Inference for the poly-Weibull model

Summary. We consider inference for the poly-Weibull model, which arises in competing risk scenarios when the risks have independent Weibull distributions and it is not known which is responsible for the failure. Real and generated data sets illustrate our approaches to inference, which in addition to standard likelihood methods include Bayesian inference by Laplace's method for analytical approximation of integrals and sampling-importance resampling. The poly-Weibull distribution is that of the minimum of several independent Weibull random variables. This arises in scenarios of competing risks (Cox and Oakes (1984), chapter 7), where failure may be due to several causes, each supposed to follow a Weibull law. As we shall see, its advantage over the Weibull model is that it allows not only increasing, constant or decreasing hazard functions with zero or non-zero asymptotes but also non-monotone hazard functions, for example with a 'bathtub' shape. This is an attractive property because such hazards are not uncommon in practice; they correspond to an initial high failure rate a 'bum-in' period followed by lower odds of failure, which eventually increase. See Blackstone et al. (1986) for an application of such hazards in the context of survival after heart surgery. This paper discusses inference for the model, which deserves to be better known. We show by example that standard likelihood or Bayesian inference is straightforward, despite the assertion by Berger and Sun (1993) that a computation of the likelihood is 'typically not feasible', a belief that led them to apply Gibbs sampling for Bayesian inference. In fact the likelihood is so simply obtained that Markov chain simulation is unnecessary, and either Laplace approximation (Tiemey and Kadane, 1986) or the Bayesian bootstrap (Smith and Gelfand, 1992) may be applied; neither has the difficulties of assessing convergence that are associated with Markov chain methods. This reduces considerably the computational burden in obtaining posterior probabilities, densities or moments and widens the class of prior densities that can easily be used because they need not be chosen mainly for computational convenience. The chief difficulty in using the likelihood is possible non-identifiability of the model parameters, but we give some practical suggestions for detecting this.