Long-term survival models with overdispersed number of competing causes

We introduce a new class of long-term survival models by assuming that the number of competing causes, say NN, belongs to a class of mixed Poisson distributions, which are overdispersed. More specifically, we suppose that N|ZN|Z follows a Poisson distribution with mean λZλZ, with λ>0λ>0, and ZZ is a positive continuous random variable belonging to the exponential family. With this, we obtain a general class for NN, which includes, for example: negative binomial, Poisson-inverse gaussian and Poisson generalized hyperbolic secant distributions. Therefore, our long-term survival models can be viewed as heterogeneous promotion models. We present some statistical properties of our models and show that the promotion model is obtained as a limiting case. Some special models of the proposed class are discussed in details. We consider the expected number of competing causes depending on covariates, so allowing to a direct modeling of the cure rate through covariates. Estimation by maximum likelihood and inference for the parameters of models are discussed. In particular, we state sufficient conditions for the maximum likelihood estimators to be consistent and asymptotically normally distributed. A small simulation study is presented in order to check the finite-sample behavior of the maximum likelihood estimators and to illustrate the importance of our models when significant covariates are non-observed. We analyze a real data set from a melanoma clinical trial to illustrate the potential for practice of our proposed models.