Competing risks arise in studies in which individuals are exposed to two or more mutually exclusive failure events, denoted by . When a failure occurs, you observe the time T and the cause of failure . The cumulative incidence function (CIF), also known as the subdistribution function, for failures of cause j is the probability

The nonparametric analysis of competing-risks data consists of estimating the CIF and comparing the CIFs of two or more groups.

For a set of competing-risks data with causes of failure, let be the distinct uncensored times. For each , let be the number of subjects at risk at , and let be the number of failures of cause j at . Let be the Kaplan-Meier estimator that would have been obtained by assuming that all failure causes are of the same type. Denote .

The nonparametric maximum likelihood estimator of the CIF of cause j is

PROC LIFETEST provides two standard error estimators of the CIF estimator: one is based on the theory of counting processes (Aalen 1978), and the other is based on the delta method (Marubini and Valsecchi 1995). You use the ERROR= option in the PROC LIFETEST statement to choose the standard error estimator. The default is the Aalen estimator (ERROR=AALEN). Denote .

Let K be the number of groups. Consider failure of type 1 to be the failure type of interest. Let be the cumulative incidence function of type 1 in group k. The null hypothesis to be tested is

Gray (1988, Section 2) gives the following K-sample test procedure for testing . Let be the observed data in the kth group. Without loss of generality, assume that there are only two types of failure (). The number of failures of type j by t is

and the number of subjects at risk just before t in group k is

For group k, let be the Kaplan-Meier estimator of the survivor function that you obtain by assuming that all failure causes are of the same type. The cumulative incidence function of type j in the kth group is estimated by

Let be the largest uncensored time in group k. Define

The cumulative hazard of the subdistribution for group k, , is estimated by

Under the null hypothesis , you can estimate the null value of , denoted by , by

The K-sample test is based on , where

You can estimate the asymptotic covariance matrix as

where

Because , only scores are linearly independent. The K-sample test statistic is formed as a quadratic form of the first components of and the inverse of the estimated covariance matrix. Under the null hypothesis , this K-sample test statistic has approximately a chi-square distribution with degrees of freedom.

If you specify the GROUP= option in the STRATA statement, you can obtain a stratified version of the test by computing the contributions to and for each stratum, summing the contributions over the strata, and proceeding as before.