Why don’t we just model the hazard ratio directly as a linear function of the covariates in a cox regression model? Came across this question in my notes and I wish for some confirmation in my answer. The full question is : why don't we just model the hazard ratio $\phi= \frac{\lambda_{1}(t)}{\lambda_{0}(t)}$, directly as a linear function of the covariates Z using the $\log(\phi)$? I think the answer lies in the scenario where $\lambda_{0}(t)$ follows a known distribution. Then when we log the hazard ratio we cannot assume that $\log(\lambda_{0}(t))$ follows that same distribution or any distribution at all. Can anyone confirm or correct my answer? Thank you in advance.

The hazard is the expected number of events per time period, so it must be non-negative. The ratios of two hazards, i.e. two non-negative numbers, must thus also be non-negative. A linear function can result in any positive or negative real value. So by modeling the hazard ratio as a linear function you can easily end up with impossible predictions. The logarithm of the hazard ratio can take negative and positive numbers, so modeling that linearly will not result in impossible predictions.
