The relationship between multilevel models and nonparametric multilevel mixture models

Abstract

Multilevel data structures are common in the social sciences. Often, such nested data are analyzed with multilevel models (MLMs) wherein heterogeneity between clusters is modeled by continuously-distributed random intercepts and/or slopes. Alternatively, nonparametric multilevel regression mixture modeling (NPMM) can accommodate the same nested data structures through discrete latent class variation. The purpose of this thesis is to delineate analytic relationships between NPMM and MLM parameters that are useful for understanding the indirect interpretation of NPMM as a nonparametric approximation of MLM. I define how seven standard and nonstandard MLM specifications are indirectly approximated by particular NPMM specifications. I provide formulas showing how NPMM can serve as an approximation of MLM in terms of intraclass correlation, random effect variances, and heteroscedasticity of residuals at level-1 and level-2. The specific relationships are illustrated with simulated graphical demonstrations, and direct and indirect interpretations of NPMM classes are contrasted. I demonstrate how the indirect relationships can be utilized in inferential testing of implied MLM parameters and provide Mplus code to implement these tests. I also provide and describe an R function to aid in implementing and visualizing an indirect interpretation of NPMM classes. An empirical example is presented and future directions are discussed.