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Demystifying Adequate Growth Percentiles

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Title: Demystifying Adequate Growth Percentiles
Language: English
Authors: Katherine E. Castellano (ORCID 0000-0002-3695-5955), Daniel F. McCaffrey (ORCID 0000-0003-1196-5273), Joseph A. Martineau
Source: Educational Measurement: Issues and Practice. 2025 44(1):31-43.
Availability: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
Peer Reviewed: Y
Page Count: 13
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Descriptors: Student Evaluation, Growth Models, Student Educational Objectives, Educational Indicators, Federal Programs, Academic Achievement, Academic Standards, Progress Monitoring, Achievement Gains, Test Reliability, Predictor Variables, Predictive Validity
DOI: 10.1111/emip.12635
ISSN: 0731-1745
1745-3992
Abstract: Growth-to-standard models evaluate student growth against the growth needed to reach a future standard or target of interest, such as proficiency. A common growth-to-standard model involves comparing the popular Student Growth Percentile (SGP) to Adequate Growth Percentiles (AGPs). AGPs follow from an involved process based on fitting a series of nonlinear quantile regression models to longitudinal student test score data. This paper demystifies AGPs by deriving them in the more familiar linear regression framework. It further shows that unlike SGPs, AGPs and on-track classifications based on AGPs are strongly related to status. Lastly, AGPs are evaluated in terms of their classification accuracy. An empirical study and analytic derivations reveal AGPs can be problematic indicators of students' future performance with previously not proficient students being more likely incorrectly flagged as not on-track and previously proficient students as on track. These classification errors have equity implications at the individual and school levels.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1460444
Database: ERIC
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  Value: <anid>AN0183983941;ems01mar.25;2025Mar26.05:51;v2.2.500</anid> <title id="AN0183983941-1">Demystifying Adequate Growth Percentiles </title> <p>Growth‐to‐standard models evaluate student growth against the growth needed to reach a future standard or target of interest, such as proficiency. A common growth‐to‐standard model involves comparing the popular Student Growth Percentile (SGP) to Adequate Growth Percentiles (AGPs). AGPs follow from an involved process based on fitting a series of nonlinear quantile regression models to longitudinal student test score data. This paper demystifies AGPs by deriving them in the more familiar linear regression framework. It further shows that unlike SGPs, AGPs and on‐track classifications based on AGPs are strongly related to status. Lastly, AGPs are evaluated in terms of their classification accuracy. An empirical study and analytic derivations reveal AGPs can be problematic indicators of students' future performance with previously not proficient students being more likely incorrectly flagged as not on‐track and previously proficient students as on track. These classification errors have equity implications at the individual and school levels.</p> <p>Keywords: adequate growth; classification accuracy; school accountability</p> <p>Measuring student growth for federal accountability has been of interest since the Growth Model Pilot Program (GMPP) was announced in November 2005. This program allowed states to use student growth information as part of federal reporting under the No Child Left Behind (NCLB) Act. With the focus on universal proficiency, these growth models were used to identify students on track toward the proficiency standard. Accordingly, "growth‐to‐standard" models, or models that compare a student's growth to the needed growth to reach a future standard, such as proficiency or college and career readiness, came into the limelight. The simplest such model is the trajectory model, which was adopted by several states in the GMPP (Hoffer et al., [<reflink idref="bib13" id="ref1">13</reflink>]).</p> <p>The trajectory model is an extension of the simple gain score model and provides a means of evaluating the adequacy of a student's gain score, assuming test scores across grade levels are vertically scaled. A student's gain score—current grade level score minus prior grade level score—is compared against the minimum annual needed gain score to be proficient in some timeframe, such as within 3 years or by grade 8, whichever comes first. The trajectory model helps interpret the size of students' growth scores, as, for example, a gain score of 15 points in and of itself may be difficult to interpret. Similarly, a school's average gain score may not imbue much meaning. A negative average gain score or an average gain of essentially zero may signal that the school is not serving its students well but interpreting the magnitude of positive gain scores may not be so straightforward. A school's average gain could be interpreted normatively against other schools' gain scores: my school gained 2 more points than the neighboring school. But such an interpretation would not convey whether the school was providing a valuable amount of learning to its students. We faced a similar conundrum with norm‐referenced test scores, which led to the advent of criterion‐referenced tests with performance levels indicating different degrees of achievement. NCLB's goal of universal proficiency pushed interpretations of "good enough" progress or learning toward the growth needed to reach proficiency. Students on track to proficiency were then counted as if they were proficient in schools' annual progress toward universal proficiency.</p> <p>Although it is not expected or even desired for classifications of students as making sufficient growth toward proficiency to be deterministic (i.e., 100% accurate), large misclassification error rates can problematize the use of on/off‐track indicators in resource allocations for students and the use of the percentage of students making adequate growth toward proficiency in accountability or educator evaluation systems. For instance, it would be undesirable for schools to get credit for students making progress toward proficiency if they rarely actually reached proficiency. Moreover, classification error rates that vary with student group membership could also lead to fairness issues as shown in Lakin and Young ([<reflink idref="bib19" id="ref2">19</reflink>]).</p> <p>The climate in education has changed since the early days of the GMPP. NCLB has been replaced with the more recent reauthorization of the Elementary and Secondary Education Act—the Every Student Succeeds Act (ESSA, [<reflink idref="bib10" id="ref3">10</reflink>])—which no longer requires states to achieve universal proficiency, but it still promotes the inclusion of student growth in school accountability by requiring a measure of student growth or the use of another academic indicator. Nearly all states include growth measures in their accountability systems. One reason for their popularity is that growth measures are often touted as being fairer for school accountability than status scores because they are less strongly correlated with student demographic variables, such as the percentage of socioeconomically disadvantaged students, than average achievement scores or proficiency rates (e.g., Figlio & Loeb, [<reflink idref="bib11" id="ref4">11</reflink>]; Houston, [<reflink idref="bib14" id="ref5">14</reflink>]; The GreatSchools Editorial Team, [<reflink idref="bib12" id="ref6">12</reflink>]). Moreover, most states have moved away from the use of gain scores and the simple trajectory model in part due to concerns of over‐relying on the interval properties of vertically scaled grade‐level assessments and on the often‐found negative correlation between gain scores and prior status. The most popular growth model is the Student Growth Percentile (SGP) model (Betebenner, [<reflink idref="bib2" id="ref7">2</reflink>]) being in use by 23 states as of 2019 (Data Quality Campaign, [<reflink idref="bib9" id="ref8">9</reflink>]).</p> <p>SGPs are a norm‐referenced measure of individual students' growth. They are the (quantile‐regression‐based) percentile ranks of students' current scores given their prior scores. An SGP of 60 can be interpreted as meaning that the student's score (this year) is as high or higher than 60% of students with the same score history (prior scores). For brevity, we refer to students with the same score history as <emph>academic peers</emph>. Unlike gain scores, SGPs do not require a vertical scale and they are uncorrelated with prior status, allowing both high‐ and low‐performing students to demonstrate high growth. However, they only provide a normative interpretation of student growth. For example, for a student with an SGP of 60 grew more than 60% of their academic peers, the question remains whether that growth was adequate or signifies a sufficient amount of learning. Likewise, a school with an average SGP of 55 has students that on average performed better than 55% of their academic peers, but the question remains whether students in that school on average demonstrated adequate growth.</p> <p>To address this limitation, in his seminal paper on SGPs, Betebenner ([<reflink idref="bib2" id="ref9">2</reflink>]) also developed a corresponding criterion‐referenced growth‐to‐standard component—SGP trajectories and Adequate Growth Percentiles (AGPs). Indeed, Betebenner states "because achievement levels are of primary concern, the most natural way to create standards for growth is to quantify what level of growth is necessary to reach prescribed levels of achievement" (p. 47). AGPs are growth targets that reflect the minimum annual SGP needed to achieve a standard of interest, such as proficiency within a given timeframe. Just as with the trajectory model, a student's observed SGP would be compared against their SGP target (or AGP) to classify them as making adequate growth. Despite the move away from universal proficiency goals, several states still use AGPs to help interpret SGPs and aggregates of AGPs in teacher evaluations or school accountability systems (e.g., Michigan Department of Education, [<reflink idref="bib23" id="ref10">23</reflink>]; Nevada Department of Education, [<reflink idref="bib26" id="ref11">26</reflink>]; South Dakota Department of Education, [<reflink idref="bib30" id="ref12">30</reflink>]). States also use AGPs to fulfill the ESSA requirement of measuring whether English learners are making progress toward attaining English language proficiency (e.g., Indiana Department of Education, [[<reflink idref="bib17" id="ref13">17</reflink>]]; Michigan Department of Education, [<reflink idref="bib23" id="ref14">23</reflink>]; Nevada Department of Education, [<reflink idref="bib26" id="ref15">26</reflink>]).</p> <p>Although the SGP methodology is in popular use and there is extensive research on the properties of the SGP scores themselves (Akram et al., [<reflink idref="bib1" id="ref16">1</reflink>]; Castellano & Ho, [<reflink idref="bib5" id="ref17">5</reflink>]; Castellano & McCaffrey, [<reflink idref="bib7" id="ref18">7</reflink>]; Lockwood & Castellano, [<reflink idref="bib20" id="ref19">20</reflink>]; McCaffrey et al., [<reflink idref="bib22" id="ref20">22</reflink>]; Monroe & Cai, [<reflink idref="bib24" id="ref21">24</reflink>]; Shang et al., [<reflink idref="bib29" id="ref22">29</reflink>]), there is limited research on AGPs. For example, there is little, if any, exploration of the relationship between aggregated AGPs and the demographics of schools, even though achieving future standards would appear likely to be related to students' current status, potentially creating a correlation between AGPs and demographics associated with achievement levels. A handful of studies have explored the classification or predictive accuracy of AGPs (e.g., Colorado Department of Education, [<reflink idref="bib8" id="ref23">8</reflink>]; Lakin & Young, [<reflink idref="bib19" id="ref24">19</reflink>]; Murphy & Gaertner, [<reflink idref="bib25" id="ref25">25</reflink>]). Generally, they find that they provide some or minimal improvement over using status in accurately identifying students who will meet proficiency targets.</p> <p>In this paper, we explore the characteristics of AGPs by (<reflink idref="bib1" id="ref26">1</reflink>) demonstrating how they can be derived in the more familiar linear regression framework, (<reflink idref="bib2" id="ref27">2</reflink>) showing they are essentially status measures, and (<reflink idref="bib3" id="ref28">3</reflink>) extending earlier work on classification accuracy to highlight problematic classification errors if AGPs are used in accountability. Moreover, by situating AGPs within the linear regression framework, we can show how under assumptions of multivariate normal test scores and normally distributed residuals, classification accuracy results can be analytically derived using properties of the multivariate normal distribution. This jump away from the intricacies of the quantile‐regression‐based AGPs facilitates investigations into the impact of manipulations of relevant factors, such as the location of cut scores and years to proficiency.</p> <hd id="AN0183983941-2">Growth‐to‐Standard Models</hd> <p>Growth‐to‐Standard models address questions like "What growth must be maintained over <emph>n</emph> years to reach a future achievement level target?". The simplest growth‐to‐standard model is the trajectory model (e.g., Castellano & Ho, [<reflink idref="bib6" id="ref29">6</reflink>]; Hoffer et al., [<reflink idref="bib13" id="ref30">13</reflink>]). The SGP trajectory model follows a similar procedure as the simple trajectory model. To help understand the SGP trajectory model and AGPs, we first describe the trajectory model.</p> <hd id="AN0183983941-3">The Trajectory Model</hd> <p>The trajectory model builds off the gain score model. It involves computing a growth target that equals the minimum needed annual gain score to reach a standard of interest within a timeframe of interest. Take, for example, a current grade 4 student with a 3‐year timeframe to proficiency—proficient by grade 7. For illustrative purposes, we focus on a grade 4 student who scored 500 on the previous year's grade 3 reading assessment and consider a proficiency cut score in grade 7 of 620. This student's annual growth target is (620–500)/4 = 30 points. That is, the student needs to gain at least 30 points each year (for grade 4 to grade 7) to score at or above the proficiency cut score in grade 7 as seen by the thick black arrow in Figure 1. If the student's grade 3 to grade 4 gain score is 30 points or more, they made adequate growth in grade 4. Our example student only scores 515 in grade 4, gaining 15 points from grade 3 to grade 4; consequently, they did not meet their growth target and are not on track to reaching proficiency in the timeframe of interest.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/EMS/01mar25/emip12635-fig-0001.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="emip12635-fig-0001.jpg" title="1 Illustration of the Trajectory Model" /> </p> <p></p> <p>Equivalently, the trajectory model could be formulated as addressing the question: "If a student continues growing at the same (linear) rate, are they on target to reach a future standard?" In this formulation, the trajectory model involves linearly extrapolating the student's observed gain to the target grade of interest. For our example student with an observed gain of 15 score points from grades 3 to 4, their extrapolated or projected score assuming constant growth over each of the 4 years from grade 3 to grade 7 would be 560 (= 500+(15 × 4)), as shown by the dashed arrows in Figure 1. Because this projected grade 7 score of 560 is less than the future standard of interest—the grade 7 proficiency cut score of 620—we would again classify the student as not making adequate growth or being off‐track. Using either procedure (comparing observed growth to needed growth or comparing the projected score based on observed growth to the future standard) will always result in the same classification.[<reflink idref="bib1" id="ref31">1</reflink>]</p> <hd id="AN0183983941-5">The SGP and PRR Trajectory Models</hd> <p>The SGP trajectory model is analogous to the gain‐score‐based trajectory model. Instead of addressing the question "What <emph>absolute</emph> growth must be maintained over <emph>n</emph> years to reach a future achievement level target?", it addresses the question: "What <emph>normative</emph> growth must be maintained over <emph>n</emph> years to reach a future achievement level target?". Similarly, instead of assuming constant linear growth each year, the SGP model assumes constant conditional or norm‐referenced growth each year; the underlying assumption is that students maintain their same SGP or position relative to their academic peers.[<reflink idref="bib2" id="ref32">2</reflink>] Figure 2a illustrates this assumption with an analogous plot as shown in Figure 1. Our example student needs to maintain 58th percentile growth relative to their peers each year to reach the grade 7 proficiency standard, as indicated by the thick black arrow, but their score of 515 in grade 4 only results in 9th percentile growth relative to their academic peers (students with a score of 500 in grade 3). As indicated by the dashed arrows in Figure 2a, if the student continues to make 9th percentile conditional growth each year, they will only score 484 in grade 7—well below the grade 7 proficiency cut of 620 and noticeably below the projected score of 560 from the gain‐score‐based trajectory model (see Figure 1).</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/EMS/01mar25/emip12635-fig-0002.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="emip12635-fig-0002.jpg" title="2 Illustration of SGP‐ and PRR‐Based Trajectories" /> </p> <p></p> <p>Figure 2a shows the minimum needed SGP for our example student to reach proficiency in the terminal grade in the timeframe of interest—grade 7. However, in practice, states typically find the SGP targets for every year from the current year to the terminal year in the timeframe of interest to determine a student's AGP or final SGP target. For our example student, we would need SGP targets for reaching proficiency in grade 4, grade 5, grade 6, and grade 7. These SGP targets are 79, 58, 54, and 58, respectively. For instance, to be at or above the grade 4 proficiency cut score, our example student must make at least 79th percentile growth from grade 3 to grade 4 relative to their academic peers, while the student must make at least 58th percentile growth from grade 3 to grade 4 and grade 4 to grade 5 to reach or exceed the grade 5 proficiency cut score. Similarly, the student must make at least 54th percentile growth each year to meet/exceed the grade 6 proficiency cut score and 58th percentile growth each year for the grade 7 cut score. Typically, for previously not proficient students, these targets decrease for further out grade levels. For this student, there is a large drop from the needed target from grade 4 to later grades 5, 6, and 7, but then the target levels off with the targets for grades 5, 6, and 7 all being highly similar.</p> <p>Figure 2a makes it clear that the underlying foundation of SGP trajectories is similar to that of gain‐based trajectories shown in Figure 1. However, the mechanics of constructing SGP trajectories and deriving AGPs, such as the 58th percentile grade 7 growth target for our example student, are much more involved than the simple linear extrapolations used in the trajectory model. SGPs are estimated by fitting 100 (nonlinear) quantile regressions modeling the 1st to 100th conditional percentiles of current scores given prior scores. To compute an SGP target for a particular grade level, we first need to map out all 100 SGP trajectories for that grade level. Expressing this key step with the quantile regressions used in SGPs is cumbersome given the use of b‐spline parameterizations of the predictors (prior test scores). Castellano and Ho ([<reflink idref="bib5" id="ref33">5</reflink>]) demonstrated that percentile ranks of residuals (PRRs) from linear regression—that models the conditional mean of the current score—are highly comparable to quantile‐regression‐based SGPs. We extend their methodology to obtain PRR targets that we will show are highly comparable to SGP targets. Comparing panel (b) to panel (a) of Figure 2 gives a preview of this comparability. Our example student has a PRR of 8 compared to an SGP of 9 and a grade 7 PRR target of 52 compared to an SGP target of 58. They differ somewhat due to the smaller sample size (<emph>N</emph> ≈ 6,500) of this example dataset, which was chosen so that we could include the code to reproduce these plots (see Supplemental Online Appendix A). We also see that the projected first to third conditional percentiles (bottom‐most grey lines in Figure 2a and b) are less extreme under the linear assumptions of the PRR (Figure 2b) approach than the nonlinear b‐spline SGP approach (Figure 2a), which has them all land on the very low score of 300 for grade 7.</p> <p>The process for estimating SGP or PRR targets for any given grade level follows the same three steps of model, project, and look‐up as shown in Table 1. We walk through these steps using the simpler and more familiar linear regression framework to illustrate how conditional growth trajectories and AGPs are computed. The PRR framework not only allows for demystifying the mechanics of AGPs, but also facilitates empirical or simulated data investigations, such as changing locations of cut scores or the timeframe to meeting the standard, given its ease of implementation.</p> <p>1 Table The Steps to Compute SGP Targets for Current Grade 4 Students</p> <p> <ephtml> <table><thead><tr><th>Target Grade Level</th><th>Step</th><th>SGP Approach</th><th>PRR Approach</th></tr></thead><tbody><tr><td>Grade 4</td><td>Model:Using the grade 4 cohort, model grade 4 scores given prior grade 3 scores</td><td>Fit 100 separate quantile regressions of grade 4 given prior grade 3 scores for each pth percentile (1st to 100th)</td><td>Fit a single linear regression of the conditional mean of grade 4 scores given prior grade 3 scores</td></tr><tr><td>Grade 4</td><td>Project:Estimate the projected pth conditional percentile of current grade 4 scores given prior observed grade 3 scores</td><td>Plug the observed grade 3 score into the 100 separate quantile regressions from the grade 4 Model step to obtain the p = 1 to 100 projected conditional quantile scores</td><td>Plug the observed grade 3 score into the single linear regression from the grade 4 Model step and add the 1st to 100th percentile of (g4|g3) residuals to obtain the p = 1 to 100 projected conditional quantile scores</td></tr><tr><td>Grade 4</td><td align="left">Look up:Look up which pth conditional percentile projected score just exceeds the grade 4 proficiency cut score to determine the minimum needed conditional growth score to reach/exceed proficiency in grade 4.</td></tr><tr><td>Grade 5</td><td>Model:Using the grade 5 cohort, model grade 5 scores given prior grades 3 and 4 scores</td><td>Fit 100 separate quantile regressions of grade 5 scores given prior grades 3 and 4 scores for each pth percentile (1st to 100th)</td><td>Fit a single linear regression of the conditional mean of grade 5 scores given prior grades 3 and 4 scores</td></tr><tr><td>Grade 5</td><td>Project:Estimate the projected pth conditional percentile of current grade 5 scores given prior observed grade 3 scores and pth projected grade 4 scores</td><td>For each of the 100 pth quantile regressions from the grade 5 Model step, plug the observed grade 3 score and corresponding pth projected grade 4 score from the grade 4 Project step to obtain the p = 1 to 100 projected conditional quantile scores</td><td>Plug the observed grade 3 score and the 100 pth projected grade 4 scores (from the grade 4 Project step) into the single linear regression from the grade 5 Model step and add the corresponding 1st to 100th percentile of the (g5|g3,g4) residuals to obtain the p = 1 to 100 projected conditional quantile scores</td></tr><tr><td>Grade 5</td><td align="left">Look up:Look up which pth conditional percentile projected score just exceeds the grade 5 proficiency cut score to determine the minimum needed conditional growth score to reach/exceed proficiency in grade 5.</td></tr></tbody></table> </ephtml> </p> <p>1 <emph>Note</emph>: The cycle continues for each additional grade in the timeframe of interest.</p> <p>We start with grade 4 as the target grade of interest. The top panel of Figure 3 breaks down the process for computing the conditional grade 4 (given grade 3) growth trajectories for the 25th, 50th, and 75th percentiles using linear regression. We selected a few percentile ranks to reduce clutter in the figure, but, in practice, we would need all 100 conditional growth trajectories spanning from the 1st to the 100th percentile. First, for the model step, we model the conditional mean of the grade 4 scores given grade 3 scores (as described in Table 1). That is, we use the grade 4 cohort to regress the grade 4 scores on prior grade 3 scores. In the example, this yields a linear model with an estimated intercept of 266 and an estimated slope of .58, which are given in dashed boxes in the left‐most (blue) portion of the top grade 4 panel of Figure 3. Next, the project step involves two components. First, we use the estimated intercept and slope from the regression equation to compute the conditional mean grade 4 score given grade 3 score. The conditional mean for a student with a prior grade 3 score of 500 is 556 <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0001" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mrow><mo>=</mo><mspace width="0.33em" /><mn>266</mn><mo>+</mo><mo>.</mo><mn>58</mn><mspace width="0.33em" /><mo>×</mo><mspace width="0.33em" /><mn>500</mn></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$({ = \ 266 +.58\ \times \ 500})$</annotation></semantics></math> </ephtml> . If we were just predicting the student's grade 4 score given their grade 3 score, we would stop here. But to formulate the projected conditional <emph>p</emph>th percentile growth trajectories, we then add an additional term—the <emph>p</emph>th percentile rank of observed grade 4 residuals (the middle yellow box in the top panel). For instance, as shown in the right‐most (green) box in the top panel of Figure 3, to obtain the 25th conditional percentile grade 4 score, we would add the 25th percentile of the residuals ( <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0002" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>−</mo><mn>17</mn></mrow><annotation encoding="application/x-tex">$ - 17$</annotation></semantics></math> </ephtml> ), to the predicted score of 556 to obtain a projected score of 539 <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0003" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mrow><mo>=</mo><mspace width="0.33em" /><mn>556</mn><mo>+</mo><mspace width="0.33em" /><mo>(</mo><mrow><mo>−</mo><mn>17</mn></mrow><mo>)</mo></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$({ = \ 556 + \ ({ - 17})})$</annotation></semantics></math> </ephtml> . That is, for our student to make 25th percentile conditional growth from grade 3 to grade 4, she would need to score 539 in grade 4. Similarly, we obtain the projected 50th and 75th percentile scores of 558 and 576 in Figure 3 by adding 2 (50th percentile of residuals) and 20 (75th percentile of residuals), respectively, to the original conditional mean score of 556. To calculate the AGP we would repeat these steps with all 100 of the quantile residuals. In contrast, as described in Table 1, in the quantile‐regression‐based SGP framework, we would instead model 100 conditional quantile regressions and then plug in students' grade 3 scores to obtain 100 <emph>p</emph>th conditional percentile projected scores for each student.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/EMS/01mar25/emip12635-fig-0003.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="emip12635-fig-0003.jpg" title="3 Illustration of the Computation of Conditional Growth Trajectories at Select Percentile Ranks of p = 25, 50, and 75" /> </p> <p></p> <p>For the look‐up step, we identify the grade 4 PRR target (analogous to the quantile‐regression‐based SGP target) or the minimum needed PRR to be proficient in grade 4. We compare a students' 100 projected scores to the grade 4 proficiency cut score and find the smallest one that equals or exceeds the cut score. In our example, the grade 4 cut score is 572. As shown in the right‐most box in the top panel of Figure 3, the target score of 572 is slightly below the 75th percentile projected grade 4 score of 576 for our example student, indicating their PRR target will be slightly less than 75. By referencing all 100 projected conditional percentile scores, we find this student's grade 4 PRR is 72: The projected 71st conditional percentile score is 571.80, while the 72nd is 572.68 so a percentile rank of 72 corresponds to the minimum projected score that is as large or larger than the target score of 572. The student needs to grow as much or more than 72% of her academic peers in grade 4 to reach the grade 4 proficiency cut score. This PRR target of 72 is similar to the student's SGP target of 79.</p> <p>So far, we have illustrated how to obtain the current grade‐level conditional growth target. The process becomes more involved as we branch out to future years as illustrated in the bottom panel of Figure 3 for grade 5. In this case, using the grade 5 cohort, we first model the conditional mean of grade 5 by regressing their grade 5 scores on their previous grades 3 and 4 scores. But for the next step to find the projected scores for the grade 4 cohort, instead of plugging in the student's observed grade 4 score, we must use the full distribution of projected grade 4 scores to follow the model's assumption of continued growth at the same percentile rank. As indicated by the dotted arrow between the top and bottom panels, the projected 1st to 100th conditional percentile grade 4 scores are plugged into the regression equation in place of a single grade 4 score to obtain 100 predicted grade 5 scores (left‐most blue box), but the process does not end there. Just as with grade 4, we then must add in the <emph>p</emph>th percentile residual to obtain the projected <emph>p</emph>th conditional percentile grade 5 score (middle yellow box). For instance, for the student's 25th percentile trajectory in grade 5 conditional on grades 3 and 4 scores, we plug in the student's observed grade 3 score of 500 and projected 25th conditional percentile grade 4 score of 539 into the linear regression equation to obtain 574 <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0004" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mrow><mspace width="0.33em" /><mo>=</mo><mspace width="0.33em" /><mn>127</mn><mo>+</mo><mo>.</mo><mn>28</mn><mo>×</mo><mn>500</mn><mo>+</mo><mo>.</mo><mn>57</mn><mo>×</mo><mn>539</mn></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$({\ = \ 127 +.28 \times 500 +.57 \times 539})$</annotation></semantics></math> </ephtml> . We then add the 25th percentile of the grade 5 cohort's residuals, or –18, to obtain 556 <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0005" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo>(</mo><mrow><mo>=</mo><mspace width="0.33em" /><mn>574</mn><mo>−</mo><mn>18</mn></mrow><mo>)</mo></mrow><annotation encoding="application/x-tex">$({ = \ 574 - 18})$</annotation></semantics></math> </ephtml> , as shown in the right‐most green box.</p> <p>For the final look‐up step to determine the growth target, or the minimum needed growth percentile to be proficient in grade 5, we would then scan through the list of 100 projected scores and look up the one that equals or exceeds the grade 5 proficiency cut score of 588. We see in Figure 3 that the projected 50th percentile grade 5 score (given grades 3 and 4 scores) is 585 so we expect the PRR target to be close to 50 for our example student. From referencing the list of 100 projected scores, this student's PRR target is 53 (corresponding to a projected grade 5 score of just over 588 at 588.06); the student needs to grow at least as much as 53% of her academic peers in both grades 4 and 5 to reach proficiency in grade 5. The student's grade 5 SGP target (found through fitting quantile regressions instead of linear regression) is again similar at 58.</p> <p>As we move up to each subsequent grade level, PRR and SGP trajectories require continual chaining of previously found projected scores. For instance, assuming the use of only two prior scores as in some states (e.g., Michigan Department of Education, [<reflink idref="bib23" id="ref34">23</reflink>]), the grade 6 trajectories would be based on grades 4 and 5 scores—neither of which are treated as observed for current grade 4 students. Thus, the projected <emph>p</emph>th percentile grade 6 scores in the PRR linear regression approach would require plugging in the corresponding projected <emph>p</emph>th percentile scores for grades 4 and 5 into the initial grade 6 equation and then adding in the <emph>p</emph>th percentile of the grade 6 cohort's residuals. Similarly, the quantile‐regression‐based SGP approach would involve plugging the projected <emph>p</emph>th percentile scores for grades 4 and 5 into the corresponding grade 6 <emph>p</emph>th percentile quantile regression equations.</p> <p>Instead of adding residuals to each initial projected score, SGP trajectories involve estimating a different regression equation for each <emph>p</emph>th percentile. However, these two processes result in similar conditional growth percentile targets. The examples in Figures 1, 2, and 4 were found using a small demo dataset available in the "SGPdata" package (Betebenner et al., [<reflink idref="bib4" id="ref35">4</reflink>]) in R (R Core Team, [<reflink idref="bib27" id="ref36">27</reflink>]).[<reflink idref="bib3" id="ref37">3</reflink>] With only about 6,000 students per grade, we expect the PRR and SGP targets to differ more than they typically would for an entire state. We thus assess their comparability with a large statewide, longitudinal dataset of mathematics test scores. The data includes test scores from 2015 to 2019. We treat 2017 as the nominal "current" year, which allows us to use 2 years of prior test scores for each cohort's conditional growth percentile and to evaluate the obtained status 2 years ahead, which will be helpful for the next section. The grade‐level cohorts in 2017 each have approximately 95,000 students. We split the data by the 2017 grade‐level cohorts: grade 4 cohort to grade 7 cohort. For each cohort, we compute the SGP and PRR targets for each grade level in a 3‐year timeframe to proficiency or by grade 8, whichever comes first.</p> <p> <img src="https://imageserver.ebscohost.com/img/embimages/rdk/EMS/01mar25/emip12635-fig-0004.jpg?ephost1=dGJyMNXb4kSepq84yOvqOLCmsE6epq5Srqa4SK6WxWXS" alt="emip12635-fig-0004.jpg" title="4 Analytically Derived Error Rates by Percentile Rank of the Cut Score for Grade 4 Students with a 2‐Year Time Horizon" /> </p> <p></p> <p>Table 2 shows that the SGP and PRR targets for any projected grade level for a given cohort are almost perfectly correlated, differ by about 2.5 percentile points on average, and result in the same adequate growth classification (on‐track or off‐track) for almost all students. The growth targets based on the more familiar linear regression framework are a good approximation of the targets based on the more flexible quantile regression framework.</p> <p>2 Table Comparing SGP (Quantile‐Regression‐Based) and PRR (Linear‐Regression‐Based) Targets</p> <p> <ephtml> <table><thead><tr valign="bottom"><th>Grade‐Level Cohort</th><th>Projected Grade Level</th><th>Spearman Rank‐Order Correlation</th><th>Root Mean Squared Difference (RMSD)</th><th>Percentage of Students with Targets within 2 Percentile Points</th><th>Percentage of Students with Targets within 5 Percentile Points</th><th>Percentage of Students with Same Adequate‐Growth Classification</th></tr></thead><tbody><tr><td align="left">4(N = 100,770)</td><td>4</td><td>1.00</td><td>2.44</td><td>58</td><td>100</td><td>99</td></tr><tr><td>5</td><td>1.00</td><td>2.84</td><td>55</td><td>97</td><td>99</td></tr><tr><td>6</td><td>1.00</td><td>2.51</td><td>68</td><td>100</td><td>99</td></tr><tr><td>7</td><td>1.00</td><td>2.55</td><td>69</td><td>100</td><td>98</td></tr><tr><td>Catch‐up/keep‐up</td><td>1.00</td><td>2.58</td><td>68</td><td>100</td><td>98</td></tr><tr><td align="left">5(N = 95,058)</td><td>5</td><td>.99</td><td>2.86</td><td>74</td><td>89</td><td>99</td></tr><tr><td>6</td><td>1.00</td><td>2.11</td><td>78</td><td>100</td><td>98</td></tr><tr><td>7</td><td>1.00</td><td>2.56</td><td>74</td><td>99</td><td>98</td></tr><tr><td>8</td><td>1.00</td><td>2.74</td><td>44</td><td>100</td><td>97</td></tr><tr><td>Catch‐up/keep‐up</td><td>1.00</td><td>2.69</td><td>46</td><td>100</td><td>97</td></tr><tr><td align="left">6(N = 93,905)</td><td>6</td><td>.99</td><td>2.02</td><td>81</td><td>100</td><td>99</td></tr><tr><td>7</td><td>1.00</td><td>2.15</td><td>76</td><td>100</td><td>99</td></tr><tr><td>8</td><td>1.00</td><td>2.58</td><td>57</td><td>99</td><td>97</td></tr><tr><td>Catch‐up/keep‐up</td><td>1.00</td><td>2.51</td><td>59</td><td>99</td><td>97</td></tr><tr><td align="left">7(N = 96,393)</td><td>7</td><td>.99</td><td>3.26</td><td>64</td><td>87</td><td>99</td></tr><tr><td>8</td><td>1.00</td><td>3.45</td><td>39</td><td>95</td><td>97</td></tr><tr><td>Catch‐up/keep‐up</td><td>1.00</td><td>3.41</td><td>59</td><td>99</td><td>97</td></tr></tbody></table> </ephtml> </p> <p>Rather than fixing the goal as proficiency in the final year in the timeframe of interest, states often use more involved classification rules that differ by a student's previous proficiency level and use information about SGP targets for each year in the timeframe of interest (e.g., Colorado Department of Education, [<reflink idref="bib8" id="ref38">8</reflink>]). For instance, the standard of interest for students who are previously <emph>not</emph> proficient is to "catch up" or be proficient at least once in the timeframe of interest. These students would need to have an SGP as large or larger than the smallest grade‐level SGP target to be on track to reaching proficiency at least once in the timeframe. That is, their AGP is the minimum over all the grade‐level SGP targets in the timeframe. Our example student's SGP targets for grades 4 to 7 are 79, 58, 54, and 58, respectively. Because our example student was not proficient in grade 3 in the prior year, her catch‐up AGP is found as follows: <ephtml> <math display="block" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0006" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>catchupAGP=mingrade4SGPtarget,grade5SGPtarget,grade6SGPtarget,grade7SGPtarget=min{79,58,54,58}=54.<annotation encoding="application/x-tex">$$\begin{equation*} \def\eqcellsep{&}\begin{array}{l} {\mathrm{catch up AGP}}\\ \quad = \min \left\{ {\mathrm{grade\, }}4\,{\mathrm{ SGP\, target}},{\mathrm{ grade\, }}5\,{\mathrm{ SGP\, target}},\right.\\ \left.\qquad\quad {\mathrm{ grade\, }}6\,{\mathrm{ SGP\, target}}, {\mathrm{ grade\, }}7\,{\mathrm{ SGP target}} \right\}\\ \quad = \min\{79, 58, 54, 58\}\\ \quad = 54. \end{array} \end{equation*}$$</annotation></semantics></math> </ephtml></p> <p>To be classified as meeting her growth target and making adequate growth, our example student would need a grade 4 SGP of 54 or higher, but her SGP is only 9 (as shown in Figure 2a). Using PRR targets instead, the catch‐up target would be similar at 48 = min{72, 53, 48, 52}.</p> <p>Students who were previously proficient are held to a different standard; they are expected to "keep up" or maintain proficiency each year. Thus, their current growth (SGP) has to be as large or larger than all of their grade‐level SGP targets (i.e., AGP = maximum of grade‐level SGP targets). Typically, for previously not‐proficient students, the needed target SGPs decrease as the number of years increases, and for previously proficient students, the SGP targets increase, resulting in the AGP often corresponding to the SGP target in the terminal year regardless of prior proficiency status. For our example student, the SGP targets do not follow this monotonically decreasing pattern exactly, but the targets for grades 5, 6, and 7 are all very similar. Some evidence of this pattern is observed in Table 2 with the SGP‐ and PRR‐based catch‐up/keep‐up AGPs having similar comparability statistics to the terminal grade level in each cohort's time horizon (e.g., the RMSD for the grade 4 cohort's catch‐up/keep‐up AGPs at 2.58 is most similar to the terminal grade 7 RMSD of 2.55).</p> <hd id="AN0183983941-9">Growth‐to‐Standard Is a Status Measure</hd> <p>The goal of growth‐to‐standard measures like AGPs is to evaluate students' growth scores. However, by tying the evaluation to a criterion like a proficiency standard, growth‐to‐standard measures bring growth back to status. They reflect the distance that a student needs to travel to reach a standard of interest. Essentially, we are seeing the formula distance equals the product of rate (in this case, SGPs) and time at play. Because students with lower initial scores have further to go to reach a future standard, they have to grow faster (i.e., need a higher SGP), making growth‐to‐standard highly dependent on students' prior scores. In fact, growth‐to‐standard targets themselves are transformations of prior status that result in a perfect (or very strong) negative correlation with prior status. We first demonstrate this result using the simple gain‐score target in the trajectory model assuming our example case of grade 4 students expected to reach proficiency by grade 7. In this case, the correlation between the growth target and prior status is exactly –1: 1 <ephtml> <math display="block" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0007" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cor</mi><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><msub><mi>C</mi><mn>7</mn></msub><mo>−</mo><msub><mi>X</mi><mn>3</mn></msub></mrow><mn>4</mn></mfrac><mo>,</mo><mspace width="0.33em" /><msub><mi>X</mi><mn>3</mn></msub></mrow></mfenced><mo linebreak="badbreak">=</mo><mfrac><mrow><mi>cov</mi><mfenced separators="" open="(" close=")"><mrow><mfrac><mrow><msub><mi>C</mi><mn>7</mn></msub><mo>−</mo><msub><mi>X</mi><mn>3</mn></msub></mrow><mn>4</mn></mfrac><mo>,</mo><mspace width="0.33em" /><msub><mi>X</mi><mn>3</mn></msub></mrow></mfenced></mrow><msqrt><mrow><mfenced separators="" open="(" close=")"><mrow><msubsup><mi>σ</mi><msub><mi>X</mi><mn>3</mn></msub><mn>2</mn></msubsup><mrow><mo>∖</mo><mn>16</mn></mrow></mrow></mfenced><mfenced separators="" open="(" close=")"><msubsup><mi>σ</mi><msub><mi>X</mi><mn>3</mn></msub><mn>2</mn></msubsup></mfenced></mrow></msqrt></mfrac><mo linebreak="goodbreak">=</mo><mfrac><mrow><mo>−</mo><mfrac><msubsup><mi>σ</mi><msub><mi>X</mi><mn>3</mn></msub><mn>2</mn></msubsup><mn>4</mn></mfrac></mrow><mfrac><msubsup><mi>σ</mi><msub><mi>X</mi><mn>3</mn></msub><mn>2</mn></msubsup><mn>4</mn></mfrac></mfrac><mo linebreak="goodbreak">=</mo><mo>−</mo><mn>1</mn><mo>,</mo></mrow><annotation encoding="application/x-tex">$$\begin{equation}{\mathrm{cor}}\left({\frac{{{{C}_7} - {{X}_3}}}{4},\ {{X}_3}} \right) = \frac{{{\mathrm{cov}}\left({\frac{{{{C}_7} - {{X}_3}}}{4},\ {{X}_3}} \right)}}{{\sqrt {\left({\sigma _{{{X}_3}}^2\backslash 16} \right)\left({\sigma _{{{X}_3}}^2} \right)} }} = \frac{{ - \frac{{\sigma _{{{X}_3}}^2}}{4}}}{{\frac{{\sigma _{{{X}_3}}^2}}{4}}} = - 1,\end{equation}$$</annotation></semantics></math> </ephtml> where the grades 3 to 4 gain score target is <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0008" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mfrac><mrow><msub><mi>C</mi><mn>7</mn></msub><mo>−</mo><msub><mi>X</mi><mn>3</mn></msub></mrow><mn>4</mn></mfrac><annotation encoding="application/x-tex">$\frac{{{{C}_7} - {{X}_3}}}{4}$</annotation></semantics></math> </ephtml> , <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0009" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msub><mi>C</mi><mn>7</mn></msub><annotation encoding="application/x-tex">${{C}_7}$</annotation></semantics></math> </ephtml> is a constant representing the grade 7 proficiency cut score, and <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0010" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msubsup><mi>σ</mi><msub><mi>X</mi><mn>3</mn></msub><mn>2</mn></msubsup><annotation encoding="application/x-tex">$\sigma _{{{X}_3}}^2$</annotation></semantics></math> </ephtml> is the variance for grade 3 scores.</p> <p>This result holds for AGPs as well. Another advantage of casting AGPs in the standard regression framework is that if we further assume that the residuals are normally distributed, then we can write the AGP in a closed‐form expression using the standard normal CDF (cumulative distribution function). As we show in the Supplemental Online Appendix B, the grade 7 SGP target for a current grade 4 student can be written as <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0011" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>(</mo><mfrac><mrow><msub><mi>C</mi><mn>7</mn></msub><mo>−</mo><mi>a</mi><msub><mi>X</mi><mn>3</mn></msub></mrow><mi>d</mi></mfrac><mo>)</mo></mrow><annotation encoding="application/x-tex">${{\Phi}}({\frac{{{{C}_7} - a{{X}_3}}}{d}})$</annotation></semantics></math> </ephtml> , where Φ() is the standard normal CDF, and <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0012" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>a</mi><annotation encoding="application/x-tex">$a$</annotation></semantics></math> </ephtml> and <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0013" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi>d</mi><annotation encoding="application/x-tex">$d$</annotation></semantics></math> </ephtml> are constants that depend on the regression coefficients and residual variances, respectively, from the linear regression (PRR) framework of deriving conditional growth targets. Because the standard normal CDF is a monotonic transformation, the Spearman rank‐order correlation between the AGP (grade 7 SGP target) and prior status at the student level is also –1 (while the Pearson product‐moment correlation will be approximately –1): 2 <ephtml> <math display="block" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0014" xmlns="http://www.w3.org/1998/Math/MathML"><semantics>corrankAGP,X3=corrankΦC7−aX3d,X3=corrankC7−aX3d,X3=−1.<annotation encoding="application/x-tex">$$\begin{equation} \def\eqcellsep{&}\begin{array}{rcl} {\mathrm{co}}{{{\mathrm{r}}}_{{\mathrm{rank}}}}\left({AGP,{{X}_3}} \right) &=& {\mathrm{co}}{{{\mathrm{r}}}_{{\mathrm{rank}}}}\left({{{\Phi}}\left({\frac{{{{C}_7} - a{{X}_3}}}{d}} \right),{{X}_3}} \right)\\[3pt] &=& {\mathrm{co}}{{{\mathrm{r}}}_{{\mathrm{rank}}}}\left({\frac{{{{C}_7} - a{{X}_3}}}{d},{{X}_3}} \right) = - 1. \end{array} \end{equation}$$</annotation></semantics></math> </ephtml></p> <p>The operational quantile‐regression‐based AGPs will not equal the normal CDF expression exactly so this result only holds approximately in practice, but the correlations will be very close to –1. In addition, as we move out to growth targets in higher grade that depend on two (or more) prior scores, the correlation between the growth target and each prior score will be a strong negative correlation but not –1. For instance, the correlation between the grade 6 AGPs for the grade 5 cohort and prior grades 4 and 5 scores is –.95 and –.86, respectively. Although at the student level, by definition, the SGP and PRR growth scores will be uncorrelated with prior status, the targets with which they are evaluated against (AGPs) are a transformation of prior status, resulting in strong negative correlations with prior status.</p> <p>We note however that AGPs themselves are not the end value of interest. In evaluating a student's growth, a student's AGP is compared to their SGP to create dichotomous on‐track indicators. These on‐track indicators can then be aggregated to the school level to obtain the percentage of students who are on‐track or made adequate growth. Accordingly, we are primarily interested in the correlations between the school‐level adequate growth and status aggregates. We computed these correlations using our empirical state dataset of mathematics test scores from 2015 to 2017, where each grade‐level cohort in the nominal current year of 2017 has about 95,000 students. We used the operational catch‐up/keep‐up AGP classifications with a 3‐year timeframe or by grade 8 for grades 4, 5, 6, and 7 students in 2017.</p> <p>As shown in Table 3, similar to the student‐level results, the school mean AGP (averaged over all students in the school in grades 4, 5, 6, or 7) is strongly negatively correlated with prior status (%Proficient based on prior student scores). Moreover, the school percentage on‐track variable, which may be used in school accountability indices (e.g., Michigan Department of Education, [<reflink idref="bib23" id="ref39">23</reflink>]; Nevada Department of Education, [<reflink idref="bib26" id="ref40">26</reflink>]; South Dakota Department of Education, [<reflink idref="bib30" id="ref41">30</reflink>]) has stronger positive relationships with prior and current status scores (% Proficient) than the school mean SGP.[<reflink idref="bib4" id="ref42">4</reflink>] Indeed, the percentage on‐track variable is highly correlated (<emph>r</emph> = .92) with the percentage of proficient students in the (nominal) current year compared to the mean SGP only being moderately correlated (<emph>r</emph> = .55). School growth indices using AGPs may be recapturing status rather than providing a distinct index on student growth. Accordingly, they bring back relationships with school composition variables, such as the percentage of socio‐economically disadvantaged students, that growth scores were often adopted to overcome. As expected, school academic status (percent proficient) is strongly negatively correlated with this composition variable with a correlation of –.70. The mean SGPs reduce this correlation to –.25, but the mean AGP and percentage on‐track bring the correlation back to levels similar to that of status with correlations of +.68 and –.60, respectively.</p> <p>3 Table Correlations between School Aggregate Status and Growth Measures Using an Empirical State Dataset</p> <p> <ephtml> <table><thead><tr valign="bottom"><th /><th>% Proficient (Current)</th><th>Mean SGP</th><th>Mean AGP</th><th align="left">% On‐Track (SGP≥AGP)</th><th align="left">% Socioeconomically Disadvantaged Students</th></tr></thead><tbody><tr><td>% Proficient (Prior)</td><td>.91</td><td>.32</td><td>−.95</td><td>.77</td><td>−.67</td></tr><tr><td>% Proficient (Current)</td><td /><td>.55</td><td>−.91</td><td>.92</td><td>−.70</td></tr><tr><td>Mean SGP</td><td /><td /><td>−.31</td><td>.75</td><td>−.25</td></tr><tr><td>Mean AGP</td><td /><td /><td /><td>−.79</td><td>.68</td></tr><tr><td>% On‐Track</td><td /><td /><td /><td /><td>−.60</td></tr></tbody></table> </ephtml> </p> <hd id="AN0183983941-10">Problematic Errors in Using AGPs to Classify Students</hd> <p>By situating AGPs in the standard regression framework, it is clear that the purpose of AGPs is not to address questions typical of a growth prediction model, such as "Where a student will most likely be in X years?". The projected <emph>p</emph>th conditional percentile scores used to determine AGPs involve adding error (residuals) to shift projected scores above (for <emph>p</emph> > 50) or below (for <emph>p</emph> < 50) the expected mean score (Figure 3). Such a practice would rarely (if ever) be involved if the main goal was to compute the most accurate prediction of a student's future score given their observed score history. Indeed, AGPs were not designed to be predictions. Betebenner ([<reflink idref="bib2" id="ref43">2</reflink>]) defined AGPs as shifting the conversation from prediction ("Where will a student be?") to "What does it take to reach a future standard?". The on‐track classification further asks if the student's score appears to have what it will take.</p> <p>While it is true that AGPs are not predictions, classification of a student's SGP as a matter of being on track implies that students with such SGPs should hit the target at a higher rate than students whose SGPs are not classified as on target. These classifications may be used for making decisions about individual students or schools, for example, providing more support or resources for students who are not on track and continuing the level of support and resources for those on track. Accordingly, it is useful if those flagged as making adequate growth and thus flagged as <emph>not</emph> at‐risk of failing to meet the standard of interest do tend to meet the standard. It is similarly valuable that students identified as off‐track would tend not to meet the standard without changes to support, effort, and resources. Otherwise, AGP may lead to ineffective decisions for students and schools. Classification accuracy and error rates can serve as a reasonable evaluation of whether AGPs serve well in their intended role of signaling how challenging it will be for a student to reach the future target and whether observed SGPs reflect being on target to do so.</p> <p>Our empirical dataset has two additional years of data in it (2018 and 2019) from those that we were treating as the prior (2015 and 2016) and current (2017) years, allowing us to have 2‐year time horizons (or by eighth grade, whichever comes first) for determining growth targets. We can then compare the on/off‐track classifications to students' <emph>obtained</emph> statuses from 2017 to 2019.[<reflink idref="bib5" id="ref44">5</reflink>] In line with the definitions of catch‐up/keep‐up AGPs, we considered previously proficient students as obtaining their standard if they were proficient (or above) in every year in the time frame (2017, 2018, and 2019), while we considered previously not proficient students as obtaining their standard if they were proficient (or above) at least once in the time frame (2017, 2018, or 2019).</p> <p>Given classifications are based on prior status, we disaggregate common classification accuracy and error rates by prior status in Table 4 using quantile‐regression‐based AGPs as typically used in practice. See Supplemental Online Appendix C for the corresponding results using linear‐regression‐based AGPs—the results are comparable, differing within 0 to 3 percentage points from those reported in Table 4. The three classification rates of interest are the overall accuracy rate, false‐positive rate (FPR), and false‐negative rate (FNR). The accuracy rate is the percentage of students whose on‐track status aligns with their obtained status (on‐track students who ultimately obtain their standard plus off‐track students who ultimately do not obtain their standard) out of all students. In contrast, the error rates FPR and FNR are conditional probabilities. The FPR is the percentage of students who are classified as on‐track (SGP ≥ AGP) out of only those students who ultimately do not obtain their standard (i.e., never proficient for previously not proficient students or not proficient in at least one of the years for previously proficient students). The FNR is the percentage of students who are classified as off‐track (SGP < AGP) out of only those students who ultimately do obtain their standard.</p> <p>4 Table Classification Accuracy and Error Rates Using Catch‐Up/Keep‐Up AGPs for an Empirical State Dataset</p> <p> <ephtml> <table><thead><tr><th>Cohort</th><th>Previous Performance</th><th><italic>N</italic></th><th>Accuracy Rate (%)</th><th>FPR (%)</th><th>FNR (%)</th><th>Percent Obtained</th></tr></thead><tbody><tr><td>4</td><td>Not proficient</td><td>47,597</td><td>89</td><td>7</td><td>28</td><td>19</td></tr><tr><td>4</td><td>Proficient</td><td>43,753</td><td>78</td><td>28</td><td>17</td><td>56</td></tr><tr><td>5</td><td>Not proficient</td><td>45,449</td><td>84</td><td>14</td><td>33</td><td>15</td></tr><tr><td>5</td><td>Proficient</td><td>41,006</td><td>81</td><td>27</td><td>12</td><td>58</td></tr><tr><td>6</td><td>Not proficient</td><td>56,774</td><td>89</td><td>7</td><td>27</td><td>17</td></tr><tr><td>6</td><td>Proficient</td><td>32,434</td><td>90</td><td>22</td><td>7</td><td>75</td></tr><tr><td>7</td><td>Not proficient</td><td>59,219</td><td>90</td><td>5</td><td>32</td><td>19</td></tr><tr><td>7</td><td>Proficient</td><td>32,552</td><td>82</td><td>33</td><td>12</td><td>74</td></tr></tbody></table> </ephtml> </p> <p>2 <emph>Note</emph>. FPR = false‐positive rate = Pr(on‐track|not obtained); FNR = false‐negative rate = Pr(off‐track|obtained). For previously not proficient students, "obtained" is defined as reaching/exceeding Proficiency at least once in 2017, 2018, and 2019 (aka, catching‐up), whereas for previously proficient students, "obtained" is defined as reaching/exceeding Proficiency every year in 2017, 2018, and 2019 (aka, keeping‐up). The <emph>N</emph> counts reflect the number of students with growth scores in 2017 who also have test scores in 2018 and 2019, which represents 88% to 97% of students with growth scores in 2017.</p> <p>Although the overall accuracy rates are generally high regardless of prior status, Table 4 reveals some interesting patterns in incorrect classifications by prior status with the FNRs and FPRs. These error rates can help shed light on whether decisions about students would be accurately informed by student‐ or aggregate‐level AGPs. A high FPR indicates that students who ultimately do not meet the standard of interest tend to be falsely classified as on‐track to meeting the standard. As shown in Table 4, students who were previously proficient have FPRs ranging from 22% to 33%, which are 2 to 7 times higher than the FPRs for previously not proficient students. That is, if a system allocated resources on the basis of student‐level AGPs, about 1 in 4 previously proficient students who need additional resources to maintain proficiency each year would not receive them. On average these students had SGPs about 20 percentile points higher than their AGPs, sending the (incorrect) message that their current effort and resources are sufficient. Previously not proficient students only had FPRs of 5% to 14%. In addition, these students generally had just barely high enough SGPs to be classified as on track with average SGPs of about 85 to 89 (across the grade levels) that were 3 to 7 percentile points higher on average than their AGPs (78 to 86 on average). Although they met the bar, their high AGPs could still be an indication that they need a lot of support to reach proficiency since they have to continue to make high growth every year in the time window. Thus, if AGPs are used in resource allocation decisions, they may be less likely to be overlooked as the previously proficient students.</p> <p>The relative magnitude of the FNRs by prior proficiency status flips from that for the FPRs. Students who were previously not proficient have FNRs ranging from 27% to 33%, which are 1.6 to 4 times higher than those for previously proficient students. FNRs could be the result of resources previously not proficient students received (over the timeframe of interest) aided them in reaching the standard in the end. However, the unevenness of the errors by prior status is problematic. As prior status is correlated with other student characteristics, similar patterns likely appear across certain student groups. For instance, Lakin and Young ([<reflink idref="bib19" id="ref45">19</reflink>]) found such differential error rates by English language status. We also note that these differential error rate patterns endure if we hold all students (regardless of prior status) to the same standard of being proficient in the terminal grade in the time window (see Supplemental Online Appendix C). Thus, the differential error rates are not an artifact of the different standards by prior status of being proficient at least once versus maintaining proficiency.</p> <p>Moreover, if not being identified as being in need of greater resources is a positive outcome (as viewed from the perspective of stigma attached to such a designation, typically an adult‐centered perspective), schools with more previously proficient students may benefit from these false positives with an inflated percentage of students on track, while those with more previously not proficient students may be disadvantaged by the false negatives with deflated percentages on track. The reverse is true if being identified as being in need of greater resources is instead the positive outcome (as viewed from the perspective of greater resources offering greater opportunity, typically a student‐centered perspective).</p> <p>However, in our empirical statewide dataset, despite the observed differential error rates at the student level, we did not find that schools' percentage of on‐track students were systematically inflated or deflated (relative to the school percentage of students who ultimately obtained their standard) by their percentage of previously proficient students. At the school level, the false‐positive and false‐negative rates are weighted by the rates of proficient and not proficient students meeting or not meeting the targets (see the last column of Table 4), and this limits their impact, among our sampled schools.[<reflink idref="bib6" id="ref46">6</reflink>] This would need checking for other states or contexts.</p> <p>We were also able to recover the same patterns in the student‐level error rates using analytically derived results under the assumptions of multivariate normal (MVN) test scores and normally distributed residuals. With these parametric assumptions, the error rates can be expressed as MVN probabilities (see Supplemental Online Appendix D). Figure 4 provides such analytically derived FNRs and FPRs using the empirical variance‐covariance matrix for our empirical grade 4 cohort across a range of cut score positions, assuming the relative position of the cut score is constant for each grade and a 2‐year time frame to match that used in Table 4.</p> <p>Figure 4 clearly shows that students who were previously proficient are more likely to be falsely classified as on‐track (false positive) and those previously not proficient are more likely to be falsely classified as off‐track (false negative). Furthermore, the result is relatively insensitive to the cut score for the target proficiency in that the distance between the curves is fairly constant, so that the FPR is roughly 2.7 times larger for previously proficient versus not proficient and vice versa for the FNR across all cut score positions. The analytic derivations suggest our empirical results are likely to replicate in other states. They also allow researchers and practitioners to perform simple experiments to determine the extent shifting a factor such as the position of the cut score or number of years in the time frame for proficiency would impact the error rates.</p> <p>The differential error rates by prior status could have consequences for the appropriateness of resource allocation to individual students and/or schools (from the typically student‐centered perspective) and the fairness of school accountability indices (from the typically adult‐centered perspective). Determining the extent these issues occur in a state program in practice is straightforward once AGPs have been used for a few years and thus on‐track status at a given time point can be compared to actual obtained status over the intended time frame of interest. However, documentation of such checks is rare. The Colorado Department of Education (CDE, [<reflink idref="bib8" id="ref47">8</reflink>]) compared adequate growth status to obtained status by prior status across three subjects but focused on the overall accuracy rates. We were able to use their reported numbers to derive the false‐positive and false‐negative error rates (see Supplemental Online Appendix D) where we found similar patterns to those observed with our data.</p> <hd id="AN0183983941-11">Discussion</hd> <p>Calibrating student growth is a difficult task. The general interest in students reaching a particular standard—popularized by NCLB, strengthened by Race to the Top, and dialed back somewhat by ESSA—makes growth‐to‐standard a seemingly obvious choice to calibrate what constitutes "good enough" or "adequate" growth. However, as we demonstrated, tying a student growth measure to a future status score essentially results in another way of holding schools accountable for students' status with students starting farther from the standard having more ground to make up (higher growth target) and vice versa for students starting closer to the standard (lower growth target). Defining different overall standards by prior performance level with the use of "catch‐up" (at least once proficient) and "keep‐up" (maintaining proficiency) AGPs may appear as if it is equalizing the difficulty of meeting the standard.[<reflink idref="bib7" id="ref48">7</reflink>] However, we demonstrated catch‐up/keep‐up AGPs are highly related to status and are problematic indicators of students' future performance. Although AGPs are not intended to be predictions, they are often interpreted as such. Moreover, they were intended to convey the degree of effort needed to reach a future standard of interest (Betebenner, [<reflink idref="bib2" id="ref49">2</reflink>]). However, the high false‐positive and false‐negative rates indicate they often <emph>under</emph>state the needed effort for previously proficient students and <emph>over</emph>state it for previously not proficient students.</p> <p>Some states that use SGPs may only report AGPs as part of student/teacher/district score reports, but even if no explicit stakes are attached to their use, they should be understood and possible misinterpretations and misuses should be cautioned against. In other cases, states have developed alternatives or extensions to catch‐up/keep‐up AGPs to calibrate growth, which may dilute some of the issues we have identified but do not necessarily resolve them. These include, for instance, specifying cut scores to distinguish among levels of low/moderate/high growth on the individual SGP scale (e.g., Rhode Island Department of Education, [<reflink idref="bib28" id="ref50">28</reflink>]; Utah State Board of Education, [<reflink idref="bib32" id="ref51">32</reflink>]) or on the school mean SGP scale (e.g., Massachusetts Department of Elementary & Secondary Education, [<reflink idref="bib21" id="ref52">21</reflink>]; Wyoming Department of Education, [<reflink idref="bib33" id="ref53">33</reflink>]), taking into account both the magnitude of the student or school's SGP and its size relative to the student or school's AGP in assigning points (to students or schools) for the growth component of a state's accountability index (e.g., Idaho State Department of Education, [<reflink idref="bib15" id="ref54">15</reflink>]; Utah State Board of Education, [<reflink idref="bib31" id="ref55">31</reflink>]), or extending the catch‐up/keep‐up AGP classifications with a "very high growth" category to acknowledge previously not proficient students who make very high growth but not high enough to exceed their catch‐up AGP (e.g., South Dakota Department of Education, [<reflink idref="bib30" id="ref56">30</reflink>]).[<reflink idref="bib8" id="ref57">8</reflink>] How the choice of cut scores or point values for various growth scores connect to the program's theoretical conceptions of adequate growth are typically not well articulated. Similarly, student growth results and their impact in school accountability do not receive the technical or psychometric treatment in establishing validity evidence that the student test scores underpinning them do.</p> <p>The explicit articulation of the intended properties and impact of using growth in an accountability system can aid in specifying and selecting the best growth index for a state. For instance, Indiana articulated a key criterion for the use of student growth in its accountability system as "Growth should be a metric relatively independent of school performance status. The metric should have low correlation to performance" ([<reflink idref="bib16" id="ref58">16</reflink>].). This state assigned points for different magnitudes of SGPs that reflect low, standard, or high movement for students by each of eight prior performance levels (which are subdivisions of the state's original three performance levels). Their approach resulted in both different cut scores on the SGP scale and different point values for each level of growth by prior status. Driven by their stated criterion, their final selected metric resulted in correlations in the .3 to .4 range between school averages of student growth points and prior achievement for both mathematics and reading (Betebenner, [<reflink idref="bib3" id="ref59">3</reflink>]).</p> <p>Our evaluation of the SGP growth‐to‐standard approach focused on its relationship with status and classification accuracy. A further step in evaluating this approach (or any growth score) is whether its use results in the desired changes in behaviors or student outcomes. For instance, the Indiana Department of Education ([<reflink idref="bib16" id="ref60">16</reflink>].) also includes criteria about the intended impact of the inclusion of growth in their accountability system: "Growth should incentivize progress toward proficiency in non‐proficient students and continued growth in proficient students" and "Growth should deter a decline in individual student performance levels." These criteria drove their selection of point values for different levels of growth by prior performance levels. However, whether the state's approach for using growth in its accountability system achieved these goals is not documented. As states continue to update and reconsider their accountability systems and indices, particularly in light of ensuring they do not result in unintended biases against schools by their student composition, careful impact evaluations could be a key ingredient in their success.</p> <p>With regard to the use of AGPs, key accountability choices and impact evaluations can be aided by using the simplified standard regression analogs we presented. The standard regression framework benefits from being more familiar and allows for faster computations, which facilitates testing out different scenarios with empirical or simulated data, such as changing locations of cut scores or the timeframe to meeting the standard. Simulations can be even more efficiently conducted by using the analytic results that follow from assuming normally distributed residuals. These alternatives to the operationally used quantile‐regression‐based AGPs both demystify the mechanics of AGPs and provide practitioners and researchers powerful tools to further assess their utility.</p> <p>GRAPH: Supplemental Online Appendices</p> <ref id="AN0183983941-12"> <title> Footnotes </title> <blist> <bibl id="bib1" idref="ref16" type="bt">1</bibl> <bibtext> That is, <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0015" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>X</mi><mn>3</mn></msub><mo>+</mo><mn>4</mn><mrow><mo stretchy="false">(</mo><mrow><msub><mi>X</mi><mn>4</mn></msub><mo>−</mo><msub><mi>X</mi><mn>3</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>≥</mo><msub><mi>C</mi><mn>7</mn></msub></mrow><annotation encoding="application/x-tex">$({{{X}_3} + 4({{{X}_4} - {{X}_3}})}) \ge {{C}_7}$</annotation></semantics></math> </ephtml> if and only if <ephtml> <math display="inline" altimg="urn:x-wiley:07311745:media:emip12635:emip12635-math-0016" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mrow><mo stretchy="false">(</mo><mrow><msub><mi>X</mi><mn>4</mn></msub><mo>−</mo><msub><mi>X</mi><mn>3</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo>≥</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>C</mi><mn>7</mn></msub><mo>−</mo><msub><mi>X</mi><mn>3</mn></msub></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">$({{{X}_4} - {{X}_3}}) \ge ({{{C}_7} - {{X}_3}})/4$</annotation></semantics></math> </ephtml> .</bibtext> </blist> <blist> <bibl id="bib2" idref="ref7" type="bt">2</bibl> <bibtext> Note how this assumption contrasts with the SGP property of zero correlation with prior score history and near‐zero correlation of SGPs over time.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref28" type="bt">3</bibl> <bibtext> See Supplemental Online Appendix A for the R code to reproduce these figures.</bibtext> </blist> <blist> <bibl id="bib4" idref="ref35" type="bt">4</bibl> <bibtext> Because on‐track indicators involve comparing SGPs to AGPs, and SGPs are a reflection of the current year's score (conditional on prior year scores), we observe a stronger relationship between the growth‐related measures and current status than prior status.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref17" type="bt">5</bibl> <bibtext> For grade 6, we were only able to have a 1‐year time horizon because the grade 8 test in 2019 was different than the grade 8 test in 2017, but it is the regression models estimated from the 2017 grade 8 cohort that are used to determine grade 8 SGP targets for the students who were in grade 6 in 2017. That is, the students' actual scores in grade 8 in 2019 are for a different test than on which their grade 8 SGP targets were based. Their grade 7 SGP targets (given in 2017) are for the same grade 7 test, which they then took in 2018, allowing for comparing on‐track status based on a 1‐year future time horizon to their obtained status in 2018 in grade 7.</bibtext> </blist> <blist> <bibl id="bib6" idref="ref29" type="bt">6</bibl> <bibtext> See the Supplemental Online Appendix D for additional details on the relationship between FPR and FNR and the overall school error rates.</bibtext> </blist> <blist> <bibl id="bib7" idref="ref18" type="bt">7</bibl> <bibtext> Despite these different definitions of AGPs by prior status, they often result in the AGP equaling the terminal grade SGP target, resulting in the same target regardless of prior status.</bibtext> </blist> <blist> <bibl id="bib8" idref="ref23" type="bt">8</bibl> <bibtext> Although it may not be intention that "high growth" previously not proficient students are on‐track‐to‐proficiency, we evaluated the enhanced catch‐up/keep‐up classifications + high growth (for previously not proficient students) against meeting future proficiency targets. The main impact on the classification rates in Table 4 would be substantially higher false positive rates (2–5 times larger) for previously not proficient students.</bibtext> </blist> </ref> <ref id="AN0183983941-13"> <title> References </title> <blist> <bibtext> Akram, K., Erickson, F., & Meyer, R. (2013, March). Issues in the estimation of student growth percentiles. Paper presented at the annual meeting of The Association for Education Finance and Policy, New Orleans, LA. <ulink href="http://www.aefpweb.org/">http://www.aefpweb.org/</ulink></bibtext> </blist> <blist> <bibtext> Betebenner, D. (2009). Norm‐ and criterion‐referenced student growth. Educational Measurement: Issues and Practice, 28 (4), pp. 42 – 51. https://doi.org/10.1111/j.1745‐3992.2009.00161.x</bibtext> </blist> <blist> <bibtext> Betebenner, D. W. (2019, September 23). Indiana targe range discovery for 2019. In Growth to Proficiency table target ranges [memorandum from the Indiana Department of Education to the Indiana State Board of Education]. https://<ulink href="http://www.in.gov/doe/files/sboe‐growth‐table10219.pdf">www.in.gov/doe/files/sboe‐growth‐table10219.pdf</ulink></bibtext> </blist> <blist> <bibtext> Betebenner, D. W., Van Iwaarden, A. R., Domingue, B., & Shang, Y. (2022). SGP: Student growth percentiles & percentile growth trajectories. (R package 2.1‐0.0). https://sgp.io</bibtext> </blist> <blist> <bibtext> Castellano, K. E., & Ho, A. D. (2013a). Contrasting OLS and quantile regression approaches to student "growth" percentiles. Journal of Educational and Behavioral Statistics, 38 (2), pp. 190 – 215. https://doi.org/10.3102/1076998611435413</bibtext> </blist> <blist> <bibtext> Castellano, K. E., & Ho, A. D. (2013b). A Practitioner's guide to growth models. Council of Chief State School Officers. https://scholar.harvard.edu/files/andrewho/files/a_pracitioners_guide_to_growth_models.pdf</bibtext> </blist> <blist> <bibtext> Castellano, K. E., & McCaffrey, D. F. (2017). The accuracy of aggregate student growth percentiles as indicators of educator performance. Educational Measurement: Issues and Practice, 36 (1), pp. 14 – 27. https://doi.org/10.1111/emip.12144</bibtext> </blist> <blist> <bibtext> Colorado Department of Education (CDE). (2014, May 13). The relationship between obtained adequate growth percentiles and obtained proficiency levels. <ulink href="http://www.cde.state.co.us/sites/default/files/Adequate%5fGrowth%5fPercentiles%5f2014.pdf">http://www.cde.state.co.us/sites/default/files/Adequate%5fGrowth%5fPercentiles%5f2014.pdf</ulink></bibtext> </blist> <blist> <bibl id="bib9" idref="ref8" type="bt">9</bibl> <bibtext> Data Quality Campaign. (2019). Growth data: It matters, and it's complicated [Research brief]. https://dataqualitycampaign.org/wp‐content/uploads/2019/04/DQC‐Growth‐Data‐Resources.pdf</bibtext> </blist> <blist> <bibtext> Every Student Succeeds Act, 20 U.S.C. § 6301 (2015). https://<ulink href="http://www.congress.gov/114/plaws/publ95/PLAW‐114publ95.pdf">www.congress.gov/114/plaws/publ95/PLAW‐114publ95.pdf</ulink></bibtext> </blist> <blist> <bibtext> Figlio, D., & Loeb, S. (2011). School Accountability. In E. A. Hanushek, S. Machin, & L. Woessmann (Eds.), Handbook in economics (Vol., 3, pp. 383 – 421). The Netherlands : North‐Holla nd.</bibtext> </blist> <blist> <bibtext> The GreatSchools Editorial Team. (2024, May 4). Why growth matters. GreatSchools.org blog post. https://<ulink href="http://www.greatschools.org/gk/articles/why‐student‐growth‐matters/">www.greatschools.org/gk/articles/why‐student‐growth‐matters/</ulink></bibtext> </blist> <blist> <bibtext> Hoffer, T. B., Hedberg, E. C., Brown, K. L., Halverson, M. L., Reid‐Brossard, P., Ho, A. D., & Furgol, K. E. (2011). Final report on the evaluation of the growth model pilot project. Washington, DC : U.S. Department of Education. https://www2.ed.gov/rschstat/eval/disadv/growth‐model‐pilot/gmpp‐final.pdf.</bibtext> </blist> <blist> <bibtext> Houston, D. M. (2019, December 3). Information about student growth, not just average test scores, can sway school and housing choices, an experiment shows. Education Next blog post. https://<ulink href="http://www.educationnext.org/information‐student‐growth‐not‐just‐average‐test‐scores‐can‐sway‐school‐housing‐choices‐experiment‐shows/">www.educationnext.org/information‐student‐growth‐not‐just‐average‐test‐scores‐can‐sway‐school‐housing‐choices‐experiment‐shows/</ulink></bibtext> </blist> <blist> <bibtext> Idaho State Department of Education. (2012, February 6). Superintendent of public instruction update to the state board of education. https://boardofed.idaho.gov/meetings/board/archive/2012/02_15‐16_12/sde.pdf</bibtext> </blist> <blist> <bibtext> Indiana Department of Education. (n.d.). Indiana accountability A‐F review: Growth metric Department of Education data analysis. https://<ulink href="http://www.in.gov/doe/files/summary‐growth‐component.pdf">www.in.gov/doe/files/summary‐growth‐component.pdf</ulink></bibtext> </blist> <blist> <bibtext> Indiana Department of Education. (2018). English Language Progress Indicator and WIDA ACCESS Growth. Memorandum to Indiana School Superintendents and Principals. https://<ulink href="http://www.in.gov/doe/files/English‐Language‐Progress‐Indicator‐and‐WIDA‐ACCESS‐Growth.pdf">www.in.gov/doe/files/English‐Language‐Progress‐Indicator‐and‐WIDA‐ACCESS‐Growth.pdf</ulink></bibtext> </blist> <blist> <bibtext> Indiana Department of Education. (2021). Understanding WIDA‐ACCESS English language proficiency growth. https://<ulink href="http://www.in.gov/doe/files/Understanding‐WIDA‐ACCESS‐Growth.pdf">www.in.gov/doe/files/Understanding‐WIDA‐ACCESS‐Growth.pdf</ulink></bibtext> </blist> <blist> <bibtext> Lakin, J. M., & Young, J. W. (2013). Evaluating growth for ELL students: Implications for accountability policies. Educational Measurement: Issues and Practice, 32 (3), pp. 11 – 26. https://doi.org/10.1111/emip.12012</bibtext> </blist> <blist> <bibtext> Lockwood, J. R., & Castellano, K. E. (2017). Estimating true student growth percentile distributions using multidimensional item response models and latent regression. Educational and Psychological Measurement, 77 (6), pp. 917 – 944. https://doi.org/10.1177/0013164416659686</bibtext> </blist> <blist> <bibtext> Massachusetts Department of Elementary and Secondary Education. (2023, September 14). School leader's guide to the 2023 accountability determinations. https://<ulink href="http://www.doe.mass.edu/accountability/lists‐tools/default.html">www.doe.mass.edu/accountability/lists‐tools/default.html</ulink></bibtext> </blist> <blist> <bibtext> McCaffrey, D. F., Castellano, K. E., & Lockwood, J. R. (2015). The impact of measurement error on the accuracy of individual and aggregate SGP. Educational Measurement: Issues and Practice, 34 (1), pp. 15 – 21. https://doi.org/10.1111/emip.12062</bibtext> </blist> <blist> <bibtext> Michigan Department of Education. (2022). Michigan school index system business rules: 2022. https://<ulink href="http://www.michigan.gov/mde/‐/media/Project/Websites/mde/OEAA/Accountability/Index/School‐Index‐Technical‐Business‐Rules.pdf">www.michigan.gov/mde/‐/media/Project/Websites/mde/OEAA/Accountability/Index/School‐Index‐Technical‐Business‐Rules.pdf</ulink></bibtext> </blist> <blist> <bibtext> Monroe, S., & Cai, L. (2015). Examining the reliability of student growth percentiles using multidimensional IRT. Educational Measurement: Issues and Practice, 34 (4), pp. 21 – 30. https://doi.org/10.1111/emip.12092</bibtext> </blist> <blist> <bibtext> Murphy, D. L., & Gaertner, M. N. (2014). Evaluating the predictive value of growth prediction models. Educational Measurement: Issues and Practice, 33 (2), pp. 5 – 13. https://doi.org/10.1111/emip.12031</bibtext> </blist> <blist> <bibtext> Nevada Department of Education. (2023). Nevada School Performance Framework Manual: 2022–23 school year. <ulink href="http://nevadareportcard.nv.gov/DI/MoreDownload?filename=Nevada%20School%20Performance%20Framework%20Manual%202022‐23%20School%20Year.pdf">http://nevadareportcard.nv.gov/DI/MoreDownload?filename=Nevada%20School%20Performance%20Framework%20Manual%202022‐23%20School%20Year.pdf</ulink></bibtext> </blist> <blist> <bibtext> R Core Team (2022). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://<ulink href="http://www.R-project.org/">www.R-project.org/</ulink>.</bibtext> </blist> <blist> <bibtext> Rhode Island Department of Education. (2019, November 8). 2019 RI School Accountability Technical Report. https://reportcard.ride.ri.gov/2019AccountabilityTechnicalManual.pdf</bibtext> </blist> <blist> <bibtext> Shang, Y., Van Iwaarden, A., & Betebenner, D. (2015). Covariate measurement error correction for student growth percentiles using the SIMEX method. Educational Measurement: Issues and Practice, 34 (1), pp. 4 – 14. https://doi.org/10.1111/emip.12058</bibtext> </blist> <blist> <bibtext> South Dakota Department of Education. (2023, August 22). Accountability and School Performance Index Technical Manual: 2022–23 School Year. https://doe.sd.gov/reportcard/documents/0823‐Manual.pdf</bibtext> </blist> <blist> <bibtext> Utah State Board of Education. (2020, May 6). Utah accountability technical manual: 2019. https://assessments.jordandistrict.org/wp‐content/uploads/sites/7/AccountabilityTechnicalManual‐5‐6‐2020.pdf</bibtext> </blist> <blist> <bibtext> Utah State Board of Education. (2023, September). Utah accountability technical manual: 2023–2024. https://schools.utah.gov/file/ba4f83a5‐0537‐4f0c‐b0c4‐4a34c8e7c9aa</bibtext> </blist> <blist> <bibtext> Wyoming Department of Education. (2023, July 17). 2022–23 Wyoming school accountability performance rating models implementation handbook. https://edu.wyoming.gov/wp‐content/uploads/2023/01/2023‐Performance‐Rating‐Model‐Implementation‐Handbook‐1.pdf</bibtext> </blist> </ref> <aug> <p>By Katherine E. Castellano; Daniel F. McCaffrey and Joseph A. Martineau</p> <p>Reported by Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib13" firstref="ref1"></nolink> <nolink nlid="nl2" bibid="bib19" firstref="ref2"></nolink> <nolink nlid="nl3" bibid="bib10" firstref="ref3"></nolink> <nolink nlid="nl4" bibid="bib11" firstref="ref4"></nolink> <nolink nlid="nl5" bibid="bib14" firstref="ref5"></nolink> <nolink nlid="nl6" bibid="bib12" firstref="ref6"></nolink> <nolink nlid="nl7" bibid="bib23" firstref="ref10"></nolink> <nolink nlid="nl8" bibid="bib26" firstref="ref11"></nolink> <nolink nlid="nl9" bibid="bib30" firstref="ref12"></nolink> <nolink nlid="nl10" bibid="bib17" firstref="ref13"></nolink> <nolink nlid="nl11" bibid="bib20" firstref="ref19"></nolink> <nolink nlid="nl12" bibid="bib22" firstref="ref20"></nolink> <nolink nlid="nl13" bibid="bib24" firstref="ref21"></nolink> <nolink nlid="nl14" bibid="bib29" firstref="ref22"></nolink> <nolink nlid="nl15" bibid="bib25" firstref="ref25"></nolink> <nolink nlid="nl16" bibid="bib27" firstref="ref36"></nolink> <nolink nlid="nl17" bibid="bib28" firstref="ref50"></nolink> <nolink nlid="nl18" bibid="bib32" firstref="ref51"></nolink> <nolink nlid="nl19" bibid="bib21" firstref="ref52"></nolink> <nolink nlid="nl20" bibid="bib33" firstref="ref53"></nolink> <nolink nlid="nl21" bibid="bib15" firstref="ref54"></nolink> <nolink nlid="nl22" bibid="bib31" firstref="ref55"></nolink> <nolink nlid="nl23" bibid="bib16" firstref="ref58"></nolink>
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  Group: Ti
  Data: Demystifying Adequate Growth Percentiles
– Name: Language
  Label: Language
  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Katherine+E%2E+Castellano%22">Katherine E. Castellano</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0002-3695-5955">0000-0002-3695-5955</externalLink>)<br /><searchLink fieldCode="AR" term="%22Daniel+F%2E+McCaffrey%22">Daniel F. McCaffrey</searchLink> (ORCID <externalLink term="https://orcid.org/0000-0003-1196-5273">0000-0003-1196-5273</externalLink>)<br /><searchLink fieldCode="AR" term="%22Joseph+A%2E+Martineau%22">Joseph A. Martineau</searchLink>
– Name: TitleSource
  Label: Source
  Group: Src
  Data: <searchLink fieldCode="SO" term="%22Educational+Measurement%3A+Issues+and+Practice%22"><i>Educational Measurement: Issues and Practice</i></searchLink>. 2025 44(1):31-43.
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  Data: Wiley. Available from: John Wiley & Sons, Inc. 111 River Street, Hoboken, NJ 07030. Tel: 800-835-6770; e-mail: cs-journals@wiley.com; Web site: https://www.wiley.com/en-us
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  Label: Peer Reviewed
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  Data: Y
– Name: Pages
  Label: Page Count
  Group: Src
  Data: 13
– Name: DatePubCY
  Label: Publication Date
  Group: Date
  Data: 2025
– Name: TypeDocument
  Label: Document Type
  Group: TypDoc
  Data: Journal Articles<br />Reports - Research
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Student+Evaluation%22">Student Evaluation</searchLink><br /><searchLink fieldCode="DE" term="%22Growth+Models%22">Growth Models</searchLink><br /><searchLink fieldCode="DE" term="%22Student+Educational+Objectives%22">Student Educational Objectives</searchLink><br /><searchLink fieldCode="DE" term="%22Educational+Indicators%22">Educational Indicators</searchLink><br /><searchLink fieldCode="DE" term="%22Federal+Programs%22">Federal Programs</searchLink><br /><searchLink fieldCode="DE" term="%22Academic+Achievement%22">Academic Achievement</searchLink><br /><searchLink fieldCode="DE" term="%22Academic+Standards%22">Academic Standards</searchLink><br /><searchLink fieldCode="DE" term="%22Progress+Monitoring%22">Progress Monitoring</searchLink><br /><searchLink fieldCode="DE" term="%22Achievement+Gains%22">Achievement Gains</searchLink><br /><searchLink fieldCode="DE" term="%22Test+Reliability%22">Test Reliability</searchLink><br /><searchLink fieldCode="DE" term="%22Predictor+Variables%22">Predictor Variables</searchLink><br /><searchLink fieldCode="DE" term="%22Predictive+Validity%22">Predictive Validity</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1111/emip.12635
– Name: ISSN
  Label: ISSN
  Group: ISSN
  Data: 0731-1745<br />1745-3992
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: Growth-to-standard models evaluate student growth against the growth needed to reach a future standard or target of interest, such as proficiency. A common growth-to-standard model involves comparing the popular Student Growth Percentile (SGP) to Adequate Growth Percentiles (AGPs). AGPs follow from an involved process based on fitting a series of nonlinear quantile regression models to longitudinal student test score data. This paper demystifies AGPs by deriving them in the more familiar linear regression framework. It further shows that unlike SGPs, AGPs and on-track classifications based on AGPs are strongly related to status. Lastly, AGPs are evaluated in terms of their classification accuracy. An empirical study and analytic derivations reveal AGPs can be problematic indicators of students' future performance with previously not proficient students being more likely incorrectly flagged as not on-track and previously proficient students as on track. These classification errors have equity implications at the individual and school levels.
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  Data: 2025
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  Data: EJ1460444
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      – Type: doi
        Value: 10.1111/emip.12635
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 13
        StartPage: 31
    Subjects:
      – SubjectFull: Student Evaluation
        Type: general
      – SubjectFull: Growth Models
        Type: general
      – SubjectFull: Student Educational Objectives
        Type: general
      – SubjectFull: Educational Indicators
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      – SubjectFull: Federal Programs
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      – SubjectFull: Academic Achievement
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      – SubjectFull: Academic Standards
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      – SubjectFull: Progress Monitoring
        Type: general
      – SubjectFull: Achievement Gains
        Type: general
      – SubjectFull: Test Reliability
        Type: general
      – SubjectFull: Predictor Variables
        Type: general
      – SubjectFull: Predictive Validity
        Type: general
    Titles:
      – TitleFull: Demystifying Adequate Growth Percentiles
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            NameFull: Katherine E. Castellano
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            NameFull: Daniel F. McCaffrey
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            NameFull: Joseph A. Martineau
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              Y: 2025
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            – TitleFull: Educational Measurement: Issues and Practice
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