Autoregressive and cross-lagged model for bivariate non-commensurate outcomes.

Autoregressive and cross-lagged models have been widely used to understand the relationship between bivariate commensurate outcomes in social and behavioral sciences, but not much work has been carried out in modeling bivariate non-commensurate (e.g., mixed binary and continuous) outcomes simultaneously. We develop a likelihood-based methodology combining ordinary autoregressive and cross-lagged models with a shared subject-specific random effect in the mixed-model framework to model two correlated longitudinal non-commensurate outcomes. The estimates of the cross-lagged and the autoregressive effects from our model are shown to be consistent with smaller mean-squared error than the estimates from the univariate generalized linear models. Inclusion of the subject-specific random effects in the proposed model accounts for between-subject variability arising from the omitted and/or unobservable, but possibly explanatory, subject-level predictors. Our model is not restricted to the case with equal number of events per subject, and it can be extended to different types of bivariate outcomes. We apply our model to an ecological momentary assessment study with complex dependence and sampling data structures. Specifically, we study the dependence between the condom use and sexual satisfaction based on the data reported in a longitudinal study of sexually transmitted infections. We find negative cross-lagged effect between these two outcomes and positive autoregressive effect within each outcome.