A two-sample comparison of mean survival times of uncured subpopulations

Comparing the survival times among two groups is a common problem in time-to-event analysis, for example if one would like to understand whether one medical treatment is superior to another. In the standard survival analysis setting, there has been a lot of discussion on how to quantify such difference and what can be an intuitive, easily interpretable, summary estimand. In the presence of subjects that are immune to the event of interest ('cured'), we illustrate that it is not appropriate to just compare the overall survival functions. Instead, it is more informative to compare the cure fractions and the survival of the uncured subpopulations separately from each other. Our research is mainly driven by the question: if the cure fraction is similar for two available treatments, how else can we determine which is preferable? To this end, we estimate the mean survival times in the uncured fractions of both treatment groups and develop both permutation and asymptotic tests for inference. We first propose a nonparametric approach which is then extended to account for covariates by means of the semi-parametric logistic-Cox mixture cure model. The methods are illustrated through practical applications to breast cancer and leukemia data.