Appendix A: Principles of information

Abstract

Simultaneous confidence bands for survival functions, from randomly right censored data, can be computed by inverting likelihood ratio functions based on appropriate thresholds. Sometimes, however, the requisite asymptotic distributions are intractable, or thresholds based on Brownian bridge approximations are not easy to obtain when simultaneous confidence bands over only sub-regions are possible or desired. We obtain the thresholds by bootstrapping (i) a nonparametric likelihood ratio function via censored data bootstrap and (ii) a semiparametric adjusted likelihood ratio function via a two-stage bootstrap that utilizes a model for the second stage. These two scenarios are grounded respectively in standard random censorship and its semiparametric extension introduced by Dikta. The two bootstraps, which are different in the way resampling is done, are shown to have asymptotic validity. The respective confidence bands are neighborhoods of the well-known Kaplan–Meier estimator and the more recently developed Dikta’s semiparametric counterpart. As evidenced by a validation study, both types of confidence bands provide approximately correct coverage. The model-based confidence bands, however, are tighter than the nonparametric ones. Two sensitivity studies reveal that the model-based method performs well when standard binary regression models are fitted, indicating its robustness to misspecification as well as its practical applicability. An illustration is given using real data.