A nonparametric simultaneous confidence band for biomarker effect on the restricted mean survival time

Study of prognostic and predictive biomarkers plays an important role in the design and analysis of clinical trials. The Cox proportional hazards model is often used to study the biomarker main effect and the treatment-biomarker interaction effect for survival data. The estimated effects can be biased if the proportional hazards assumption is violated. The restricted mean survival time is becoming popular in clinical studies for having a clear intuitive interpretation. In this article, we first propose nonparametric methods to make statistical inference for the one-sample problem of the biomarker effect on the restricted mean survival time; we then extend the methods to the two-sample problem for studying the difference in the biomarker effects between treatment groups in clinical trials. For a given biomarker, the restricted mean survival time is estimated by kernel smoothing methods with the inverse probability of censoring weights. We prove the consistency for the estimates and develop simultaneous confidence bands for the biomarker effects on the restricted mean survival time. The simultaneous confidence bands are evaluated in extensive simulation studies and are found to have good finite sample performance. We then apply the proposed methods to a breast cancer study conducted by the Breast International Group (BIG) to illustrate how the Ki67 biomarker, a protein marker of cell proliferation, affects the survival time of patients, compared between the treatment groups.