Survival Analysis: Cox Proportion Assumption

Question

I know i'm missing something here, please help me understand the cox proportion assumption. What is the point of having a hazard rate function over time if it first has to meet the cox proportion assumption. i.e. surgery risk is higher immediately after surgery and declines over time, and therefore this scenario will not meet the cox proportional hazard assumption. How then can the hazard rate function (instantaneous probability of surviving at specific point in time) be useful? isn't that the point of hazard rate? to see change in hazard over time?

Is this assumption only violated if i include categorical predictors (flags of high risk or low risk) into the model? What about continuous variables like age? Will every increase in unit (years) has to be in the same proportion to other relative years?

thanks

Answer 1

Fomite does not really answer your question and is partially false.

First, what is proportional between the exposure groups is NOT the survival probability (as Fomite mentioned) but the hazard rate. It is important to distinguish the two. If you plot two survival probabilities with a proportional rate, you will immediately see that they are not proportional.

Second, for the proportional hazard assumption, suppose $\lambda_0$ is the rate of lung cancer for a non-smoker and $\lambda_1$ for a smoker. With proportional hazard assumption, $\lambda_1 = \alpha \lambda_0$, that is the rate of lung cancer for a smoker is proportional to the rate for a non-smoker by $\alpha$. The same argument applies with quantitative variables. Say $\lambda_x$ represents rate for individuals who smoke $x$ cigarettes. Then $\lambda_1 = \alpha \lambda_0$, $\lambda_2 = \alpha \lambda_1$, $\lambda_3 = \alpha \lambda_2 ...$

Now, it is true that the proportion of the rates may change over the range of covariates. In my earlier example with number of cigarettes smoked, this can be $\lambda_1 = \alpha_1 \lambda_0$, $\lambda_2 = \alpha_2 \lambda_1$, $\lambda_3 = \alpha_3 \lambda_2 ...$ where $\alpha_1 \neq \alpha_2$ and so on. Why we fix the proportion to a constant is based on statistical reasoning. By assuming a proportional hazard over the range of the covariates, we are only concerned with one parameter, the proportion. Can you imagine you have to estimate the proportion at each value of the covariate? This would yield a highly unreliable estimates because you will likely have few events occuring at a single value of the covariate. Please note that we apply the same reasoning for the logistic regression in which we assume the odds are proportional, making the model parsimonious.

Lastly, the surgery example is a bit different topic. This is something called a time-dependent variable. Obviously, the rate of death would increase considerably after surgery so you account for the surgery in your regression as a time-dependent covariate. Now, going back to the proportional assumption, the rate of death for no surgery is proportional to the rate for surgery.

Answer 2

The Proportional Hazards Assumption is an assumption that the hazard function is proportional between two (or more) groups.

So, for example, if you have "High Risk" and "Low Risk" patients, their survival functions can change over time, but as long as they change in a way where they are still proportional to one another, the assumption is met.

What you're describing the assumption that the hazard is constant, which is something the Cox model does not assume, but some other models do.