Fitting parametric frailty and mixture models under biased sampling

Biased sampling from an underlying distribution with p.d.f.f (t),t > 0, implies that observations followthe weighted distribution with p.d.f. f w(t) = w(t)f (t)/E[w(T )] for a known weight functionw. In particular, the function w(t) = tα has important applications, including length-biased sampling (α = 1) and areabiased sampling (α = 2).We first consider here the maximum likelihood estimation of the parameters of a distribution f (t) under biased sampling from a censored population in a proportional hazards frailty model where a baseline distribution (e.g.Weibull) is mixed with a continuous frailty distribution (e.g. Gamma).A right-censored observation contributes a term proportional tow(t)S(t) to the likelihood; this is not the same as Sw(t), so the problem of fitting the model does not simply reduce to fitting the weighted distribution. We present results on the distribution of frailty in the weighted distribution and develop an EM algorithm for estimating the parameters of the model in the importantWeibull–Gamma case.We also give results for the case where f (t) is a finite mixture distribution. Results are presented for uncensored data and for Type I right censoring. Simulation results are presented, and the methods are illustrated on a set of lifetime data.