Abstract
In contextual studies, group compositions are often extracted from individual data in the sample, in order to estimate the group compositional effects [e.g., school socioeconomic status (SES) effect] controlling for interindividual differences in multilevel models. As the same variable is used at both group level and individual level, an appropriate decomposition of between and within effects is a key to providing a clearer picture of these organizational and individual processes. The current study developed a new approach with within-group finite population correction (fpc). Its performances were compared with the manifest and latent aggregation approaches in the decomposition of between and within effects. Under a moderate within-group sampling ratio, the between effect estimates from the new approach had a lesser degree of bias and higher observed coverage rates compared with those from the manifest and latent aggregation approaches. A real data application was also used to illustrate the three analysis approaches.
Highlights
In contextual models, the group compositional effects on individual development or outcomes and their underlying organizational processes have attracted a large amount of attention (Mayer et al, 2014)
To evaluate the performances of the manifest aggregation approach, the latent aggregation approach, and the new approach with within-group fpc in the decomposition of between and within effects, the model convergence rate, relative bias, root mean square error (RMSE), and observed coverage rate for the within and between effects were first obtained across the 1,000 replications under each simulation condition for each analysis approach
As there were 160 simulation conditions for each model using each analysis approach, instead of proving the raw evaluation estimates under each simulation condition, the means and standard deviations of the convergence rate, relative bias, RMSE, and coverage rate across the 160 simulation conditions for each analysis approach were first provided for each parameter
Summary
The group compositional effects on individual development or outcomes and their underlying organizational processes have attracted a large amount of attention (Mayer et al, 2014). The individual-level constructs and their aggregated group compositions often show different effects on individual outcomes, which reflect different theoretical meanings (Lau and Nie, 2008; Marsh et al, 2012). The school-level effect of achievement on student academic self-concept reflected the way schools were structured and their effects on individuals (Marsh et al, 2009). The effects of aggregated individual characteristics, like socioeconomic status (SES), gender, and ethnicity, etc., have drawn attention in contextual studies. The study on student and school SES effects is one good example, which examines the between-group effect of group compositions and the within-group effect of individual characteristics (Raudenbush and Bryk, 2002; Lüdtke et al, 2008). M, the mean of observed coverage rate across the 160 simulation conditions; SD, the standard deviation of observed coverage rate across the 160 simulation conditions; Manifest, manifest aggregation approach; Latent, latent aggregation approach; FPC Latent, the new approach with within-group fpc.
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