Estimand-based inference in the presence of long-term survivors

In this article, we develop nonparametric inference methods for comparing survival data across two samples, beneficial for clinical trials of novel cancer therapies where long-term survival is critical. These therapies, including immunotherapies and other advanced treatments, aim to establish durable effects. They often exhibit distinct survival patterns such as crossing or delayed separation and potentially leveling-off at the tails of survival curves, violating the proportional hazards assumption and rendering the hazard ratio inappropriate for measuring treatment effects. Our methodology uses the mixture cure framework to separately analyze cure rates of long-term survivors and the survival functions of susceptible individuals. We evaluated a nonparametric estimator for the susceptible survival function in a one-sample setting. Under sufficient follow-up, it is expressed as a location-scale-shift variant of the Kaplan-Meier estimator. It retains desirable features of the Kaplan-Meier estimator, including inverse-probability-censoring weighting, product-limit estimation, self-consistency, and nonparametric efficiency. Under insufficient follow-up, it can be adapted by incorporating a suitable cure rate estimator. In the two-sample setting, in addition to using the difference in cure rates to measure long-term effects, we propose a graphical estimand to compare relative treatment effects on susceptible subgroups. This process, inspired by Kendall's tau, compares the order of survival times among susceptible individuals. Large-sample properties of the proposed methods are derived for inference and their finite-sample properties are evaluated through simulations. The methodology is applied to analyze digitized data from the CheckMate 067 trial.