Multivariate and shared parameter mixed-effects models for intensive longitudinal data

Intensive longitudinal data are increasingly common in many research areas. Such data are often collected using ecological momentary assessment, experience sampling methods, internet studies, wearable devices, and diary methods. In such studies, typically there are a large number of repeated observations per subject, and often many variables are measured. In this chapter, we first describe a multivariate mixed-effects model that simultaneously considers several dependent variables as joint outcomes. This model includes several random subject effects for each outcome and considers these random effects to be correlated. In this way, one can assess the correlation of the outcomes conditionally adjusting for model covariates. Furthermore, the multivariate model allows one to test if covariates have the same or different effects on the outcomes. We then describe how the covariance parameters of the random effects can be reformulated as regression effects, leading to a shared parameter modeling of the joint outcomes. This then allows one to consider interactions with the regression versions of the covariance parameters. For example, one can examine whether the association of the outcomes varies by other model covariates (e.g., subject sex or age). Furthermore, these association parameters can interact with each other (e.g., does the association of two outcomes vary as a function of another outcome). This flexibility of the shared parameter approach is highlighted and offers data analysts the possibility of considering novel research questions for intensive longitudinal data. To illustrate these approaches, we use a study investigating weight loss in which subjects provided daily weight measurements over a treatment intervention period of 3 months and a follow-up period of 9 months. Since subjects have varying numbers of weight measurements in both the treatment and follow-up periods, the use of the mixed model, which does not assume complete data across time, is attractive. In supplemental materials, we present syntax for both frequentist and Bayesian approaches to estimate the parameters of such models using standard statistical software packages.