A copula model for joint modeling of longitudinal and time-invariant mixed outcomes.

Motivated by a preclinical study in a mouse model of breast cancer, we suggest a joint modeling framework for outcomes of mixed type and measurement structures (longitudinal versus single time/time-invariant). We present an approach based on the time-varying copula models, which is used to jointly model longitudinal outcomes of mixed types via a time-varying copula, and extend the scope of these models to handle outcomes with mixed measurement structures. Our framework allows the parameters corresponding to the longitudinal outcome to be time varying and thereby enabling researchers to investigate how the response-predictor relationships change with time. We investigate the finite sample performance of this new approach via a Monte Carlo simulation study and illustrate its usefulness by an empirical analysis of the motivating preclinical study, comparing the effect of various treatments on tumor volume (longitudinal continuous response) and the number of days until tumor volume triples (time-invariant count response). Through the real-life application and the simulation study, we demonstrate that, compared with marginal modeling, the joint modeling framework offers more precision in the estimation of model parameters.