Two-sample tests of the equality of two cumulative incidence functions

Typically, differences in the effect of treatment on competing risks are compared by a weighted log-rank test. This test compares the cause-specific hazard rates between the groups. Often the test does not agree with impressions gained from plots of the cumulative incidence functions. Here, we discuss two-sample tests of the equality of two cumulative incidence functions. The first test, based on a suggestion of Lin [1997. Non-parametric inference for cumulative incidence functions in competing risks studies. Statist. Med. 16, 901–910], compares the maximum difference between the two cumulative incidence functions. A Monte Carlo method is used to find p-values for the test. The second test, based on a suggestion of Pepe [1991. Inference for events with dependent risks in multiple endpoint studies. J. Amer. Statist. Assoc. 86, 770–778], compares the integrated difference between the functions. A new variance estimator is proposed for this statistic. A small simulation study is used to compare the various tests. The methods are illustrated on a bone marrow transplant study.