Smoothing spline ANOVA frailty model for recurrent event data

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data.