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In Epidemiology, an incidence rate ratio is the ratio between the incidence rate in the exposed and the incidence rate in the unexposed, where an incidence rate is the number of events divided by the total person-time at risk. We can calculate incidence rates and rate ratios from longitudinal studies with time-to-event outcomes, including allowing for multiple events per individual and varying time of follow-up.

We can analyse such studies with Poisson or Cox regression (as well as Kaplan-Meier). These return an output in hazard ratios (HR). I'm aware that a hazard [rate] is evaluated at a specific time point.

Rate ratios and HRs feel conceptually similar in my mind. Is there any substantial difference that I should be aware about when interpreting the results of a Poisson or Cox model?

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You have to be careful with the time period that is used for evaluating the incidence rate. The Wikipedia page discusses the usual interpretation of "incidence rate":

Use of this measure implies the assumption that the incidence rate is constant over different periods of time... When this assumption is substantially violated, such as in describing survival after diagnosis of metastatic cancer, it may be more useful to present incidence data in a plot of cumulative incidence, over time, taking into account loss to follow-up...

A simple Poisson model makes the same assumption of a constant baseline rate/hazard over time, with covariate-outcome associations also assumed to be constant over time. If the assumption is correct, then a Poisson model directly estimates incidence rates and their ratios.

A Cox model makes no assumption about constant rates over time, just that hazard ratios associated with covariate differences are constant over time. This page explains how the different assumptions between Poisson and Cox models can lead to differences between incidence rate ratios and hazard ratios.

It's possible, however, to approximate a Cox model with a series of Poisson models. That's called a "piecewise exponential model." You break up the time axis into a series short time periods. You assume a constant baseline hazard within each individual time period and fit a Poisson model with regression coefficients for covariates assumed to be the same for all periods.

In that context, if you think of something approaching an instantaneous incidence rate ratio, it would be related to a hazard ratio. But I don't think that's the usual interpretation of "incidence rate" or "incidence rate ratio"; the assumption of constant rates over time seems much more typical.

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