Abstract
Estimating a patient's disease trajectory as defined by clinical measures is an essential task in medicine. Given multiple biomarkers, there is a practical choice of whether to estimate the joint distribution of all biomarkers in a single model or to model the univariate marginal distribution of each marker separately ignoring the covariance structure among measures. To fully utilize all trajectory-relevant information in multiple longitudinal markers, a joint model is required, but its complexity and computational burden may only be warranted when joint estimates of trajectories are substantially more efficient than separate estimates. This paper derives general expressions for the inefficiency of univariate or "separated" estimates of population-average trajectories and individual's random effects as compared to the fully efficient multivariate or "combined" estimates. Then, in two settings: (1) a general bivariate case; and (2) our motivating clinical case study with 5 measures, we find that separated estimates of fixed effects are nearly fully efficient. However, joint estimates of random effects can be meaningfully more efficient for measures with substantial missing data when other strongly correlated measures are observed more frequently. This increased efficiency of the joint model derives more from joint shrinkage of random effects in multivariate space than from improved estimates of the subject-specific trajectories obtained when accounting for correlations in measurements. These findings have application to a diverse array of chronic diseases where biomarkers' trajectories guide clinical decisions.