A stratified log-rank test with missing stratum information and dependent censoring

The log-rank test is often used to compare randomized treatment groups with respect to the distribution of a failure time outcome. The so-called stratified log-rank test can be used when it is necessary to adjust for the effect of some discrete covariate that may be predictive of the outcome. In many applied situations, this discrete covariate is missing for some of the patients and moreover, the distribution of the censoring time depends on the treatment group. In this paper, we introduce a modified version of the stratified log-rank test, which accommodates both these problems simultaneously. The asymptotic distribution of this new test under the null hypothesis of equality of the randomized treatment groups is established. A numerical study is conducted to examine the finite-sample behavior of this test under both the null and alternative hypotheses.