Compound Polynomial growth model

Research Method · All research methods
Evidence · 12 claims (10 for, 1 mixed, 1 against) · 5 studies (4 associational, 1 design), q1–q2 · 1 of 5 report an effect size · 12 claims rest on one study

Description

The CP model, developed by Thum and Hauser (2015), is a flexible functional form for describing growth trends with marked seasonal patterns. It combines "a polynomial growth in performance (scores) that occurs within a segment" with "a second polynomial growth model for one or more within-segment growth components," so that the model "includes parameters capturing between- and within-year growth." In this article it is applied in a school evaluation context to estimate school contributions to student growth from fall and spring MAP Growth data across grades two through six, with three levels (time points within students within schools).

Accounts

• The Compound Polynomial (CP) growth model: The CP model, developed by Thum and Hauser (2015), is a flexible functional form for describing growth trends with marked seasonal patterns. It combines "a polynomial growth in performance (scores) that occurs within a segment" with "a second polynomial growth model for one or more within-segment growth components," so that the model "includes parameters capturing between- and within-year growth." In this article it is applied in a school evaluation context to estimate school contributions to student growth from fall and spring MAP Growth data across grades two through six, with three levels (time points within students within schools). (Soland et al. (2019))
• Compound polynomial model for estimating school effectiveness in program evaluation contexts: The page lists an associated working paper titled "School effectiveness, summer loss, and federal accountability: Applying the compound polynomialmodel in a program evaluation context." The compound polynomial model is thus a growth-modeling approach applied to estimate school effectiveness while accounting for summer loss in program evaluation settings. The page provides no further description of the model's structure or mechanics, so it is recorded only as a named modeling approach linked to seasonality-aware school effectiveness estimation. (NWEA Research (2019))
• A general statistical model for interim achievement data that jointly represents year-to-year growth and within-year seasonal gains and losses: The article introduces a general statistical model motivated by the seasonal character of interim achievement data. It is "designed to combine features from traditional polynomial models that estimate year-to-year growth but ignore within-year gains and losses with those from piecewise models, which directly estimate within-year gains/losses but do not include terms for year-to-year growth." The model is intended for monitoring progress of students, classrooms, and schools within and across school years. (James Soland (2022))
• Compound Polynomial (CP) growth model for jointly estimating within-year and between-year growth: The CP model is a multilevel growth model whose design matrix combines a within-year (fall-to-spring) matrix with a between-year (spring-to-spring) polynomial matrix, so that both kinds of growth are estimated in one model. The authors use it "to simultaneously estimate fall-to-spring (within-year) and spring-to-spring (between-year) growth over five years using a single model," producing conditional Z-scores via contrast matrices that allow school effectiveness comparisons across timespans on a consistent sample. The preferred specification treats all coefficients as random except the within-year quadratic, which is fixed at the school level. (Soland et al. (2019))

Claims

• The CP model fits seasonal achievement data better than a traditional polynomial growth model on RMSE, AIC, and BIC while reducing residual autocorrelation [+M]
• Across 90 grade-subject-level conditions, the three-piece piecewise model showed the lowest deviance for 64% of student-level and 78% of school-level models, outperforming both compound polynomial candidates [-W]
• Interim achievement data exhibit distinctly seasonal patterns that few existing statistical models can approximate [+W]
• Ignoring summer loss can impact which schools are identified as low performers under federal accountability [+W]
• School effectiveness estimates from fall-to-spring versus spring-to-spring growth correlate only moderately to strongly (.399 to .705), below the .90 threshold at which rank orderings diverge in accountability [+W]
• More of the variance in linear growth lies at the school level for fall-to-spring than for spring-to-spring estimates, in some cases twice or three times as much [+W]
• Within-year (fall-to-spring) achievement gains decelerate linearly as students move through school, at roughly 1.4 to 4 RIT per year depending on subject and centering grade [+W]
• In the study data, mean spring grade-2 math RIT was 190.6 with linear spring-to-spring growth of 14.5 RIT, and mean fall-to-spring gain in grade-2 math was 13.4 RIT [+W]
• The proposed seasonal growth model combines polynomial terms for year-to-year growth with piecewise terms for within-year gains and losses [+W]
• Seasonal patterns are discrepant across national data sets [~W]
• School contributions to within-year (single-year fall-to-spring) growth correlate at roughly .10 or below with school contributions to growth across elementary school (grades 2–6) [+M]
• School growth estimates based on two years of data correlate around .55–.58 with estimates based on all five elementary years, below the .90 threshold at which ESSA accountability determinations would likely change [+M]

Related Research Methods

-

Key Sources

• Soland, J. & Thum, Y.M. (2019). School effectiveness, summer loss, and federal accountability: Applying the compound polynomial model in a program evaluation context. The Collaborative for Student Growth at NWEA Working Paper. https://www.nwea.org/research/publication/school-effectiveness-summer-loss-and-federal-accountability-applying-the-compound-polynomialmodel-in-a-program-evaluation-context/
• NWEA Research. (2019). School effectiveness, summer loss and federal accountability. Research brief. https://www.nwea.org/research/publication/school-effectiveness-summer-loss-and-federal-accountability/
• James Soland, Yeow Meng Thum. (2022). Estimating and comparing growth using longitudinal interim achievement data with seasonal trends. Journal of Research on Educational Effectiveness 15(3) 635-654. https://doi.org/10.1080/19345747.2021.2018744
• Soland, J., Thum, Y.M., & King, G. (2019). Reconciling long-term education policy goals with short-term school accountability models: Evidence and implications from a longitudinal study. (The Collaborative for Student Growth at NWEA Working Paper). https://www.nwea.org

Page metadata

Field	Value
type	research-method
id	compound-polynomial-growth-model
aliases	compound-polynomial-cp-growth-model
title	Compound Polynomial growth model
description	The CP model, developed by Thum and Hauser (2015), is a flexible functional form for describing growth trends with marked seasonal patterns.
status	review
moved_from	theory/compound-polynomial-growth-model, theory/compound-polynomial-model-school-effectiveness, theory/seasonal-growth-model-interim-achievement-data
generated	by process:sweep-kinds on 2026-10-10
trust	unverified — no one has confirmed this page's content
sources	3 — soland-2019, nwea-research-2019, james-soland-2022

Frontmatter is the page's machine-readable record — see the Schema & Guide.