Likelihood ratio tests for and against ordering of the cumulative incidence functions in multiple competing risks and discrete mark variable models

In this paper we consider the problem of testing the equality of r (r⩾2) cumulative incidence functions against an ordered alternative, using the likelihood ratio approach. We assume a discrete time framework and obtain maximum likelihood estimators of the r cumulative incidence functions under the restriction that they are uniformly ordered. The asymptotic null distribution of the derived likelihood ratio test statistic for testing the equality of the cumulative incidence functions against the alternative they are uniformly ordered is of the chi-bar square (χ¯2) type. In addition to applications within the competing risks setting our methods are also applicable to investigating the association between failure time and a discretized or ordinal mark variable that is observed only at time of failure. We give examples in both the competing risks and mark variable settings and discuss details concerning the implementation of our methods.