Cox proportional hazards. What does constant over time mean?

Question

I am studying the Cox PH model. But there is one thing I cannot seem to wrap my head around. In the Cox PH model, we want the hazard ratio to be constant over time. But what is time in this case?

For example, when I study the impact of a rapid change in blood pressure on survival, I could argue that the impact of this change is large at the moment of this change, but becomes smaller over time for that individual. Could this possibly imply non-proportionality? Or is non-proportionality that the impact changes over the years. That is, people five years ago reacted differently on this change and over time the ratio of people with rapid change in blood pressure and without a rapid change in blood pressure is non-constant over these five years.

Related, what does $t$ mean in the Cox PH model:

$$\lambda(t|x)=\lambda_0(t)\text{exp}(\beta'x)$$.

Is it the absolute time? For example, the study was from 2000 to 2002. Then t denotes the time between 200 and 2002. Or is it the time the patient spends in the study. For example, the patient started participating in the study starting in 2001 and died in 2002, hence t=1 year.

Answer 1

I am a little bit confused by your question. I hope the following points can help you though they are not strictly said.

1. Time alignment is important in survival analysis. You can either align participants by (1) entry of study, (2) biological clock (e.g., age), or (3) clinical events (e.g., onset of disease).

2. For the example you mentioned, are you thinking of the change in blood pressure $x$ as a time-varying covariate/predictor? If so, you need to modify the model by letting $x$ be a function of time as well: $$ \lambda(t)=\lambda_0(t)\exp(\beta x(t)). $$

3. Hazard rate can be interpreted as the instantaneous probability of survival/event outcomes in the next small time interval. For example, if the time interval is from [0 day,100 day], given that a participant has survived until the 30th day, the probability of having survival outcome is 0.0005 on the 30th day. Then $\lambda(30)$ is approximately 0.0005/1day=0.0005.

4. Proportional hazards mean that for a fixed time $t$, the log of hazard increase linearly as $x$ (or say $x(t)$) increase.