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I have some time-to-event data with right-censoring.

Details Added: Each subject is followed up until first event (no recurrent events) or loss to follow-up. Let's assume there are no competing risks.

Given that rates are constant (and other Poisson assumptions hold), and censoring is non-informative, am I correct in thinking that Poisson regression gives unbiased estimates of the incidence rate ratio in the following model?

glm(status ~ trt, offset = log(time), family = poisson(link = "log"), data = df)

A few test datasets have yielded very similar results between the Cox HR and Poisson IRR, but I want to understand whether this is general.

My logic is that any bias in the estimation of the rates is balanced between both groups, leaving ratios unaffected?

ZKA
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  • It's not clear to me from your description where the right censoring comes from and how you are handling right-censored observations in your model. Is this a situation where an individual can have at most one event, and you are simply not recording an event at the last observation time? What determines that last observation time? Or might an individual experience more than one event, and you are recording the number of events over an observation duration? – EdM May 15 '23 at 11:34
  • @EdM Good point - added more info for clarity. Hope that helps. – ZKA May 15 '23 at 17:36
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    If censoring is non-informative, how exactly is either model providing a biased estimation of rates? How is the Cox model providing an estimate of rates at all? – AdamO May 15 '23 at 17:44
  • I think that this page provides the theoretical justification that you're looking for. Please look that over and edit this question further if it doesn't do so. @AdamO I think it's the rate ratio between trt groups that's being estimated in the Cox model. – EdM May 15 '23 at 18:40
  • @EdM a hazard is an instantaneous rate. If the hazards are non-proportional, I can believe that these models estimate fundamentally different things, but otherwise they should be consistent for the same value. Correct? – AdamO May 15 '23 at 18:47
  • @AdamO, Yes, given the Poisson assumption of constant incidence rate, hazards will be proportional => the HR and the IRR should be consistent (and I understand that Cox does not estimate the hazard function, but rather the parametric component; i.e., the betas). Modelling the hazard parametrically with an exponential distr yields the exact same values as the Poisson IRR, but Cox yields ever so slightly larger effect sizes in my simulations. Therefore, I became unsure of whether Poisson / parametric survival modelling is vulnerable to censoring, or if this is just a random finding. – ZKA May 15 '23 at 20:53
  • And thanks @EdM. I think I found something closer to what I was looking for: "Limited Dependent Poisson Regression" by Kurt Brännäs. Bias from censoring shouldn't be a big problem with large lambda -- but I'd love to know if anyone has any resources on individually varying censoring (i.e., where the censoring threshold isn't the same for everyone, which it often isn't with time-to-event). Otherwise, I guess I'll have to start simulating a bit harder. – ZKA May 15 '23 at 21:18
  • @ZKA you still haven't clarified how bias would show up due to non-informative censoring. I have my ideas of what it might be, but I'm curious what you are expecting or projecting from your particular analysis. – AdamO May 15 '23 at 21:56
  • @AdamO Non-mathematically, large values are more affected, so marginal effects will be attenuated. E.g., assume all subjects experience the event at some point, one group has (on avg) longer time-to-event (lower rate) than the other, and the censoring condition has equal and constant rate. Then, both groups will be equally censored for counts but the low-rate group'll have more follow-up time shaved off. Therefore, low-rate group's rate is overestimated relative to high-rate group => bias towards the null

    See also here

     – ZKA May 15 '23 at 23:18
  • I suppose reading these papers* gave me my answer: there will be bias, but magnitude all depends. Wanna be completely sure, perform censored PR.

    *Also: Famoye, Trinh, and Saffari

     – ZKA May 15 '23 at 23:19

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McCullagh and Nelder show in Section 13.3.1 of "Generalized Linear Models" (Second Edition; Chapman & Hall/CRC, 1989) that an exponential survival model is equivalent to a Poisson model with an offset, like you show. If the Poisson model is correct, there should be no inherent bias other than what's already the case for fitting a finite data set by maximum likelihood.

Several of the references you cite note additional bias problems (often important in practice) that arise in count models with things like over-dispersion. The risk of informative censoring can't be forgotten in the context of survival analysis. And I haven't completely thought through the implications of finite-sample bias if there are different numbers of events among treatment groups.

Nevertheless, there is a very close connection between a Cox model and a Poisson model with piecewise-constant baseline hazards. See these course notes by Rodríguez. That should help to explain your observations.

EdM
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