A joint model of recurrent events and a terminal event with a nonparametric covariate function

We extend the shared frailty model of recurrent events and a dependent terminal event to allow for a nonparametric covariate function. We include a Gaussian random effect (frailty) in the intensity functions of both the recurrent and terminal events to capture correlation between the two processes. We employ the penalized cubic spline method to describe the nonparametric covariate function in the recurrent events model. We use Laplace approximation to evaluate the marginal penalized partial likelihood without a closed form. We also propose the variance estimates for regression coefficients. Numerical analysis results show that the proposed estimates perform well for both the nonparametric and parametric components. We apply this method to analyze the hospitalization rate of patients with heart failure in the presence of death.