Multi-parameter regression survival models

In general, parametric regression models can be motivated by allowing the parameters of a probability distribution to depend on covariates. Furthermore, it is standard practice to relate covariates to one parameter of particular interest; we will refer to this approach as single parameter regression (SPR). In these SPR models, the role of the other (covariate independent) parameters is often little more than to provide the model with sufficient generality to adapt to data. A more flexible approach is to also regress these other parameters on covariates; we call this multi-parameter regression (MPR). The primary focus of this thesis is the development of MPR models in the setting of survival analysis (of course, MPR models are not limited to the field of survival analysis). In Chapter 1 we review some basic concepts of survival analysis - these are standard and may be skipped by the reader familiar with the area. Chapter 2 is largely concerned with developing likelihood theory for survival data which, again, is quite standard and may be skipped. However, in Section 2.3.2 we propose a method - m.l.e. simulation - for calculating the standard error / confidence intervals for functions of parameters. M.l.e. simulation, which competes with the well-known delta method and method of bootstrapping, is based on simulating a sample of ˆθ vectors, {ˆθ(1), . . . , ˆθ(m)}, from ˆθ(b) ∼ N(ˆθ, ˆ ) and is used throughout the thesis. In Chapter 3 we discuss a method for simulating survival data and, furthermore, we extend this method to handle models that support a cured proportion (Section 3.5). This is followed by some interesting simulation studies (Section 3.6) where, among other things, we compare the delta method to m.l.e. simulation and investigate how reliably the cured proportion can be estimated (if it exists). We consider standard regression models for survival data in Chapter 4; in particular, Section 4.18 contains a brief review of some commonly used SPR survival models. Chapter 5 contains our development of MPR survival models: we display the flexibility of MPR (relative to SPR) and discuss the consequences of the approach in terms of interpreting covariate effects (via the hazard ratio), carrying out hypothesis tests (on regression coefficients) and variable selection procedures. Motivated by the need to enhance interpretability of MPR models (and indeed any regression model), in Chapter 6 we propose a least squares approximation to covariate-dependent model quantities, e.g., the hazard function. The proposed method allows straightforward interpretation of covariate effects in terms of the quantity in question but, of course, depends on the adequacy of the approximation. In Chapter 7 we consider frailty modelling - an area of survival analysis concerned with the analysis of unexplained variation (or heterogeneity). In particular, we go through the straightforward algebra of multiplicative gamma frailty which can be used to generalise any parametric model, e.g., Weibull MPR model with multiplicative gamma frailty. Furthermore, using gamma frailty as our starting point, we propose some extensions which combine the ideas of multi-parameter regression and frailty. Finally, we close with a discussion in Chapter 8.