Efficient semiparametric mixture inferences on cure rate models for competing risks

Cancer patients may die from causes other than the diagnosed cancer. In a study of patients treated for soft tissue sarcoma, the patients may die from the disease or die without experiencing disease recurrence. In addition, a substantial proportion of the patients will remain cancer‐free after surgical resection of the tumour, and therefore will not be at increased risk of any type of failure. Our goal is to describe the effect of adjuvant chemotherapy simultaneously on the probabilities of long‐term survival, death from cancer, or death from other causes. To this end, we propose a semiparametric mixture model to determine the effects of factors on the probability of occurrence, allowing the surviving fraction, and the hazard rate conditional on each of the failure types. These quantities are combined in the mixture approach using a multinomial logistic model and a class of semiparametric transformation models. Estimation of the regression and nonparametric parameters is achieved with a novel nonparametric maximum likelihood approach. Statistical inferences can be conveniently made from the inverse of the observed information matrix. Simulation studies show that the procedures work well in practical settings. The methodology is illustrated with data from the soft tissue sarcoma study. The Canadian Journal of Statistics 43: 420–435; 2015 © 2015 Statistical Society of Canada