Testing for the Functional Form of a Continuous Covariate in the Shared-Parameter Joint Model

Shared-parameter joint modeling is a useful technique for properly associating longitudinal and time-to-event data. When the interest is in the survival outcome, the conditional logarithm of the hazard function for an event is conventionally presumed to be linearly related over time to a set of explanatory covariates, among other terms. However, this hypothesis is quite restrictive and may yield misleading results. Our objective here is to easily check such a modeling assumption for any continuous fixed covariate. For this purpose, we examine the appropriateness of a nonparametric test criterion based on a penalty-modified version of the Akaike information criterion. An extensive numerical study is conducted to check the validity of the test within the joint modeling framework, while determining the extent to which the function embedding the continuous covariate deviates from linearity. Furthermore, once a deviation from linearity is detected, the improvement in the model's predictive performance is examined. The usefulness of the testing procedure is illustrated using a clinical trial with HIV-infected subjects. Specifically, our example focuses on properly accounting for the effect of nadir CD4 cell count within a predictive joint model for the time to immune recovery.