><th align="left" rowspan="2">Number</th><th align="left" rowspan="2">Model</th><th align="left" colspan="2">Model applied</th></tr><tr><th align="left">18–19</th><th align="left">21–22</th></tr></thead><tbody><tr><td>1a</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em" xmlns=""><mtr><mtd><mrow><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></mrow><mo>=</mo></mtd><mtd><mspace width="0.25em" /><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><mi>β</mi><mi>X</mi><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>0</mn><mi>j</mi></msub></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>1</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>2</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>3</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>0</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub><mspace width="0.25em" /><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>3</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></math></p></td><td>Yes</td><td>Yes</td></tr><tr><td>1b</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em" xmlns=""><mtr><mtd><mrow><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></mrow><mo>=</mo></mtd><mtd><mspace width="0.25em" /><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>3</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>×</mo><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>4</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>×</mo><msub><mrow><mi mathvariant="normal">Spanish</mi><mspace width="0.25em" /><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>β</mi><mn>5</mn></msub><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>×</mo><mrow><mi mathvariant="normal">Spanish</mi><mspace width="0.25em" /><mi mathvariant="normal">MA</mi></mrow><msub><mrow><mi mathvariant="normal">P</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>6</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>×</mo><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>×</mo><mrow><mi mathvariant="normal">Spanish</mi><mspace width="0.25em" /><mi mathvariant="normal">MA</mi></mrow><msub><mrow><mi mathvariant="normal">P</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><mi>β</mi><mi>X</mi><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>0</mn><mi>j</mi></msub></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>1</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>2</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>3</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>4</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>5</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>0</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub><mspace width="0.25em" /><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>3</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>4</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>5</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>6</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></math></p></td><td>Yes</td><td>No</td></tr><tr><td>1c</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em" xmlns=""><mtr><mtd><mrow><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></mrow><mo>=</mo></mtd><mtd><mspace width="0.25em" /><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><msub><mi>β</mi><mn>2</mn></msub><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>+</mo><mi>β</mi><mi>X</mi></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>0</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>1</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>2</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>0</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub><mspace width="0.25em" /><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></math></p></td><td>Yes</td><td>Yes</td></tr><tr><td>2a</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em" xmlns=""><mtr><mtd><mrow><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></mrow><mo>=</mo></mtd><mtd><mspace width="0.25em" /><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub><mo>+</mo><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mspace width="0.25em" /><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mi>X</mi></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>0</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>1</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>2</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>3</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>0</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub><mspace width="0.25em" /><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>3</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></math></p></td><td>Yes</td><td>Yes</td></tr><tr><td>2b</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em" xmlns=""><mtr><mtd><mrow><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></mrow><mo>=</mo></mtd><mtd><mspace width="0.25em" /><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><msub><mn>1</mn><mrow><mi mathvariant="normal">Spanish</mi><mspace width="0.25em" /><mi mathvariant="normal">MAP</mi></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><mspace width="0.25em" /><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mi>X</mi></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>0</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>1</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>2</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>3</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>0</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub><mspace width="0.25em" /><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>3</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></math></p></td><td>Yes</td><td>No</td></tr><tr><td>2c</td><td><p><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right left" columnspacing="thickmathspace" displaystyle="true" rowspacing=".5em" xmlns=""><mtr><mtd><mrow><mrow><msub><mrow><mover><mrow><mi>S</mi><mi>S</mi><mi>A</mi></mrow><mo>~</mo></mover></mrow><mrow><msub><mi>c</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mrow></mrow><mo>=</mo></mtd><mtd><mspace width="0.25em" /><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub><mo>+</mo><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mspace width="0.25em" /><msub><mi>f</mi><mn>2</mn></msub><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo><mo>+</mo><mi>β</mi><mi>X</mi></mtd></mtr><mtr><mtd /><mtd><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>0</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>1</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>u</mi><mrow><msub><mn>2</mn><mi>j</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>0</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub><mspace width="0.25em" /><mo>+</mo><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub><mo>+</mo><msub><mi>e</mi><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></math></p></td><td>Yes</td><td>Yes</td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note</emph>. SSA = state summative assessment; MAP = Measures of Academic Progress; Spanish MAP = indicator for students taking Spanish MAP. In 2021–2022, Spanish MAP was included as a covariate in vector <emph>X</emph> along with binary indicators for home language [indicator for Spanish, indicators the two other most frequent (Polish, Japanese), and an indicator for missing language (only in 2018–2019) ], IEP status, free/reduced-price lunch status, race/ethnicity (indicator for Hispanic/Latino and indicator for Black, Asian, NHPI, AI/AN, or two or more races), EL status, and gender. All covariates are grade and school mean-centered ([<reflink idref="bib56" id="ref61">56</reflink>]). In Models 2a–2c, the varying effects <emph>u</emph> (school) and <emph>r</emph> (grade) refer to the same varying effects specification as equations 1a–1c (i.e., smooth functions <emph>f</emph> are estimated for MAP and ACCESS constant effects, but traditional linear varying effects are estimated in <emph>u</emph> and <emph>r</emph>). The varying effects numerical subscripts refer to the numerical subscript for each <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> (i.e., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mrow><msub><mn>1</mn><mi>k</mi></msub></mrow></msub></math> </ephtml> refers to the grade varying slope of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>1</mn></msub><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></math> </ephtml> , <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>r</mi><mrow><msub><mn>2</mn><mi>k</mi></msub></mrow></msub></math> </ephtml> refers to the grade varying slope of <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mn>2</mn></msub><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></math> </ephtml> , etc.). This is the case across all equations, except there are no varying effects of smooths (<emph>f</emph>). <emph>e</emph> is fixed to 1 (the default) across all models. All varying effects are multivariate normally distributed within level. Subscripts in parantheses indicate partial cross-classification ([<reflink idref="bib30" id="ref62">30</reflink>]).</p> <p>We standardized MAP and ACCESS to a <emph>M</emph>(<emph>SD</emph>) = 0(<reflink idref="bib1" id="ref63">1</reflink>) scale within grade and school, and all categorical covariates mean-centered within school and grade ([<reflink idref="bib56" id="ref64">56</reflink>]). SSA performance data were missing for approximately 3% of students. It is possible to impute these missing values in posterior prediction based on the model parameters obtained from the non-missing data ([<reflink idref="bib7" id="ref65">7</reflink>]), but this would not alter our conclusions.</p> <p>In all models, <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>β</mi><mrow><msub><mn>0</mn><mi>c</mi></msub></mrow></msub></math> </ephtml> represents the four thresholds (intercepts) which correspond to <emph>c</emph>-1 ordinal categories of the SSA performance levels. <emph>e</emph> (the latent scale) is fixed to 1 in all models. All <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> </ephtml> s represent regression coefficients on the log-odds scale. The linear log-odds coefficients can be interpreted as the increase in log-odds of reaching the next highest performance level per one-unit increase in the predictors (e.g., a 1 SD increase in MAP or ACCESS). <emph>u</emph> and <emph>r</emph> represent school and grade varying effects, respectively.</p> <p> <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mrow><mi mathvariant="normal">ACCESS</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow></math> </ephtml> ) represents a tensor product smooth, which simultaneously estimates nonlinear functions of MAP, ACCESS, and their interaction ([<reflink idref="bib55" id="ref66">55</reflink>]). <emph>f</emph> subscripted with Spanish MAP indicates that different smooth functions were estimated for English and Spanish MAP. A smooth with a single variable [e.g., <ephtml> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mo stretchy="false">(</mo><mrow><msub><mrow><mi mathvariant="normal">MAP</mi></mrow><mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msub></mrow><mo stretchy="false">)</mo></math> </ephtml> ] would indicate that a smooth function was implemented for one variable at a time, though multiple single smooths can be estimated within the same model. Smooth functions do not have a clear interpretation on their own and are best represented visually; however, we provide their posterior parameter estimates where relevant (these are not useful to interpret). In this case, smooth functions were estimated as <emph>SD</emph> parameters along with a constant slope parameter. A higher <emph>SD</emph> for a smooth function would indicate there is more nonlinearity.</p> <p>We compared models based on the difference in expected log predictive pointwise density (ELPD) between models and the standard error (<emph>SE</emph>) of the ELPD. The ELPD is based on leave-one-out cross-validation ([<reflink idref="bib54" id="ref67">54</reflink>]). Higher ELPD values indicate better fit (multiplying the ELPD by −2 converts it to the leave-one-out information criterion). We conducted all analyses in R ([<reflink idref="bib38" id="ref68">38</reflink>]) using the brms package ([<reflink idref="bib6" id="ref69">6</reflink>]). We used the loo package for model comparisons ([<reflink idref="bib54" id="ref70">54</reflink>]).</p> <hd id="AN0184489646-13">Bayesian Priors</hd> <p>Bayesian modeling requires encoding of priors, which are probability distributions representing the underlying assumptions of model parameter estimates ([<reflink idref="bib23" id="ref71">23</reflink>]). These priors are then updated based on the raw data to produce updated estimates, called posterior estimates. We used weakly informative priors in all models, with the strongest source of prior information coming from strong evidence of normative predictive validity of MAP Math ([<reflink idref="bib31" id="ref72">31</reflink>]). Bayesian inference hinges on the updating of priors in the face of new information ([<reflink idref="bib23" id="ref73">23</reflink>]), so we used the model estimates from 2018 to 2019 as priors for the models in 2021–2022. When presenting results, we provide priors along with the posterior estimates.</p> <p>Bayesian modeling offers several advantages, including improving model estimation (especially complex models) and quantifying the uncertainty in parameter estimates through prior distributions ([<reflink idref="bib16" id="ref74">16</reflink>]). <emph>Quantifying uncertainty</emph> in this context refers to this process of formulating and updating priors considering available evidence ([<reflink idref="bib23" id="ref75">23</reflink>]). We estimated several complex models, which had a combination of both overlapping and different priors. Consequently, these Bayesian model comparisons reflect an accumulation of information from new data in combination with our prior assumptions, which we contend is advantageous given that we were focused on extracting information on predictive strength of MAP screening across ELP when considering the prior information available on screener predictive validity and separate years of analysis, i.e., 2018–2019/2021–2022 ([<reflink idref="bib18" id="ref76">18</reflink>]). Finally, sensitivity analyses of Bayesian models with different priors are recommended ([<reflink idref="bib11" id="ref77">11</reflink>]). In the current study, our conclusions were unchanged when using uninformative priors (the defaults in brms); we observed only minor differences in the posterior estimates when using default/uninformative priors compared to our weakly informative or weak priors.</p> <hd id="AN0184489646-14">Results</hd> <p>All model parameters are reported with their posterior mean estimate, their posterior standard deviation (<emph>pSD</emph>), and their 95% credible interval (CI).</p> <hd id="AN0184489646-15">Descriptive Statistics</hd> <p>Descriptive statistics of the measures are provided in Table 3. As illustrated, in 2018–2019, students who were assessed in Spanish scored lower on ACCESS, which was expected given that the criteria for taking Spanish MAP was based on an ACCESS ELP level of <5. We observed several substantial differences in English MAP scores between 2018 and 2021 (e.g., >.5 <emph>SD</emph> difference), with the most pronounced difference occurring in fourth grade. ACCESS scores show similar patterns.</p> <p>Table 3. MAP, ACCESS, and SSA Performance Level Descriptive Statistics.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /></colgroup><thead><tr><th align="left" colspan="2" rowspan="3">Grade</th><th align="center" colspan="3"><italic>M</italic>(<italic>SD</italic>)</th></tr><tr><th align="center" colspan="2">2018–19</th><th align="center">2021–22</th></tr><tr><th align="center">English</th><th align="center">Spanish</th><th align="center">All</th></tr></thead><tbody><tr><td>3</td><td /><td>188.97 (13.74)</td><td>184.46 (9.36)</td><td>180.08 (16.12)</td></tr><tr><td>4</td><td /><td>200.24 (14.54)</td><td>193.06 (10.77)</td><td>189.66 (15.63)</td></tr><tr><td>5</td><td>MAP</td><td>206.91 (15.65)</td><td>199.86 (11.41)</td><td>198.86 (15.95)</td></tr><tr><td>6</td><td /><td>207.49 (15.27)</td><td>203.39 (8.61)</td><td>202.19 (14.85)</td></tr><tr><td>7</td><td /><td>209.97 (19.03)</td><td>206.23 (9.34)</td><td>205.34 (13.21)</td></tr><tr><td>8</td><td /><td>209.73 (18.47)</td><td>208.35 (10.94)</td><td>208.23 (15.06)</td></tr><tr><td>3</td><td /><td>318.53 (35.7)</td><td>300.31 (29.12)</td><td>307.40 (37.63)</td></tr><tr><td>4</td><td /><td>337.97 (32.20)</td><td>322.50 (29.97)</td><td>325.23 (33.83)</td></tr><tr><td>5</td><td>ACCESS</td><td>366.12 (29.70)</td><td>350.43 (21.96)</td><td>350.43 (30.50)</td></tr><tr><td>6</td><td /><td>361.40 (34.18)</td><td>354.29 (28.06)</td><td>357.01 (34.75)</td></tr><tr><td>7</td><td /><td>351.42 (31.56)</td><td>346.09 (21.23)</td><td>351.49 (30.01)</td></tr><tr><td>8</td><td /><td>349.98 (31.04)</td><td>348.43 (36.40)</td><td>355.71 (30.85)</td></tr></tbody></table> </ephtml> </p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /></colgroup><tbody><tr><td /><td /><td align="center" colspan="10">Performance level proportions</td></tr><tr><td /><td /><td align="center" colspan="5">2018–19</td><td align="center" colspan="5">2021–22</td></tr><tr><td /><td /><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Overall</td><td /><td>.27</td><td>.34</td><td>.21</td><td>.16</td><td>.02</td><td>.36</td><td>.34</td><td>.18</td><td>.11</td><td>.01</td></tr></tbody></table> </ephtml> </p> <hd id="AN0184489646-16">2018–2019 Results</hd> <p>Model comparison results using the ELPD are presented in Supplementary Table S1. The model with the best fit was Model 2b; however, the margin of this difference between the next-best model was not substantial (approximately ≤Δ4; [<reflink idref="bib45" id="ref78">45</reflink>]; [<reflink idref="bib53" id="ref79">53</reflink>]). With a lack of clear evidence for a "best" model in this case, we selected the simpler linear model of 1b because allowing nonlinearity in the MAP × ACCESS interaction was not advantageous. Moreover, Model 1b allows the ACCESS, MAP, and Spanish MAP as well as their interactions to vary across grade levels, which is potentially important given the developmental differences in ELP and math performance across grades.</p> <p>Table 4 presents the results for Model 1b. The log-odds coefficient for MAP (averaged across ACCESS, MAP language, and covariates) was 1.90 (<emph>pSD</emph> = 0.17, 95% CI = [1.59–2.26], <emph>OR</emph> = 6.69). Based on the <emph>pSD</emph>, MAP's coefficient's probability of exceeding zero was essentially 1. ACCESS demonstrated a much weaker association with SSA performance levels (<emph>b </emph>= 0.26, <emph>pSD</emph> = 0.18, 95% CI = [−0.10 to 0.62], <emph>OR</emph> = 1.30), though this is not completely unexpected given how much variation in SSA performance MAP would be expected to explain. The coefficient for the interaction of MAP and ACCESS was small but very likely to exceed zero (<emph>b </emph>= 0.17, <emph>pSD</emph> = 0.11, 95% CI = [−0.04 to 0.37]; 94% chance >0). This interaction coefficient would suggest that the log-odds of MAP for a student scoring +1 <emph>SD</emph> on ACCESS would be 1.90 + 0.17 = 2.07 [<emph>OR</emph> = exp(2.07) = 7.92]. This shows that MAP's prediction strength is greater at higher ELP (averaging over both languages).</p> <p>Table 4. Parameters of Selected Models in 2018–2019 and 2021–2022.</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="left" /><col align="left" /><col align="left" /></colgroup><thead><tr><th align="left" rowspan="3">Parameter (prior 2018-19/prior 2021-22)</th><th align="left" colspan="2">2018-19</th><th align="left" colspan="2">2021-22</th></tr><tr><th align="left" colspan="2">Model 1b</th><th align="left" colspan="2">Model 2c</th></tr><tr><th align="left">Est. (<italic>SD</italic>)</th><th align="left">95% CI</th><th align="left">Est. (SD)</th><th align="left">95% CI</th></tr></thead><tbody><tr><td>Constant effectsThreshold 1 ∼<italic>N</italic>(2,2)/(-2.2, 0.6) Threshold 2 ∼<italic>N</italic>(0.5, 2)/(0.2, 0.6) Threshold 3 ∼<italic>N</italic> (2, 2)/(2.0, 0.6) Threshold 4 ∼<italic>N</italic> (4, 2)/(6.1, 0.7) MAP* ∼<italic>N</italic>(4, 2)/(15.6,3.0) ACCESS* ∼<italic>N</italic>(3,3)/(2.7,4.7) MAP X ACCESS ∼<italic>N</italic>(0, 3) MAP X Spanish MAP ∼<italic>N</italic>(0, 2) ACCESS X Spanish MAP ∼<italic>N</italic>(0, 2) MAP X ACCESS X Spanish MAP ∼<italic>N</italic>(0, 2)</td><td>-2.19 (0.61)0.21 (0.60)2.12 (0.60)6.55 (0.65)1.89 (0.18)0.28 (0.18)0.18 (0.10)-0.71 (0.23)0.40 (0.20)-0.22 (0.32)</td><td>-3.42-1.05-1.01-1.370.91-3.305.29-7.861.59-2.27-0.06-0.66-0.02-0.39-1.17- -0.290.02-0.80-0.85-0.42</td><td>-2.23 (0.30)0.10 (0.30)2.15 (0.30)6.92 (0.41)15.55 (2.39)2.56 (2.94)––––</td><td>-2.84- -1.67-0.50-0.671.54-2.736.13 - 7.7310.65 - 20.13-3.24–8.62––––</td></tr><tr><td>Smooth terms (standard deviations)<italic>f</italic>(MAP) ∼ <italic>half student-t</italic> (3, 0, 1)<italic>f</italic>(ACCESS) ∼ <italic>half student-t</italic> (3, 0, 2.8)</td><td>––</td><td>––</td><td>1.50 (0.68)1.99 (1.30)</td><td>0.52-3.120.13-4.90</td></tr><tr><td>Varying effectsGradeIntercept ∼ <italic>half student-t</italic> (3, 0, 1))/(3, 0, 1.6) MAP∼ <italic>half student-t</italic> (3, 0, 2.5))/(3, 0, 2.5) ACCESS∼ <italic>half student-t</italic> (3, 0, 2.5))/(3, 0, 2.5) MAP X ACCESS∼ <italic>half student-t</italic> (3, 0, 2.5) MAP X Spanish MAP∼ <italic>half student-t</italic> (3, 0, 2.5) ACCESS X Spanish MAP∼ <italic>half student-t</italic> (3, 0, 2.5) MAP X ACCESS X Spanish MAP∼ <italic>half student-t</italic> (3, 0, 2.5) SchoolIntercept ∼ <italic>half student-t</italic> (3, 0, 1))/(3, 0, 1.6) MAP ∼ <italic>half student-t</italic> (3, 0, 2.5))/(3, 0, 2.5) ACCESS ∼ <italic>half student-t</italic> (3, 0, 2.5)/(3, 0, 2.5) MAP X ACCESS ∼ <italic>half student-t</italic> (3, 0, 2.5)</td><td>1.65 (0.61)0.25 (0.21)0.32 (0.20)0.12 (0.13)0.25 (0.24)0.24 (0.22)0.53 (0.43)1.68 (0.30)0.20 (0.10)0.24 (0.11)0.14 (0.10)</td><td>0.76-3.100.02-0.740.07-0.820.00-0.410.01-0.850.01-0.830.02-1.561.21 - 2.360.02-0.410.03-0.480.01-0.37</td><td>1.21 (0.49)0.64 (0.25)0.13 (0.13)––––1.77 (0.32)0.23 (0.23)0.11 (0.08)–</td><td>0.55-2.380.30 - 1.280.00-0.46––––1.27 - 2.510.01-0.560.01-0.29–</td></tr></tbody></table> </ephtml> </p> <p>3 <bold>*</bold>Indicates this parameter in 2021-22 should not be interpreted on its own since it is a smooth function. 2018-19 priors represent assumptions based on prior evidence, including the typical distribution of SSA performance level proportions that account for the higher proportion of ELs performing in lower performance on state assessments. 2018-19 MAP prior based on the normative correlation between MAP and criterion assessments (converted to log-odds). All other priors represent weak or weakly informative assumptions about parameter estimates. Grade and school random slope priors are the brms default and do not incorporate prior information from 2018-19 given the high degree of uncertainty in these parameter estimates within GAMMs. 2021-22 priors based on posterior parameter estimates from the same model estimated in 2018-19. All models converged adequately after 2000 iterations (1000 of which were warm-up) with all parameter <emph>R-hat</emph> values < 1.05. These models estimated with default priors demonstrated similar parameter values and did not alter the conclusions.</p> <p>However, we also found that the MAP × ACCESS interaction was −0.23 smaller on average among those who took Spanish MAP. To better represent this difference, in Table 5, we present MAP log-odds coefficients at ACCESS scores of 0 and 1 for English and Spanish within each grade along with the difference in MAP log-odds between ACCESS scores of 0 and 1. There was only one instance in which the difference in Spanish MAP log-odds across ACCESS is likely >0 (Grade 6). Otherwise, the variation of MAP prediction across ACCESS occurred primarily for English MAP scores, and MAP showed stronger predictive strength among students taking the English version (compared to Spanish).</p> <p>Table 5. MAP × ACCESS × Spanish MAP Interaction Posterior Estimates by Grade (2018–2019 Only).</p> <p>Graph</p> <p> <ephtml> <table><colgroup><col align="left" /><col align="left" /><col align="char" char="(" /><col align="char" char="(" /><col align="char" char="(" /></colgroup><thead><tr><th align="left">Grade</th><th align="left">MAP language</th><th align="left">MAP log-odds <italic>M</italic>(<italic>SD</italic>) |ACCESS = 0</th><th align="left">MAP log-odds <italic>M</italic>(<italic>SD</italic>) |ACCESS = 1</th><th align="left">Difference in MAP log-odds <italic>M</italic>(<italic>SD</italic>)</th></tr></thead><tbody><tr><td rowspan="2">3</td><td>English</td><td>2.29 (0.19)</td><td>2.74 (0.24)</td><td>0.45 (0.14)</td></tr><tr><td>Spanish</td><td>1.66 (0.19)</td><td>1.61 (0.26)</td><td>−0.05 (0.18)</td></tr><tr><td rowspan="2">4</td><td>English</td><td>2.28 (0.18)</td><td>2.52 (0.23)</td><td> 0.25 (0.13)</td></tr><tr><td>Spanish</td><td>1.57 (0.18)</td><td>1.56 (0.24)</td><td>−0.01 (0.16)</td></tr><tr><td rowspan="2">5</td><td>English</td><td>2.10 (0.19)</td><td>2.40 (0.23)</td><td>0.30 (0.12)</td></tr><tr><td>Spanish</td><td>1.38 (0.18)</td><td>1.41 (0.25)</td><td>0.03 (0.15)</td></tr><tr><td rowspan="2">6</td><td>English</td><td>2.12 (0.2)</td><td>2.31 (0.28)</td><td>0.19 (0.19)</td></tr><tr><td>Spanish</td><td>1.46 (0.19)</td><td>1.67 (0.28)</td><td>0.21 (0.19)</td></tr><tr><td rowspan="2">7</td><td>English</td><td>2.23 (0.24)</td><td>2.28 (0.34)</td><td>0.04 (0.26)</td></tr><tr><td>Spanish</td><td>1.44 (0.23)</td><td>1.69 (0.35)</td><td>0.24 (0.25)</td></tr><tr><td rowspan="2">8</td><td>English</td><td>2.46 (0.31)</td><td>2.96 (0.46)</td><td>0.50 (0.33)</td></tr><tr><td>Spanish</td><td>1.71 (0.32)</td><td>1.62 (0.46)</td><td>−0.09 (0.37)</td></tr></tbody></table> </ephtml> </p> <p>4 <emph>Note</emph>. |ACCESS denotes "conditional on" ACCESS = 0 or 1 (i.e., simple slopes).</p> <p>To better represent these estimates in probability terms, Figure 1a displays average probability estimates by MAP, ACCESS, MAP language, and SSA performance category. Log-odds and <emph> OR</emph>s do not on their own pertain to the accuracy of a screener ([<reflink idref="bib37" id="ref80">37</reflink>]), so the display of these probabilities helps portray one aspect of classification accuracy at specific MAP scores. If the category membership depends on an estimated probability of >.5, then the MAP scores in which the credible interval bands exceed.5 are those that are most accurate in classifying individuals into that category. For example, a MAP score of −0.67 (corresponding to the local 25% percentile within each grade among ELs) would be most likely to classify students into either Categories 1 or 2. However, these categories are not easily differentiable at this score (but if Categories 1 and 2 were used to define "at risk" status, this may not be as consequential), so a lower MAP score would be needed to classify students in Category 1 with more certainty. Because lower ELP corresponds to a slightly weaker relation between MAP and SSA levels, a larger change in MAP scores is necessary to better differentiate the most likely categories among students with low ELP. Category classification errors would be most likely to occur in the ranges in which the credible interval bands overlap, which is a substantial portion of Categories 1 and 2 as well as Categories 2 and 3, suggesting that it may be difficult to distinguish a student's predicted performance level in certain ranges of MAP scores, and this would vary across ELP. However, this differential change in accuracy across ELP was not present among Spanish MAP (also shown in Table 5).</p> <p>MAP: Figure 1. Average posterior probability estimates and 95% credible bands by SSA performance levels (1–5), MAP, ACCESS, and MAP language.</p> <hd id="AN0184489646-17">2021–2022 Results</hd> <p>Results from model comparison in 2021–2022 indicate that Model 2c fits best (see Supplementary Table S1). Table 4 presents Model 2c parameter estimates. Model 2c includes nonlinearity for MAP and ACCESS separately but no interaction between them. This model selection indicates that MAP's prediction was constant across ACCESS. Because coefficients for GAMMs are not easily interpretable on their own, Figure 2 presents log-odds estimates across 1 <emph>SD</emph> intervals of MAP and ACCESS scores, demonstrating the variation (or lack thereof) in the size of the coefficients across the range of each variable. Consistent with 2018–2019, ACCESS was not a robust predictor of SSA performance level over and above MAP scores, grade levels, and demographic covariates [small (<.5) and often not substantially >0 log-odds]. MAP demonstrated detectable nonlinearity in predictive strength across the range of the measure.</p> <p>MAP: Figure 2. Distributions of Model 2c estimates in 2021–2022 for MAP and ACCESS.</p> <p>These results suggest that 1 <emph>SD</emph> changes in MAP at higher MAP scores were more predictive of performance levels than those at lower ranges of MAP scores. That is, increases in MAP scores in the higher range differentiated students' performance more than increases in scores in the lower range, holding ELP constant. Figure 1b presents probability estimates by MAP score, ACCESS score, and SSA performance level. The relation of MAP to probabilities was identical across ACCESS (i.e., no interaction).</p> <hd id="AN0184489646-18">EL/Non-EL Comparisons Across Years</hd> <p>As noted in Footnote 2, we also conducted an analysis to examine the degree of differential prediction across binary EL status. (We report results in full in the Supplemental material.) In both 2018–2019 and 2021–2022, we found evidence that the predictive strength of MAP was somewhat smaller among ELs. Regression estimates ranged across grades from 2.2 to 3.2 among ELs compared to 2.8 to 3.8 among non-ELs, with the difference between coefficients ranging from.6 to.9 (all estimates have near 100% chance of exceeding 0). The coefficients for ELs from this model closely aligned with those of the models reported in the main text that include only students with prior ELP scores.</p> <hd id="AN0184489646-19">Discussion</hd> <p>Although academic screening among ELs within MTSS has been a popular topic of study ([<reflink idref="bib4" id="ref81">4</reflink>]; [<reflink idref="bib9" id="ref82">9</reflink>]; [<reflink idref="bib19" id="ref83">19</reflink>]; [<reflink idref="bib24" id="ref84">24</reflink>]; [<reflink idref="bib44" id="ref85">44</reflink>]), little research has examined how math screening prediction may function across a wide range of ELP. The unique confluence of academic, linguistic, and cognitive skills among ELs ([<reflink idref="bib48" id="ref86">48</reflink>]), particularly those with or at risk for learning disabilities ([<reflink idref="bib28" id="ref87">28</reflink>]; [<reflink idref="bib49" id="ref88">49</reflink>]), necessitates more consideration of how universal screening in math operates in combination with ELP. A better understanding of how math screening operates among ELs will help allocate appropriate instructional resources and strategies to culturally and linguistically diverse students ([<reflink idref="bib28" id="ref89">28</reflink>]).</p> <p>The current study addressed this gap in the literature by examining whether and to what degree math screening (using MAP math) was differentially accurate across ELs' ELP scores, language of MAP Math administration (Spanish or English), and pre/post-COVID-19 learning disruptions. Three primary findings emerged.</p> <p>First, the models in 2018–2019 demonstrated a moderate (and detectably >0) degree of differential predictive strength across MAP, but this occurred almost exclusively among students assessed in English. We found little to no variation in Spanish MAP's prediction across ELP. This suggests that English MAP prediction was more sensitive to ELP than Spanish MAP prediction; however, Spanish MAP predictive strength was generally lower overall, regardless of ELP. This is consistent with the relative English language load of English compared to Spanish math assessment. This should be interpreted carefully, however, because the highest ELP among students taking Spanish was substantially lower as a result of the ELP criterion for administering Spanish MAP. These differences could be artifacts of the truncated distribution. Nevertheless, within the range of ELP among Spanish MAP takers, there was essentially no differential predictive strength.</p> <p>Our second primary finding was that MAP was not differentially predictive across ELP in 2021–2022. This is striking given that the priors incorporated into the 2021–2022 models (based on 2018–2019 estimates) encoded the assumption of differential prediction across ELP, which suggests that the patterns in 2021–2022 were strong enough to override more informative priors about the strength of the interaction between MAP and ELP. Several factors may contribute to this difference. The criteria for administering MAP Spanish changed, resulting in mostly only third graders taking Spanish MAP. However, based on the 2018–2019 findings, we would have expected similar evidence for differential prediction in 2021–2022 since this occurred primarily in English MAP in 2018–2019.</p> <p>Another issue relates to changes in base rates of performance levels pre- and post-COVID. As discussed earlier, screening is sensitive to base rates ([<reflink idref="bib12" id="ref90">12</reflink>]). We assumed COVID-19 was a main factor in shifting base rates lower in 2021–2022 in addition to lowering MAP and ACCESS scores beyond what would be expected from 2018 to 2019. The joint changes in ACCESS, MAP, and performance level distributions would collectively alter whether ELP moderates MAP's prediction, over and above issues such as sample idiosyncrasies across years. The nonlinearity we found in MAP's prediction may be indicative of base rate differences as well: Few students reach performance levels 4–5, which may have changed the predictive properties of higher scores on MAP since those typically correspond to higher criterion performance level. Another explanation is that there might be more changes in math development across the school year among students who perform lower on MAP in the fall, which would attenuate MAP's ability to accurately predict these students' performance. These patterns of findings demonstrate both the importance of considering these screening prediction issues across educational contexts and the utility of Bayesian methods in terms of propagating prior assumptions into similar analyses ([<reflink idref="bib17" id="ref91">17</reflink>], [<reflink idref="bib18" id="ref92">18</reflink>]).</p> <p>Our third primary finding was that screening among ELs was somewhat less accurate, but it is unclear what this may indicate about the assessments themselves given the myriad factors confounded with EL status and academic performance on the screener and the outcome. MAP math remained a strong predictor of state test performance across groups, but there may be a slight advantage to predictive strength among non-ELs.</p> <hd id="AN0184489646-20">Implications</hd> <p>Our results provide at best inconsistent evidence for differential predictive strength of MAP across ACCESS scores, with the more recent data from 2021 to 2022 providing minimal support. Based on our findings, therefore schools should carefully consider how math screeners and criterion math outcomes interrelate with ELP before assuming there are (or are not) substantial differences in screener predictive strength (and relatedly, predictive accuracy). A key factor to consider in understanding differential prediction is the inherent difference in base rates between groups (EL/non-EL) or across ELP scores. This may in part explain the differences between these groups in terms of predictive strength because the "signals" (i.e., performance levels) being detected are quite different, which could on its own change alter the predictive strength and accuracy of MAP, all else being equal ([<reflink idref="bib12" id="ref93">12</reflink>]).</p> <p>As [<reflink idref="bib12" id="ref94">12</reflink>] recommended, multiple-gated screening may be one way to avoid inaccurate decisions with low base rates; indeed, research has demonstrated the utility of multiple gating in screening, particularly using posttest probabilities ([<reflink idref="bib27" id="ref95">27</reflink>]). Multiple gating may be useful among ELs to differentiate their relative math strengths and difficulties, especially with more targeted subskill measures such as math calculation and math concepts/applications ([<reflink idref="bib22" id="ref96">22</reflink>]). These more targeted measures may help identify areas of math performance that are more directly linked to ELP (e.g., word problems vs math calculation fluency). For example, gated screening could involve administering a broad measure like MAP and then, based on the student's MAP score (e.g., below the "at risk" criterion), followed by a second stage of more targeted assessment, for example, curriculum-based measures that target specific content areas, like math computation versus word-problem solving (see [<reflink idref="bib27" id="ref97">27</reflink>], for an example of multiple-gated screening in the context of interval likelihood ratios). This type of process could help distinguish the various types of math skills with which students are struggling relative to their ELP, which could more accurately identify those actually at risk for math difficulties and disability ([<reflink idref="bib28" id="ref98">28</reflink>]). However, when implementing a multi-gated screening approach with ELs, it is crucial to carefully evaluate its cost-effectiveness relative to its ability to detect actual math difficulties ([<reflink idref="bib3" id="ref99">3</reflink>]; [<reflink idref="bib36" id="ref100">36</reflink>]).</p> <p>Our findings indicate a clear need for schools to understand how math screening captures relevant math skills (potentially in multiple languages and how these different language skills interact with respect to math; [<reflink idref="bib46" id="ref101">46</reflink>], [<reflink idref="bib47" id="ref102">47</reflink>]) and how criterion indicators are defined for data-based decisions in order to avoid making erroneous decisions about academic risk among culturally and linguistically diverse students. The same screener score may result in different outcome classifications between groups due to the confounding of ELP and academic skills as well as the sensitivity of screeners to different base rates across groups, even within the math domain. However, we also found that math screening prediction was less sensitive to ELP in 2021–2022, although we are unable to determine the underlying cause of this difference in results.</p> <p>Collectively, schools need to understand the processes generating their observed math screening data among ELs, including the interacting factors of social and instructional context, language background and proficiencies, and the qualities of the specific math screeners being implemented ([<reflink idref="bib1" id="ref103">1</reflink>]; [<reflink idref="bib28" id="ref104">28</reflink>]; [<reflink idref="bib51" id="ref105">51</reflink>]). For culturally and linguistically diverse students with and at risk for math learning disabilities, [<reflink idref="bib28" id="ref106">28</reflink>] recommended assessment tools (to identify challenges and progress monitor) that tease apart language and reading from math skills in addition to instructional supports that directly support learning math language as well as leverage students' cultural background knowledge and experiences.</p> <hd id="AN0184489646-21">Limitations and Future Directions</hd> <p>Several limitations in the current study must be considered. First, although computer-adaptive tests are a popular screening method, additional research on the interaction of language and math screening is needed on more specific math measures, including calculation and word-problem solving, to better determine the item conditions under which language proficiency has the most impact on math screening performance. Second, the current data come from a single midwestern school district. Despite the linguistic diversity of the district, the generalizability of the results is limited to districts with similar achievement patterns and demographic characteristics. Third, we do not have data on the specific types of language instruction available to ELs or the specific accommodations available during assessment. Future research should devote specific attention to the interactions between math screening performance, language proficiency, and language instructional programs. Last, the inherent differences in who took Spanish or English MAP present problems for assessing moderation in this context because there were differences in ACCESS, MAP, and SSA scores across groups, which presents additional challenges for determining the sensitivity of MAP screening to ELP and differences in base rates. More methodological research is needed to better understand how to disambiguate these interacting factors when examining the invariance of screening predictions across groups or other moderating variables.</p> <hd id="AN0184489646-22">Conclusions</hd> <p>Math screening research among ELs has gained attention in recent years, but there remains a paucity of research addressing math data-based decisions in screening among this student population. The current findings suggest that math screening in Grades 3–8 is similarly predictive across the range of ELP, even considering the slight variation of prediction across ELP and language of math assessment. Considering the wide range of sociocultural and linguistic assets these students bring to their math learning, schools should carefully analyze the unique strengths and needs of math learning among linguistically diverse students, how different kinds of screeners represent their knowledge, and how those qualities of screeners impact equitable data-based decisions regarding academic risk and intervention provision.</p> <hd id="AN0184489646-23">Supplemental Material</hd> <p>Graph: Supplemental material, sj-docx-1-ldr-10.1177_09388982241309118 for Predictive Strength of Math Screening Across English Language Proficiency Scores in Third to Eighth Grade by Garret J. Hall, PhD, Emma Doyle, MS, and Edgardo Mejias Vazquez, MS in Learning Disabilities Research & Practice</p> <hd id="AN0184489646-24">Acknowledgments</hd> <p>The authors thank the partnering school district for providing the data used in this study.</p> <ref id="AN0184489646-25"> <title> References </title> <blist> <bibl id="bib1" idref="ref2" type="bt">1</bibl> <bibtext> Albers C. A., Martinez R. (2015). Promoting success with English language learners: Best practices for RTI. Guilford Press.</bibtext> </blist> <blist> <bibl id="bib2" idref="ref18" type="bt">2</bibl> <bibtext> American Educational Research Association, American Psychological Association, & National Council on Measurement in Education. (2014). Standards for educational and psychological testing. American Educational Research Association.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref99" type="bt">3</bibl> <bibtext> Barrett C. A., Johnson L. J., Truckenmiller A. J., VanDerHeyden A. M. (2024). 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Psychological Methods, 28(3), 613–630. https://doi.org/10.1037/met0000434</bibtext> </blist> </ref> <ref id="AN0184489646-26"> <title> Footnotes </title> <blist> <bibtext> The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.</bibtext> </blist> <blist> <bibtext> The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by an Early Career Research Award from the Society for the Study of School Psychology to GJH. Opinions expressed herein do not represent the funding organization.</bibtext> </blist> <blist> <bibtext> Garret J. Hall https://orcid.org/0000-0002-8285-3239 Edgardo Mejias Vazquez https://orcid.org/0009-0006-4339-4531</bibtext> </blist> <blist> <bibtext> Supplemental material for this article is available online.</bibtext> </blist> <blist> <bibtext> We use the term "English learner" to reflect the federal criteria for students who (a) speak a language other than English and (b) have not yet reached benchmarks for English language proficiency as defined by their state education agency.</bibtext> </blist> <blist> <bibtext> We also conducted an analysis to examine the differential predictive strength of MAP across EL status. This analysis is discussed further in the Supplemental material.</bibtext> </blist> </ref> <aug> <p>By Garret J. Hall; Emma Doyle and Edgardo Mejias Vazquez</p> <p>Reported by Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib51" firstref="ref3"></nolink> <nolink nlid="nl2" bibid="bib21" firstref="ref4"></nolink> <nolink nlid="nl3" bibid="bib29" firstref="ref5"></nolink> <nolink nlid="nl4" bibid="bib14" firstref="ref6"></nolink> <nolink nlid="nl5" bibid="bib48" firstref="ref7"></nolink> <nolink nlid="nl6" bibid="bib47" firstref="ref8"></nolink> <nolink nlid="nl7" bibid="bib49" firstref="ref9"></nolink> <nolink nlid="nl8" bibid="bib19" firstref="ref12"></nolink> <nolink nlid="nl9" bibid="bib10" firstref="ref14"></nolink> <nolink nlid="nl10" bibid="bib20" firstref="ref15"></nolink> <nolink nlid="nl11" bibid="bib42" firstref="ref20"></nolink> <nolink nlid="nl12" bibid="bib13" firstref="ref21"></nolink> <nolink nlid="nl13" bibid="bib25" firstref="ref24"></nolink> <nolink nlid="nl14" bibid="bib24" firstref="ref26"></nolink> <nolink nlid="nl15" bibid="bib26" firstref="ref29"></nolink> <nolink nlid="nl16" bibid="bib32" firstref="ref30"></nolink> <nolink nlid="nl17" bibid="bib12" firstref="ref32"></nolink> <nolink nlid="nl18" bibid="bib41" firstref="ref35"></nolink> <nolink nlid="nl19" bibid="bib46" firstref="ref36"></nolink> <nolink nlid="nl20" bibid="bib43" firstref="ref41"></nolink> <nolink nlid="nl21" bibid="bib39" firstref="ref45"></nolink> <nolink nlid="nl22" bibid="bib40" firstref="ref46"></nolink> <nolink nlid="nl23" bibid="bib55" firstref="ref49"></nolink> <nolink nlid="nl24" bibid="bib52" firstref="ref50"></nolink> <nolink nlid="nl25" bibid="bib35" firstref="ref52"></nolink> <nolink nlid="nl26" bibid="bib33" firstref="ref53"></nolink> <nolink nlid="nl27" bibid="bib31" firstref="ref54"></nolink> <nolink nlid="nl28" bibid="bib34" firstref="ref55"></nolink> <nolink nlid="nl29" bibid="bib17" firstref="ref57"></nolink> <nolink nlid="nl30" bibid="bib15" firstref="ref58"></nolink> <nolink nlid="nl31" bibid="bib50" firstref="ref60"></nolink> <nolink nlid="nl32" bibid="bib56" firstref="ref61"></nolink> <nolink nlid="nl33" bibid="bib30" firstref="ref62"></nolink> <nolink nlid="nl34" bibid="bib54" firstref="ref67"></nolink> <nolink nlid="nl35" bibid="bib38" firstref="ref68"></nolink> <nolink nlid="nl36" bibid="bib23" firstref="ref71"></nolink> <nolink nlid="nl37" bibid="bib16" firstref="ref74"></nolink> <nolink nlid="nl38" bibid="bib18" firstref="ref76"></nolink> <nolink nlid="nl39" bibid="bib11" firstref="ref77"></nolink> <nolink nlid="nl40" bibid="bib45" firstref="ref78"></nolink> <nolink nlid="nl41" bibid="bib53" firstref="ref79"></nolink> <nolink nlid="nl42" bibid="bib37" firstref="ref80"></nolink> <nolink nlid="nl43" bibid="bib44" firstref="ref85"></nolink> <nolink nlid="nl44" bibid="bib28" firstref="ref87"></nolink> <nolink nlid="nl45" bibid="bib27" firstref="ref95"></nolink> <nolink nlid="nl46" bibid="bib22" firstref="ref96"></nolink> <nolink nlid="nl47" bibid="bib36" firstref="ref100"></nolink>
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  Label: Title
  Group: Ti
  Data: Predictive Strength of Math Screening across English Language Proficiency Scores in Third to Eighth Grade
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Garret+J%2E+Hall%22">Garret J. Hall</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-8285-3239">0000-0002-8285-3239</externalLink>)<br /><searchLink fieldCode="AR" term="%22Emma+Doyle%22">Emma Doyle</searchLink><br /><searchLink fieldCode="AR" term="%22Edgardo+Mejias+Vazquez%22">Edgardo Mejias Vazquez</searchLink> (ORCID <externalLink term="https://orcid.org/0009-0006-4339-4531">0009-0006-4339-4531</externalLink>)
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Learning+Disabilities+Research+%26+Practice%22"><i>Learning Disabilities Research & Practice</i></searchLink>. 2025 40(2):71-85.
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  Label: Availability
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  Data: SAGE Publications. 2455 Teller Road, Thousand Oaks, CA 91320. Tel: 800-818-7243; Tel: 805-499-9774; Fax: 800-583-2665; e-mail: journals@sagepub.com; Web site: https://sagepub.com
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  Label: Peer Reviewed
  Group: SrcInfo
  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 15
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Audience
  Label: Education Level
  Group: Audnce
  Data: <searchLink fieldCode="EL" term="%22Elementary+Education%22">Elementary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Junior+High+Schools%22">Junior High Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Middle+Schools%22">Middle Schools</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Early+Childhood+Education%22">Early Childhood Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+3%22">Grade 3</searchLink><br /><searchLink fieldCode="EL" term="%22Primary+Education%22">Primary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+4%22">Grade 4</searchLink><br /><searchLink fieldCode="EL" term="%22Intermediate+Grades%22">Intermediate Grades</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+5%22">Grade 5</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+7%22">Grade 7</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+8%22">Grade 8</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Screening+Tests%22">Screening Tests</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Tests%22">Mathematics Tests</searchLink><br /><searchLink fieldCode="DE" term="%22Achievement+Tests%22">Achievement Tests</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Achievement%22">Mathematics Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Language+Proficiency%22">Language Proficiency</searchLink><br /><searchLink fieldCode="DE" term="%22Predictive+Validity%22">Predictive Validity</searchLink><br /><searchLink fieldCode="DE" term="%22English+Learners%22">English Learners</searchLink><br /><searchLink fieldCode="DE" term="%22Elementary+School+Students%22">Elementary School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Middle+School+Students%22">Middle School Students</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+3%22">Grade 3</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+4%22">Grade 4</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+5%22">Grade 5</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+6%22">Grade 6</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+7%22">Grade 7</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+8%22">Grade 8</searchLink><br /><searchLink fieldCode="DE" term="%22Spanish%22">Spanish</searchLink><br /><searchLink fieldCode="DE" term="%22Summative+Evaluation%22">Summative Evaluation</searchLink>
– Name: SubjectThesaurus
  Label: Assessment and Survey Identifiers
  Group: Su
  Data: <searchLink fieldCode="SU" term="%22Measures+of+Academic+Progress%22">Measures of Academic Progress</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1177/09388982241309118
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0938-8982<br />1540-5826
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Using data from students in Grades 3-8 in school years 2018-2019 (N = 1,871) and 2021-2022 (N = 1,740), we examined the strength of fall math screening using Measures of Academic Progress (MAP) for predicting end-of-year state math assessment performance levels and whether this prediction varied across English language proficiency (ELP). In addition, we examined whether differential predictive strength varied across screenings using English or Spanish MAP math (2018-2019 only). MAP math was similarly predictive across the ELP continuum (odds ratios between 6 and 15; log-odds between 1.5 and 2.5). However, in 2018-2019, there was more variation in screening prediction across ELP for students assessed using the English MAP math compared to Spanish MAP math. For 2021-2022, there was little evidence of differential screening prediction across ELP. Our findings suggest that MAP math is moderately predictive across ELP and school years. We discuss limitations, future directions, and implications for research and practice.
– Name: AbstractInfo
  Label: Abstractor
  Group: Ab
  Data: As Provided
– Name: DateEntry
  Label: Entry Date
  Group: Date
  Data: 2025
– Name: AN
  Label: Accession Number
  Group: ID
  Data: EJ1469552
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      – Type: doi
        Value: 10.1177/09388982241309118
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 15
        StartPage: 71
    Subjects:
      – SubjectFull: Screening Tests
        Type: general
      – SubjectFull: Mathematics Tests
        Type: general
      – SubjectFull: Achievement Tests
        Type: general
      – SubjectFull: Mathematics Achievement
        Type: general
      – SubjectFull: Language Proficiency
        Type: general
      – SubjectFull: Predictive Validity
        Type: general
      – SubjectFull: English Learners
        Type: general
      – SubjectFull: Elementary School Students
        Type: general
      – SubjectFull: Middle School Students
        Type: general
      – SubjectFull: Grade 3
        Type: general
      – SubjectFull: Grade 4
        Type: general
      – SubjectFull: Grade 5
        Type: general
      – SubjectFull: Grade 6
        Type: general
      – SubjectFull: Grade 7
        Type: general
      – SubjectFull: Grade 8
        Type: general
      – SubjectFull: Spanish
        Type: general
      – SubjectFull: Summative Evaluation
        Type: general
      – SubjectFull: Measures of Academic Progress
        Type: general
    Titles:
      – TitleFull: Predictive Strength of Math Screening across English Language Proficiency Scores in Third to Eighth Grade
        Type: main
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            NameFull: Garret J. Hall
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            NameFull: Emma Doyle
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            NameFull: Edgardo Mejias Vazquez
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          Dates:
            – D: 01
              M: 05
              Type: published
              Y: 2025
          Identifiers:
            – Type: issn-print
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              Value: 1540-5826
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              Value: 40
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            – TitleFull: Learning Disabilities Research & Practice
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