# Hazard function for proportional odds model

The Cox proportional hazards model for survival data with covariate ${\bf z}$ is defined through the hazard function $h(t,{\bf z})$ by $$h(t,{\bf z}) = h_0(t)~\cdot\theta~~,~~~\theta = \theta(\beta, {\bf z}).$$The proportional odds model is sometimes given by

$$O(t,{\bf z}) = O_0(t)\cdot\theta$$ where

$O_0(t) = O(t,{\bf 0})$ and $$O(t,{\bf z}) = \frac{1-S(t,{\bf z})}{S(t,{\bf z})}$$ is the odds of an event occurring in the time interval $(0,t)$ for an individual with covariate ${\bf z}.$

My question is this: Is there a way to define the proportional odds model via its hazard function?

• Relevant: the complementary log-log link can be used to model discrete survival, cases where "time-to-event" is either aggregated over intervals (such as number of drug injections until infection from hepatitis C, or such). – AdamO Jan 10 '18 at 20:04

Yes. You can use the relationship between the survival and hazard function: $$\lambda(t) = \frac{d}{dt} -\ln S(t) \iff S(t) = e^{-\int_0^t \lambda(t)\,dt}.$$
This gives $$O(t) = \frac1{S(t)} - 1 = -1 + e^{\int_0^t \lambda(t)\,dt}$$ which you can simplify however you want.