3.7 Interpretation of parameter estimates

In the proportional hazards model the parameter estimates are logarithms of risk ratios relative to the baseline hazard. The precise interpretations of the coefficients for the two types of covariates are discussed. The conclusion is that \(e^\beta\) has a more direct and intuitive interpretation than \(\beta\) itself.

3.7.1 Continuous covariate

If \(x\) is a continuous covariate, and \(h(t; x) = h_0(t)e^{\beta x}\), then

\[\begin{equation*} \frac{h(t; x+1)}{h(t; x)} = \frac{h_0(t)e^{\beta (x+1)}} {h_0(t)e^{\beta x}} = e^\beta. \end{equation*}\] so the risk increases with a factor \(e^\beta\), when \(x\) is increased by one unit. In other words, \(e^\beta\) is a relative risk (or a hazard ratio, which often is a preferred term in certain professions).

3.7.2 Factor

For a factor covariate, in the usual coding with a reference category, \(e^\beta\) is the relative risk compared to that reference category.