Hazard Regression

In survival analysis * , proportional hazards models (see PROPORTIONAL HAZARDS MODEL, COX'S) are commonly used to estimate covari-ate effects. Two advantages of this approach are that the interpretation of the results is similar to that for ordinary linear models, and that the effects are estimated regardless of the baseline hazard function. Inspired by the success of polynomial splines and their tensor * products in adaptive multiple regression (MARS, Friedman [6]), Kooperberg et al. [11] developed a similar adaptive hazard regression (HARE) methodology for estimating the conditional log-hazard function based on possibly censored survival data with one or more covariates. This methodology circumvents the propor-tionality used in proportional hazards models while still retaining the usual interpretation of the estimated effects. HARE also provides greater flexibility in modeling these effects through the use of polynomial splines (see SPLINE FUNCTIONS) and stepwise addition and deletion of basis functions. Early attempts to use splines in survival analysis are described in Abra-hamowicz et al. Sleeper and Harrington [15], and Whitte-more and Keller [17]. Other nonparametric methods, such as kernel estimates, have been used to test for nonproportionality [7]. Intra-tor and Kooperberg [10] compare the use of trees and splines in survival analysis. In this entry, hazard regression refers to the HARE methodology [11]. Some authors use ''hazard(s) regression'' or ''proportional hazard(s) regression'' to refer to the Cox proportional hazards model [3]. Let T be a (nonnegative) survival time whose distribution may depend on a vector x = (x 1 ,. .. , x M) of covariates ranging over a subset χ = χ 1 × · · · × χ M of R M. Suppose f (t|x), F(t|x) = t 0 f (u|x)du, λ(t|x) = f (t|x)/[1 − F(t|x)], and α(t|x) = log λ(t|x) denote the corresponding conditional density, distribution, hazard, and log-hazard functions , respectively. Let G be a p-dimensional linear space of functions on [0, ∞) × χ , and let B 1 ,. .. , B p be a basis, i.e., a collection of basis functions, of G. The HARE model for the log-hazard function is given by α(t|x; β) = p j=1 β j B j (t|x), t 0. (1) The coefficient vector β = (β 1 ,. .. , β p) in (1) is estimated by the maximum likelihood * method. Specifically, consider n randomly selected individuals. For 1 i n, let T i be the survival time, C …