A generalized self-consistency approach to semiparametric survival models

Introducing a random effect into the Cox model is a useful tool for building hierarchical families of univariate semiparametric regression survival models. Hougaard (1984) used the Laplace transform to build frailty models with explicitly defined survival functions and random effects. The family derived from stable distributions was then extended (Aalen, 1992) to frailty variables following a Discrete--Continuous compound (Poisson--Gamma) structure. Still, in this form the techniques applies only to a subset of frailty models.; In the first part of this paper we extend the idea of compounding, first to arbitrary, frailty models and then to non-frailty Nonlinear Transformation Models (NTM). The EM algorithm can be used to provide inference with frailty models. Motivated by second moment properties of frailty models, Tsodikov (2003) generalized the EM algorithm into a non-frailty frame represented by the Quasi-EM algorithm (QEM). We derive a chain rule showing that QEM will fit any model constructed using the new composition technique, provided it is applicable to the submodels. Simulations, real data and a variety of models are used to illustrate the composition technique. The non-identifiability aspect of semiparametric frailty models is discussed. Many important modelling issues and links are highlighted.; A bivariate distribution function H(x, y) with marginals F(x) and G( y) is said to be generated by an Archimedean copula if it can he expressed in the form H(x, y) = o -1 [o{lcub}F(x){rcub} + o{lcub} G(y){rcub}] for some convex, decreasing function o defined on (0,1] in such a way that o(1) = 0. Frailty models also fall under this general prescription and therefore Archimedean copulas can be interpreted as NTMs. We extend the univariate QEM algorithm to bivariate QEM algorithm to fit bivariate frailty models. Under some conditions, we can verifying the bivariate QEM approach is applicable to any Archimedean copulas. By using the composition technique, we can incorporate other covariates into the model and similar to the univariate case a chain rule for the bivariate case still exists.