Hazard Ratios and Survival Curves

The hazard rate is the instantaneous risk of the event:

$\lambda(t) = \lim_{\Delta t \to 0} \frac{P(t \leq T < t + \Delta t | T \geq t)}{\Delta t}$

Interpretation: Rate of events among those still at risk.

The hazard ratio compares hazards:

$HR = \frac{\lambda_1(t)}{\lambda_0(t)}$

Cox model assumes this ratio is constant over time.