  A fundamental concept in two-arm non-parametric survival analysis is the
comparison of observed versus expected numbers of events on one of the
treatment arms (the choice of which arm is arbitrary), where the expectation is
taken assuming that the true survival curves in the two arms are identical.
This concept is at the heart of the counting-process theory that provides a
rigorous basis for methods such as the log-rank test. It is natural, therefore,
to maintain this perspective when extending the log-rank test to deal with
non-proportional hazards, for example by considering a weighted sum of the
"observed - expected" terms, where larger weights are given to time periods
where the hazard ratio is expected to favour the experimental treatment. In
doing so, however, one may stumble across some rather subtle issues, related to
the difficulty in ascribing a causal interpretation to hazard ratios, that may
lead to strange conclusions. An alternative approach is to view non-parametric
survival comparisons as permutation tests. With this perspective, one can
easily improve on the efficiency of the log-rank test, whilst thoroughly
controlling the false positive rate. In particular, for the field of
immuno-oncology, where researchers often anticipate a delayed treatment effect,
sample sizes could be substantially reduced without loss of power.
