  Delayed treatment effects on time-to-event outcomes have often been observed
in randomized controlled studies of cancer immunotherapies. In the case of
delayed onset of treatment effect, the conventional test/estimation approach
using the log-rank test for between-group comparison and Cox's hazard ratio to
estimate the magnitude of treatment effect is not optimal, because the log-rank
test is not the most powerful option, and the interpretation of the resulting
hazard ratio is not obvious. Recently, alternative test/estimation approaches
were proposed to address both the power issue and the interpretation problems
of the conventional approach. One is a test/estimation approach based on
long-term restricted mean survival time, and the other approach is based on
average hazard with survival weight. This paper integrates these two ideas and
proposes a novel test/estimation approach based on long-term average hazard
(LT-AH) with survival weight. Numerical studies reveal specific scenarios where
the proposed LT-AH method provides a higher power than the two alternative
approaches. The proposed approach has test/estimation coherency and can provide
robust estimates of the magnitude of treatment effect not dependent on
study-specific censoring time distribution. Also, the proposed LT-AH approach
can summarize the magnitude of the treatment effect in both absolute difference
and relative terms using ``hazard'' (i.e., difference in LT-AH and ratio of
LT-AH), meeting guideline recommendations and practical needs. This proposed
approach can be a useful alternative to the traditional hazard-based
test/estimation approach when delayed onset of survival benefit is expected.
