Statistical Inference in Survival Analysis via Wild Bootstrap

The research described in the thesis concerns statistical inference in survival analysis for general counting process-based statistics and, in particular, for the estimators involved in the Fine-Gray model. The main goal of the research was to provide the theoretical foundation for the use of the wild bootstrap to approximate the unknown distribution of counting process-based statistics. First, the asymptotic distribution of general counting process-based statistics is analyzed, as the sample size goes to infinity. The underlying models may be parametric, semiparametric, or non-parametric. A general wild bootstrap resampling scheme is proposed, which can be used to define the corresponding wild bootstrap statistic. In order to justify the applicability of the wild bootstrap, rigorous proofs implying the asymptotic equivalence of the distribution of the counting process-based statistic and the distribution of its wild bootstrap counterpart are presented. The proofs rely on weak regularity conditions and are developed in a novel way based on martingale theory. Next, the focus is on the Fine-Gray model in the competing risks setting with censoringcomplete data. Based on the theory developed for general counting process-based statistics, the validity of the wild bootstrap for the estimators involved in the Fine-Gray model is verified. Furthermore, the aforementioned result is extended to the level of the cumulative incidence function. Based on this extension, asymptotically valid time-simultaneous (1−α)- confidence bands for the cumulative incidence function are constructed and their small sample performance is analyzed in a simulation study. In addition, the proposed method is illustrated by investigating the impact of pneumonia for intensive care unit patients on the probabilities of hospital death competing with alive discharge. Finally, for the situation where the data are not censoring-complete, the wild bootstrap is combined with multiple imputation methods to obtain a novel time-simultaneous confidence band for the cumulative incidence function under the Fine-Gray model with incomplete data. Under regularity conditions, a proof of the validity of the proposed approach is provided. Furthermore, its reliability is numerically assessed and compared with the wild bootstrap confidence bands based on censoring-complete data as well as confidence bands obtained by bootstrapping the inverse-probability-of-censoring-weighting estimator. The approach is illustrated by re-analyzing the above-mentioned pneumonia data set.