Is the effect of a 1-unit change in a covariate always constant in Cox- regression?

Question

I need an explanation. For Cox regression, is it true that:

• an effect of one year of age is the same in 50-years old as in 70-years old?
• an effect of one year of age is the same a year after the diagnosis as 5 years after the diagnosis.

Thanks.

Answer 1

The Cox model assumes proportional hazards. That means that the risk of an event at time t is proportional to the risk at baseline (time t=0). This is an assumption that should be tested looking at the data. The model assumption does not make anything be true.

Answer 2

This is going to depend on the situation you are working with and the functional form that you are using for your covariates. Thus, more information about your situation, data and model would be ideal for helping you with this question. However, I do not think that Cox regression vs. OLS regression vs. logistic regression (etc.) has anything to do with it. If you regress $Y$ on $\text{age}$ (where $Y$ is: survival time, a continuous response, a yes/no outcome (etc.), respectively), then each year of $\text{age}$ is associated with the same amount of increase [decrease] (indexed by the beta) in the hazard, value, log odds (etc.). If you were to add a squared term, $\text{age}^2$, then the rate of change would differ depending on the current value of $\text{age}$ from which the change is being assessed. You could also use $\text{ln(age)}$, or any number of other functional forms. Similarly, if you wanted to add a spline term (i.e., $\text{yearsSinceDiagnosis}$), it would yield a constant rate of change unless a squared version of the spline term (for example) were added as well. Thus, if you believe (hypothesize, or know from prior research) that in your situation there is such an effect, you should add additional terms with the appropriate functional forms to your model to assess or capture this effect. However, it has nothing to do with Cox regression per se.