Gamma failure‐time mixture models: yet another way to establish efficacy

Using a Yamaguchi‐type generalized gamma failure‐time mixture model, we analyse the data from a study of autologous and allogeneic bone marrow transplantation in the treatment of high‐risk refractory acute lymphoblastic leukaemia, focusing on the time to recurrence of disease. We develop maximum likelihood techniques for the joint estimation of the surviving fractions and the survivor functions. This includes an approximation to the derivative of the survivor function with respect to the shape parameter. We obtain the maximum likelihood estimates of the model parameters. We also compute the variance‐covariance matrix of the parameter estimators. The extended family of generalized gamma failure‐time mixture models is flexible enough to include many commonly used failure‐time distributions as special cases. Yet these models are not used in practice because of computational difficulties. We claim that we have overcome this problem. The proposed approximation to the derivative of the survivor function with respect to the shape parameter can be used in any statistical package. We also address the issue of lack of identifiability. We point out that there can be a substantial advantage to using the gamma failure‐time mixture models over nonparametric methods. Copyright © 2003 John Wiley & Sons, Ltd.