As the title says, I'm trying to figure out how the length of follow-up affects the value of the hazard ratio (e.g. from a Cox model). If proportional hazard holds, then presumably the hazard ratio point estimate should be identical regardless of whether we truncate followup at 1 year, 2 year etc. If it does not hold and the hazard ratio is time varying, only then should the length of follow-up affect the value of the hazard ratio. Is this correct?
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There are a couple of issues to disentangle.
First, if proportional hazards (PH) strictly hold, then the expected value of the log-hazard ratio (model coefficient) should be independent of the duration of observations. But the precision of the estimate will depend on the number of events. If you have a shorter duration of observations, you will have less precision and might end up with sample-based point estimates that differ depending on that duration. In general, the more events you have the better.
Second, a lack of PH might not necessarily lead to differences in estimated hazard ratios over different durations of observations, depending on the details of the deviation from PH. As @AdamO explains on this page, if PH is violated than you get a sort of event-averaged hazard ratio from a Cox model. Depending on the pattern of deviations from PH, two particular choices of observation durations could (in principle) thus lead to similar event-averaged hazard ratios.
Also, note that perfect adherence to the PH assumption is unlikely. You have to evaluate whether any violation of PH is big enough to matter in practice.
A final warning about terminology: in survival analysis the word "truncated" has a particular meaning. It refers to event times that could not have been observed, given the study design.
So it's risky to say that you "truncate followup at 1 year, 2 year etc." That's correct English usage, but it can lead to some confusion. The individuals still event-free after you "truncate followup" have right-censored event times. It's safer to say something like "stop follow-up." See Klein and Moeschberger, or many pages on this site, for the distinction between truncation and censoring in survival analysis.
