Event History Analysis with R

Chapter 3

Cox Regression

3.1 Introduction

We move slowly from the two-sample and k-sample cases, where the logrank test is introduced, to the general regression situation. There the logrank test is generalized to Cox regression (Cox 1972). The fundamental concept of proportional hazards is introduced.

3.2 Proportional Hazards

The property of proportional hazards is fundamental in Cox regression. It is in fact the essence of Cox’s simple yet ingenious idea. The definition is as follows.

Definition 1 Proportional hazards

If h1(t) and h0(t) are hazard functions from two separate distributions, we say that they are proportional if

$\begin{array}{ll}\begin{array}{ll}{h}_1\left(t\right)=\psi h_0\left(t\right),\hfill & \text{for all}t\ge 0,\hfill \end{array}\hfill & (3.1)\hfill \end{array}$

for some positive constant ψ and all t ≥ 0. Further, if (3.1) holds, then the same property ...