Gibbs sampling for long-term survival data with competing risks

Many researchers use the 5-year survival probability as a measurement of cure for long-term survival data. In addition to this probability, a mixture model with possibility of cure provides a predictive probability of cure given that a patient has survival to a period of time. Such probabilities are more realistic in measuring the effectiveness of the treatment and predicting the long-term survival of the patient than the 5-year survival rate. An extension of the cure model to the competing risks data is developed. The extended model is a finite mixtures model, where the independence of cause-specific failure times is not assumed. A data set consisting of leukemia with bone marrow transplant is used for illustration. Patients have three possible statuses after transplant: cured by the treatment, relapse of leukemia, or non-relapse-related death. Only the last two events are observable. Patients observed with these endpoints are uncensored cases and the transplant is not successful for them. A case is censored if the case is relapse-free and still alive at the end of its follow-up. Only censored cases have the possibility of being cured, but cure is not assumed to be observable. The status of cure is imputed by the posterior predictive probability of cure given the lifetime and is implemented in the Gibbs sampling. Cure is defined by assuming the risk for failure of a cured patient to be approximately zero. The probability of cure for the leukemia patients after the bone marrow transplant is about 27% for patients with the acute graft-versus-host disease (GVHD) and 46% for the non-GVHD group. The probability of relapse, given that one is not cured, is 0.50 for the non-GVHD group and 0.34 for the GVHD group. The non-GVHD group has a better chance of survival, while the GVHD group has a lower chance for relapse. This is known as the GVHD-versus-leukemia effect.