Frailty Model with Spline Estimated Nonparametric Hazard Function

Frailty has been introduced as a group-wise random effect to describe the within-group dependence for correlated survival data. In this article, we propose a penalized joint likelihood method for nonparametric estimation of hazard function. With the proposed method, the frailty variance component and the smoothing parameters become the tuning parameters that are selected to minimize a loss function derived from the Kullback-Leibler distance through delete-one cross-validation. Confidence intervals for the hazard function are constructed using the Bayes model of the penalized likelihood. Combining the functional ANOVA decomposition and the Kullback-Leibler geometry, we also derive a model selection tool to assess the covariate effects. We establish that our estimate is consistent and its nonparametric part achieves the optimal convergence rate. We investigate finite sample performance of the proposed method with simulations and data analysis.