Moderation With a Latent Class Variable: Specification, Visualization, and Specification

• In Event: Methodological Advances in Longitudinal Mixture Modeling

Abstract

Traditional regression moderation is a widely utilized and useful tool in the area of social science research. Applied researchers use moderation analyses to determine whether the relationship between variables depends on (or are moderated by) the different levels of another variable (moderator). In recent research, using regression methods to identify this relationship is no longer sufficient given that more researchers are using latent variables to measure constructs. As such, moderation analyses should be expanded to include latent variables in order to understand the more complex relationships in social science research. More specifically, researchers interested in applying mixture models, or latent class analysis (LCA), may incorporate auxiliary variables (e.g., covariates and distal outcomes) to identify whether the different classes moderate the relationship between two variables.
Mixture models are person-centered statistical approaches used to identify and model unobserved subgroups of a population using a set of items in a measure. Moderation with mixture models, or regression mixture models, applies a three-step method (i.e., ML three-step or BCH) to estimate the relationships of the auxiliary variables with the latent class variable while leaving the measurement parameters of the latent class variable unchanged. Auxiliary variables used in the analyses may be observed, latent, or treatment variables. The current paper builds on the foundation of moderation, extending to mixture modeling.
Because moderation with mixture models using auxiliary variables is a relatively new method, interpretation of results of mixture regressions may be challenging to many researchers unfamiliar with the model. Additionally, the amount of detail in applied papers does not lend itself to new users, despite interest in fitting the model for their research. In addition to the interpretation of the results, visualization of mixture models may be difficult when including covariates and distal outcomes. Data visualization in mixture regression models can appear as trivial tasks to some methodologists (e.g., conditional item probability plots and bar chart of the covariates conditional on class), but is often less visible to others. The very few conventions that do exist have significant limitations that are rarely addressed.
This paper aims to provide a pedagogical approach to understanding moderation in the context of mixture modeling, as well as interpretation and visualization of the relationship between exogenous and endogenous variables among the moderating latent class variables. An applied example using the Longitudinal Survey of American Life (LSAL) will demonstrate the use of regression mixture models and provide examples of how to interpret models graphically. Demonstrations of the statistical methods are using the R and the MplusAutomation package within R and R Studio.

Authors

• Dina Naji Arch, University of California - Santa Barbara

• Karen L. Nylund-Gibson, University of California - Santa Barbara

• Marsha M. Ing, University of California - Riverside