C on x: Modeling the Covariance Between Independent Variables and Latent Classes in Regression Mixture Models

This presentation addresses one of the issues that applied researchers face when implementing regression mixture models to examine differential effects. The typical specification of the regression mixture model includes effects of the predictor on the outcome variable(s) for each latent class (the differential effects), and excludes the relation between predictor variables and the latent class variable itself. There exists some controversy in the regression mixture literature about whether this relationship between the predictor (x) and the latent classes (C) should be estimated.  In this presentation, we demonstrate that failure to include this relationship results in assuming that the means of the predictor are equal across latent classes. We then examine the effects of misspecifying the model by assuming constant means of the predictor across latent classes when, in reality, they are not equal. We present findings of a Monte Carlo simulation study in which we evaluated the effects of violating this common, implicit assumption (i.e., that independent variables in the model that are not directly related to latent classes). The constraint of equal means was relaxed in these simulations by regressing the latent class variable (C) on the covariates (x) in the model; and used simulations to examine the effects of the inclusion or exclusion of this path on class enumeration and parameter estimates under various model conditions.  Across conditions, findings indicate that the major risk of excluding the relationship between predictor and latent class in model estimation was an increase in the probability of selecting additional latent classes and biased class proportions.