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 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 (Zhang, Zyphur, & Preacher, 2009). Two analysis approaches, the manifest and latent aggregation approaches, were proposed in previous studies, which were based on different assumptions about the natures of group-level constructs, populations of research interest, and sampling designs. The current study developed a new approach with within-group finite population correction (fpc) to decompose the between and within effects. Its performances were compared with the manifest and latent aggregation approaches for the two-level random intercept model, 2-1-1 mediation model, and 1-1-1 mediation model through mathematical derivation and simulation. The simulation conditions included the between-to-within-effect ratio, intraclass correlation coefficient (ICC) of the decomposed predictor or mediator, number of groups in the sample, balanced or unbalanced design, group size in the population, and within-group sampling ratio. When the within-group sampling ratio was moderate (i.e., from 10% to 90%), which is commonly seen in education research, the mathematical derivation and simulation results indicated that, in general, the between effect estimates from the new approach with within-group fpc were of less degrees of biases and higher observed coverage rates compared to those from the manifest and latent aggregation approaches. Although the between effect estimates from the manifest aggregation approach were less variable, the trade-offs were their lower observed coverage rates. For the within effects, there was no significant difference among the three analysis approaches in the relative biases, absolute relative biases, or root mean square errors (RMSE) on average, or across different simulation conditions. However, under the unbalanced design, the standard errors for the within effects from the new approach were underestimated. A real data application was also used to illustrate the three analysis approaches.
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