Evaluating Differential Effects Using Regression Interactions and Regression Mixture Models

Abstract

Research increasingly emphasizes understanding differential effects. This paper focuses on understanding regression mixture models, a relatively new statistical methods for assessing differential effects by comparing results to using an interactive term in linear regression. The research questions which each model answers, their formulation, and their assumptions are compared using Monte Carlo simulations and real data analysis. The capabilities of regression mixture models are described and specific issues to be addressed when conducting regression mixtures are proposed. The paper aims to clarify the role that regression mixtures can take in the estimation of differential effects and increase awareness of the benefits and potential pitfalls of this approach. Regression mixture models are shown to be a potentially effective exploratory method for finding differential effects when these effects can be defined by a small number of classes of respondents who share a typical relationship between a predictor and an outcome. It is also shown that the comparison between regression mixture models and interactions becomes substantially more complex as the number of classes increases. It is argued that regression interactions are well suited for direct tests of specific hypotheses about differential effects and regression mixtures provide a useful approach for exploring effect heterogeneity given adequate samples and study design.

Children exposed to intimate partner violence: Identifying differential effects of family environment on children's trauma and psychopathology symptoms through regression mixture models

“…Through regression mixture models, differential effects can be found in an empirical manner; however, Van Horn et al (2009, 2015) warn that this approach is largely data driven. Therefore, it is critical that the results of studies are replicated in other samples to explore whether different samples yield different results.…”

“…These values can range from 0 to 1, with higher scores representing greater classification accuracy. It is important to note, however, that Van Horn et al (2015) argue that entropy should not be considered a class enumeration criterion when evaluating regression mixtures.…”

Repeated measures regression mixture models

“…We also hypothesized that the benefit of having multiple outcomes would be greater when the measurement error variance was larger, primarily due to jointly modeling these repeated measures outcomes. The reason for this was that the regression mixture model with a single outcome relies strongly on assumptions about the conditional distribution of that outcome for parameter estimation (Van Horn et al, 2015); with multiple outcomes, estimation is informed by the joint distribution of all outcomes, which contains more information than does the univariate distribution.…”

“…We anticipated that the regression mixture models would perform poorly when the Class 1 effect size was medium, given the reduced class separation from Class 2 (i.e., 0.40 vs. 0.70). We used the current setting to generate the two latent classes because we believe that if regression mixtures are to be useful for finding individual differences, they should be able to detect a difference of at least this size (Kim, Vermunt, Bakk, Jaki, & Van Horn, 2016;Van Horn et al, 2015). Although the entropy is regarded as a model fit index to show the accuracy of class assignment in general finite mixture models, it is not necessarily indicative of goodness of fit in regression mixture models.…”

Penalized likelihood for logistic-normal mixture models with unequal variances

“…Kang, Janes and Huang (2014) relied on a novel boosting algorithm to choose an optimal treatment. Besides interaction models, methods based on mixture models have been proposed in Shen and He (2015) and Van Horn et al (2015). They showed that regression mixture models can be effective in evaluating differential treatment effects.…”