Abstract
Longitudinal data tracking repeated measurements on individuals are highly valued for research because they offer controls for unmeasured individual heterogeneity that might otherwise bias results. Random effects or mixed models approaches, which treat individual heterogeneity as part of the model error term and use generalized least squares to estimate model parameters, are often criticized because correlation between unobserved individual effects and other model variables can lead to biased and inconsistent parameter estimates. Starting with an examination of the relationship between random effects and fixed effects estimators in the standard unobserved effects model, this article demonstrates through analysis and simulation that the mixed model approach has a “bias compression” property under a general model for individual heterogeneity that can mitigate bias due to uncontrolled differences among individuals. The general model is motivated by the complexities of longitudinal student achievement measures, but the results have broad applicability to longitudinal modeling.
Highlights
Longitudinal data are highly valued in all areas of social science research
We consider a generalization of the structural model in Equation 1 that allows for multiple time-invariant individual effects that can be related to the measurements in ways that vary across time
The multi-factor formulation is perfectly suited to joint longitudinal modeling of outcomes from different tested subjects by treating the models for different subjects as a set of seemingly unrelated regressions (SUR) [51] where the factors can represent different ability attributes relevant for different subjects
Summary
Longitudinal data are highly valued in all areas of social science research. In education research in particular, the rapidly growing availability of data tracking student achievement over time has made longitudinal data analysis increasingly prominent. The goal of this article is to demonstrate that the bias from the mixed model approach can be small under a model that generalizes the standard unobserved effects model by allowing for multiple time-invariant individual parameters that are related in time-varying ways to the observed measurements. Such a model is motivated by the complexities of longitudinal student achievement data but has broad applicability to other outcome variables in other research areas.
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