Next, we re-specified the MNLFA to include effects of predictors on the mean and variance of depression. We specified a linear model for the factor mean and a log-linear model for the factor variance (to prevent negative predicted variances). Not all predictors must appear in the model for the factor mean and variance; rather, in our experience, the variance model is often simpler than the mean model. For our depression example, we detected a cubic age trend on the factor mean, differing by study and gender, and a main effect of parental history of alcoholism. This indicates that changes in depression scores across age follow a cubic trend which differs across study and gender, and that having an alcoholic parent increases mean levels of depression. Regarding the factor variance, we detected an interaction between age and study membership, such that depression levels increased in variability with age in AHBP but not in MLS or AFDP. Bringing these predictors into the model not only makes the specification of the model more realistic (by explicitly incorporating known sources of heterogeneity), but it
