In studies involving nonparametric testing of the equality of two or more survival distributions, the survival curves can exhibit a wide variety of behaviors such as proportional hazards, early/late differences, and crossing hazards. As alternatives to the classical logrank test, the weighted Kaplan-Meier (WKM) type statistic and their variations were developed to handle these situations. However, their applicability is limited to cases where the population membership is available for all observations, including the right censored ones. Quite often, failure time data are confronted with missing population marks for the censored observations. To alleviate this, a new WKM-type test is introduced based on imputed population marks for the censored observations leading to fractional at-risk sets that estimate the underlying risk for the process. The asymptotic normality of the proposed test under the null hypothesis is established, and the finite sample properties in terms of empirical size and power are studied through a simulation study. Finally, the new test is applied on a study of subjects undergoing bone marrow transplantation.
Keywords: Competing risk; Fractional risk set; Logrank test; Right censoring; Weighted Kaplan-Meier.