Chapter 37

Relative Risk Modeling

Duncan C. Thomas

37.1 Introduction

Risk models are used to describe the hazard function λ(t, z) for time-to-failure data as a function of time t and covariates Z = Z1, …, Zp, which may themselves be time dependent. The term “relative risk models” is used to refer to the covariate part r(.) of a risk model in a proportional hazards form:

(1) equation

where β represents a vector of parameters to be estimated. In the standard proportional hazards model, the relative risk term takes the loglinear form r(Z, β) = exp(Z′β). This has the convenient property that it is positive for all possible covariate and parameter values, since the hazard rate itself must be nonnegative. However, in particular applications, some alternative form of relative risk model may be more appropriate.

First, an aside on the subject of time is warranted. Time can be measured on a number of different scales, such as age, calendar time, or time since start of observation. One of these must be selected as the time axis t for use of the proportional hazards model. In clinical trials, time since diagnosis or start of treatment is commonly used for this purpose, since one of the major objectives of such studies is to make statements about prognosis. In epidemiological studies, however, age is the preferred time axis, because it is usually a powerful determinant of disease rates, but it is not ...

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