Interpreting and Testing Interactions in Conditional Mixture Models

Abstract

Mixture modeling applications in psychology often include covariates to explain class membership and aid in construct validation of the latent classification variable. These applications tend to use between-class models involving only main effects of predictors. However, a variety of developmental theories posit interactions among risk and protective variables in predicting membership in trajectory classes or behavioral symptom profiles. This article bridges this disconnect between substantive theory and methodological practice by presenting and comparing two approaches for testing interactive effects of predictors on class membership: product term (PT) and multiple group (MG) approaches. For each approach, we discuss alternative interpretation strategies involving predicted probabilities and odds ratios; we also discuss when the approaches provide equivalent inferences. Published longitudinal and cross-sectional mixture model applications that had originally allowed for only additive effects on class membership are re-analyzed to illustrate the testing and interpretation of interactive effects on class membership using both PT and MG approaches.

Notes

¹Long (2012) describes the persistence generally of “limit[ing] interpretation to a table of coefficients with a brief discussion of the signs and significance” or “simply presenting odds ratios without information that allows determination of the magnitude of the effect in terms of changes in probabilities” (p. 28).

²The three-step approach has not yet been presented for multiple group specifications, though this is a feasible extension. An alternative two-step (classify-analyze) approach performs worse than either the one-step or three-step approaches (e.g., Clark & Muthén, 2009; Clogg, 1995) and is not considered here.

³If there were only two classes, K = 2, the multinomial logistic specification would simplify and would provide equivalent results to a binary logistic specification.

⁴Suppose that a K = 2 mixture model is fit where the first class evidences low delinquency whereas the second (reference) class evidences high delinquency. Suppose the researcher had instead desired the low delinquency class to be the reference class. One way to achieve this is to refit the model, placing estimates from the low delinquency class as starting values for the parameters of the last class. Such reordering of the classes does not alter model fit (e.g., McLachlan & Peel, 2000).

⁵Subsequently, we use the term “baseline odds” as shorthand to refer to the odds where all covariates are equal to 0.

⁶This extension would involve regressing outcomes within-class (here, y_ij) on x_2i as well as allowing x_2i to interact with each within-class predictor.

⁷Exact ages were not made available for participants in this dataset, at www.icpsr.umich.edu. The analysis sample of N = 438 consists of young adults with at least one outcome and all covariate data present. Under missing-at-random assumptions, missing outcomes were accommodated with full information maximum likelihood, implemented with the Expectation-Maximization algorithm.

⁸Endorsement of an aggressive conduct offense occurred if an adolescent over the past 12 months: fought in a group, shot or stabbed, pulled a knife or gun, badly injured someone, or threatened with a weapon. It would be possible to adopt other representations of this aggression construct.

⁹The previous study found K = 3 classes to be best fitting using a more complex within-class model that involved time-varying covariates, which were not used here. Yet the K = 3 conditional LCGM used here had marginal class trajectories nearly identical to the previous study and was also found best fitting according to information criteria (for AIC, K = 2: 1397.88, K = 3: 1344.82, K = 4; 1348.16; for BIC, K = 2: 1442.78; K = 3: 1426.46; K = 4: 1466.55).

¹⁰If the reader is interested in comparing the appearance of predicted probability plots assuming additive effects of predictors on class (in Muthén, 2002) vs. allowing interactive effects of predictors on class (here, in Figure 2), note that in the corresponding additive-effect plot in Muthén (2002), class 2 and 3 labels were switched.