Adaptive bi-level variable selection for multivariate failure time model with a diverging number of covariates

In this study we propose an adaptive bi-level variable selection method to analyze multivariate failure time data. In the regression setting, we treat the coefficients corresponding to the same predictor variable as a natural group and then consider variable selection at the group level and individual level simultaneously. By imitating the group variable selection procedure with adaptive bi-level penalty, the proposed variable selection method can select a predictor variable at two different levels allowing different covariate effects for different event types: the group level where the predictor is important to all failure types, and the individual level where the predictor is only important to some failure types. An algorithm based on cycle coordinate descent is developed to carry out the proposed method. Based on the simulation results, our method outperforms the classical penalty methods, especially in removing unimportant variables for different failure types. We obtain the asymptotic oracle properties of the proposed variable selection method in the case of a diverging number of covariates. We construct a generalized cross-validation method for the tuning parameter selection and assess model performance using model errors. We also illustrate the proposed method using a real-life data set.