I am new to Cox regression / survival analysis, and I wonder whether the coefficient in Cox regression is invariant to changes of scale in the survival variable. That is, do I get the same coefficient estimates regardless of whether the survival variable is measured in years or days, for example?
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Yes, any transformation of the times that preserves the order of the events leaves the estimated regression coefficients (and their standard errors etc.) unchanged. Some software will enforce some conventions such as that the time to event or censoring times has to be positive and records with event times $\leq 0$ will often be ignored.
The reason why the numerical event times do not matter is because the partial likelihood that gets maximized does not actually make use of the numerical event times. Instead it involves a comparison at each time, at which at least one event has occurred, of the records that have an event at that time and those that do not (or at least not until later) amongst those records that are still at risk (i.e. have not had an event at a previous time or were censored before).
However, even in many parametric models a rescaling by a multiplicative factor (say, $c$) will not really matter. E.g. if you have an exponential time to event model (in fact, this should be the case for any proportional hazards model) with the log-hazard rate given by $$\log \lambda_i = \alpha + \boldsymbol{X\beta},$$ where $\alpha$ is an intercept, then changing the scale of the event times will just change the estimated $\hat{\alpha}$ by an addition $\log c$, while the regression coefficients for the covariates would stay unchanged.
