Regression models and subject heterogeneity

Abstract

We consider several models that describe survival in the presence of observable covariates, these covariates measuring subject heterogeneity. The most general situation can be described by a model with a parameter of high, possibly unbounded, dimension. Proportional hazards models, partially proportional hazards models (O'Quigley and Stare 2002), stratified models or models with frailties or random coefficients all arise as special cases of this model (O'Quigley and Xu 2000). One useful parameterization (O'Quigley and Pessione 1991, O'Quigley and Prentice 1991) can be described as a non proportional hazards model with intercept. Changepoint models are a particular form of a non proportional hazards model with intercept (O'Quigley and Natarajan 2004). Any model can be viewed as a special case of the general model, lying somewhere on a conceptual scale between this general model and the most parametric extreme, which would be the simple exponential model. Models can be placed on this scale according to the extent of model constraints and, for example, a random effects model would lie strictly between a stratified model and the simple exponential model. Relative risk models used in epidemiology come under these headings. For relative risk models the time component is usually taken to be age and great generalization, e.g., period or cohort analysis is readily accomplished. Time-dependent covariates, Z(t), in combination with the at-risk indicator, Y(t),can be used to describe states. Multistate models in which subjects can move in and out of different states, or into an absorbing state such as death, can then be analyzed using the same methodology.