I am having trouble linking these concepts together:
- Here is a Cox PH (Proportional Hazards) Regression Model:
$$h(t|X) = h_0(t) \exp(\beta^T X)$$
where:
- $h(t|X)$ is the hazard function for an individual with covariate values $X$.
- $h_0(t)$ is the baseline hazard function, representing the hazard for an individual with $X=0$. -$\beta$ is a vector of regression coefficients.
- Here is a Martingale:
A martingale $M_t$ is a stochastic process satisfying the following property:
$$E[M_{t+1} | M_1, M_2, ..., M_t] = M_t$$
- Here is the Proportional Hazards assumption for a Cox PH model:
The hazard ratio between any two individuals is constant over time:
$$\frac{h(t|X_1)}{h(t|X_2)} = \exp(\beta^T (X_1 - X_2))$$
where $X_1$ and $X_2$ are two sets of covariate values, and $\beta$ is the vector of regression coefficients. This implies that the effect of covariates on the hazard rate is constant over time.
But what I don't understand is that how/why are martingales useful at validating the proportional hazards assumption in Cox PH?
The only thing which comes to mind is that residuals have to be without trends and a martingale encapsulates this idea via the expectation. But this makes me wonder why martingales are not used in basic regression models (e.g. OLS) to validate assumptions.
In Cox-PH, is it possible to mathematically see:
- why the Proportional Hazards assumption is important (e.g. what happens when it is not met ... perhaps coefficients become biased, large variance, not consistent, etc.)
- what residual distribution/properties are indicative of the Proportional Hazards assumption (e.g. Expected Residual at time t+1 equals expected residual at time t given all histories)
- why martingales are useful at validating if the Proportional Hazards assumption is being met?