Joint models: When are treatment estimates improved?

Treatments affect many aspects of disease for example, a drug may improve symptoms, prolong survival, and cause serious side effects. A broader perspective on clinical effectiveness, considering multiple outcomes, requires analyses that account for relationships among outcomes. So-called joint modeling induces such relationships via shared parameters. Practical questions arise, including "When do we require a joint model?" and "How much do we gain by its use?" Motivated by these questions, we compare Gaussian joint models with shared latent parameters to separate models for each outcome individually. When we assume a single longitudinal measurement, known error variances, and no censoring, joint and separate treatment effect posteriors converge as the priors become improper. This result still holds when we add multiple longitudinal measurements and unknown error variance, but not when we make the prior informative for at least one treatment effect (longitudinal or survival). Joint models also improve inference under some censoring scenarios. Our results suggest that joint models are most useful when an information imbalance allows abundant information in one outcome to compensate for a paucity of information in another.