In educational settings, a central question is whether the collective class characteristics influence individual student achievement over and above students’ own characteristics (Televantou et al., 2015). These influences are commonly referred to as compositional effects (e.g., Becker et al., 2022). The effects are detected by aggregating student characteristics, such as prior achievement or socioeconomic status, to classroom or school level and estimating whether the aggregated measures influence the individual outcomes after controlling for the student-level variable on which aggregation is based (Lüdtke et al., 2008; Televantou et al., 2015).

Findings on composition effects of class-average prior achievement are, however, inconsistent. While many studies report positive composition effects (e.g., Nilsen et al., 2025; Stäbler et al., 2017), others find none (e.g., Dicke et al., 2026), and effect sizes vary widely and are sensitive to model specification (e.g., Becker et al., 2022; Televantou et al., 2015). One reason for this inconsistency is that composition effects are defined as the difference between within-class and between-class achievement relations, both of which are distorted when prior achievement is measured with error. Standard multilevel models may overestimate composition effects because they ignore both measurement error and sampling bias (Marsh et al., 2009).

Prior research shows that composition effects are sensitive to selection bias and measurement error (Becker et al., 2022; Marsh et al., 2009, 2023). Selection bias arises when models fail to adequately capture important aspects of student composition, causing between-class differences to reflect preexisting individual differences rather than true contextual effects. Becker et al. (2022) demonstrate that controlling for prior achievement substantially reduces this bias and that including covariates, such as socioeconomic background and migration status, further attenuates estimated composition effects. Measurement error introduces a second source of bias. When achievement is measured unreliably, multilevel models tend to underestimate individual-level effect and, as a result, overestimate the group-level effect because the regression coefficient for the aggregated score picks up variance left unexplained due to measurement error. Becker et al. (2022) show that correcting for measurement error reduces spurious inflation of composition effects, whereas correcting for sampling error in class means prevents underestimation due to imperfect aggregation.

Against this background, the present study builds on this line of research by estimating a sequence of three multilevel models that impose progressively stronger corrections for measurement error and aggregation bias. Applying this methodological approach allows us to assess the models across multiple national contexts. In this respect, the study contributes by extending and validating the method beyond a single-country setting. In addition, as socioeconomic and demographic sorting into classrooms can produce biased composition effects, we estimate three additional models where student-level and class-level background controls are introduced incrementally, to distinguish peer achievement effects from contextual SES composition. In the present study, we address the following research question:

What is the magnitude of composition effects, and how does accounting for measurement error alter these estimates?

The study utilizes data from the TALIS Video Study 2018, which links mathematics teachers and students across eight countries/jurisdictions (Chile, Colombia, England, Germany, Japan, Madrid, Mexico, and Shanghai) during a common instructional unit (quadratic equations). Excluding Japan and Shanghai because of ceiling effects in these countries, we analyse student data from six school systems, comprising 12440 students nested in 478 classrooms. Based on our interest for national variation in composition effects, we analyze each country separately. Student achievement was measured using a pre-test assessing foundational mathematical and algebraic skills and a post-test measuring students’ ability to solve quadratic equations and apply these methods in problem-solving contexts. Pre-test scores are used as predictors of post-test outcomes. To assess the robustness of within- and between-classroom associations, we employed a staged multilevel structural equation modeling strategy implemented in R that progressively accounts for measurement error and aggregation bias. Measurement reliability was estimated separately by country using IRT-based standard errors and used to fix error variances in subsequent models. Because the relative performance of the model specifications we test depends on sampling and measurement conditions, particularly when nearly all students in a class are sampled (Becker et al., 2022), we estimated all three models for comparison. First, we estimated a doubly-manifest (DM) multilevel regression model in which both pre-test and post-test scores were treated as observed variables. Student-level pre-test scores were cluster-mean centered to estimate within-classroom effects, while classroom means of prior achievement were included to estimate between-classroom (compositional) effects. This model ignores measurement error in both achievement constructs. Second, we estimated a latent-measurement/manifest-aggregation (LM/MA) model. At within-level, pre-test and post-test achievement were modeled as latent variables to correct for individual-level measurement error, while at between-level, classroom means were treated as observed aggregates. This model isolates the impact of student-level unreliability, so differences between DM and LM/MA estimates primarily reflect correction for measurement error at within-level. Third, we estimated a doubly-latent multilevel structural equation model (DLM) in which both student-level achievement and classroom-level composition were modeled as latent variables. At between-level, residual variances of class-level achievement were fixed to zero, representing true classroom means. In a final step, student-level controls were introduced incrementally to the LM/MA and the DLM models. In Model 1, gender and immigration background were included at within-level. In Model 2, parental education and home-resources were additionally included at within-level. In Model 3, parental education and home-resources included at between-level.

Our findings show that estimates differed systematically across models. Within-class effects increased from DM to LM/MA in all countries, consistent with attenuation due to measurement error at the student level. Between-class effects were more sensitive to model specification. Across countries, between-level estimates in the LM/MA were substantially reduced once within-level measurement error was accounted for. In contrast, DLM yielded larger between-level estimates than both the DM and LM/MA, consistent with corrections for aggregation bias and sampling error in class means. The DLM produced large and consistently significant between-level effects, ranging from 0.66 to 0.95. Indicating that, once measurement error and aggregation bias are addressed, compositional effects are substantial and comparable in magnitude to individual-level effects. This pattern remained after inclusion of control variables. Interestingly, the socioeconomic status explained little of the between-class achievement differences once prior achievement was controlled. This indicates that although SES-associated segregation is common, accounting for prior knowledge captures the compositional effect that may be associated with SES. Despite the inclusion of covariates, residual confounding cannot be ruled out. Because the TALIS video study did not use a randomized design, unobserved factors, such as school organizational characteristics, may remain correlated with class-average achievement and bias estimated compositional effects. Furthermore, tracking structures were not included in the data, and some observed between-class differences may therefore reflect tracking mechanisms rather than classroom composition per se.

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