  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 measure. 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 sub-populations 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 ($MST_u$) and develop permutation tests for inference. In the
first out of two connected papers, we focus on nonparametric approaches. The
methods are illustrated with medical data of leukemia patients. In Part II we
adjust the mean survival time of the uncured for potential confounders, which
is crucial in observational settings. For each group, we employ the widely used
logistic-Cox mixture cure model and estimate the $MST_u$ conditionally on a
given covariate value. An asymptotic and a permutation-based approach have been
developed for making inference on the difference of conditional $MST_u$'s
between two groups. Contrarily to available results in the literature, in the
simulation study we do not observe a clear advantage of the permutation method
over the asymptotic one to justify its increased computational cost. The
methods are illustrated through a practical application to breast cancer data.
