In the Cox model, how is modelling the hazard equivalent to modelling time to event? In the Cox model, the dependant variable is often stated as time to event, e.g.
https://www.statsdirect.com/help/survival_analysis/cox_regression.htm
even though in the regression formula we model the hazard, h. I am aware of the relationship:
h(t) = f(t) / S(t)
where f(t) is the pdf of survival time and S(t) the survival function.
I have seen lots of formulae using terms like above but these seem to be referring to rates and probabilities, not "time" itself, even though they are functions of time.
I did notice this interesting comment:
This just to clarify that the interpretation will not be different if you look at the same problem in terms of survival time.
From here:
What is the dependent variable in a coxph regression in R?
but it does not go into detail. I would be most grateful for some explanation of how modelling the hazard is equivalent to modelling time to event (if ive understood correctly). If im totally on the wrong track, I would equally appreciate somebody correcting me! Thank you!
 A: Fitting a Cox model to get regression coefficients doesn't directly involve time, only the ordering of events in time. The specific value of the hazard at each event time cancels out in the calculation. After the model has been fit, you can then estimate the underlying cumulative baseline hazard around which the covariates and their associated coefficient estimates work. See this page, for example. That re-introduces the original time scale.
The point that I think your quoted comment is making is that a covariate that leads to a higher hazard leads to a shorter estimated survival time. Don't over-interpret that. It's about direction, not specifically about the magnitude of the association. The specific form of the relationship depends on the baseline hazard as a function of time.
That's easiest to think about when the covariate values are constant over time. Drawing on this answer, for time $t$, a baseline cumulative hazard estimate $\hat{H}_0(t)$ and a set of covariate values $x_j$ with a corresponding vector of coefficient estimates $\beta$, the estimated survival as a function of time is:
$$\hat S(t;x_j) = \exp\left(-\hat{H}_0(t) \exp (\beta^T x_j)\right)$$
So covariates that increase the linear predictor $\beta^T x_j$ necessarily decrease the estimated survival as a function of time.
