Likelihood Based Inference for Cause Specific Hazard Rates under Order Restrictions

We consider the competing risks model with grouped data or with discrete Failure times where a unit is exposed to several risks, but its eventual failure is due to exactly one of the causes. Nonparametric maximum likelihood estimates of the cause specific hazard rates are obtained under the restriction that these risks are uniformly ordered. We allow for random censoring. Unlike most papers on this topic, no assumption is made about the independence of the various risks, although we do assume that the censoring mechanism is acting independently of the life distribution. We derive the likelihood ratio statistic for testing the null hypothesis of equality of cause specific hazard rates against ordered alternatives. The asymptotic null distribution of the test statistic is seen to be of the chi-bar squared (χ2) type. The procedures developed here are illustrated with the help of an example involving survival rates for mice exposed to radiation.