ABSTRACT
The Cox regression model is clearly the most used hazards model when analyzing survival data as it provides a convenient way of summarizing covariate effects in terms of relative risks. The proportional hazards assumption may not always hold, however. A typical vi-
assumption is time-changing covariate effects which is often encountered in bio-medical applications. A typical example is a treatment effect that varies with time such as treatment efficacy fading away over time due to, for example, tolerance developed by the patient. Under such circumstances, the Aalen additive hazard model (Aalen, 1980) may provide a useful alternative to the Cox model as it easily incorporates time-varying covariate effects. For Aalen’s model, the hazard function for an individual with a p-dimensional covariate X vector takes the form
α(t|X) = XTβ(t). (3.1) The time-varying regression function β(t) is a vector of locally integrable functions, and usually the X will have 1 as its first component allowing for an intercept in the model. Model (3.1) is very flexible and can be seen as a first order Taylor series expansion of a general hazard function α(t|X) around the zero covariate:
α(t|X) = α(t, 0) +XTα′(t,X∗) with X∗ on the line segment between 0 and X. As we shall see below, it is the cumulative regression function
B(t) =
β(s) ds
that is easy to estimate and its corresponding estimator converges at the usual n1/2-rate. Estimation and asymptotical properties were derived by Aalen (Aalen, 1980), and further results for this model were given in Aalen (1989), Aalen (1993), and Huffer and McKeague (1991) derived efficient estimators.
