An Algorithm for Fitting Mixtures of Gompertz Distributions to Censored Survival Data

Abstract

We consider the fitting of a mixture of two Gompertz distributions to censored survival data. This model is therefore applicable where there are two distinct causes for failure that act in a mutually exclusive manner, and the baseline failure time for each cause follows a Gompertz distribution. For example, in a study of a disease such as breast cancer, suppose that failure corresponds to death, whose cause is attributed either to breast cancer or some other cause. In this example, the mixing proportion for the component of the mixture representing time to death from a cause other than breast cancer may be interpreted to be the cure rate for breast cancer (Gordon, 1990a and 1990b). This Gompertz mixture model whose components are adjusted multiplicatively to reflect the age of the patient at the origin of the survival time, is fitted by maximum likelihood via the EM algorithm (Dempster, Laird and Rubin, 1977). There is the provision to handle the case where the mixing proportions are formulated in terms of a logistic model to depend on a vector of covariates associated with each survival time. The algorithm can also handle the case where there is only one cause of failure, but which may happen at infinity for some patients with a nonzero probability (Farewell, 1982).