  Competing risks data are common in medical studies, and the sub-distribution
hazard (SDH) ratio is considered an appropriate measure. However, because the
limitations of hazard itself are not easy to interpret clinically and because
the SDH ratio is valid only under the proportional SDH assumption, this article
introduced an alternative index under competing risks, named restricted mean
time lost (RMTL). Several test procedures were also constructed based on RMTL.
First, we introduced the definition and estimation of RMTL based on
Aalen-Johansen cumulative incidence functions. Then, we considered several
combined tests based on the SDH and the RMTL difference (RMTLd). The
statistical properties of the methods are evaluated using simulations and are
applied to two examples. The type I errors of combined tests are close to the
nominal level. All combined tests show acceptable power in all situations. In
conclusion, RMTL can meaningfully summarize treatment effects for clinical
decision making, and three combined tests have robust power under various
conditions, which can be considered for statistical inference in real data
analysis.
