# poisson vs logistic regression

I have a cohort of patients with different length of follow-up. So far I´m disregarding the time aspect and just need to model a binary outcome-disease/no disease. I usually do logistic regression in these studies, but another collegue of mine asked if Poisson regression would be just as appropriate. I´m not that into poisson and was left uncertain as to what the benefits and disadvantages of doing poisson in this setting would be compared logistic regression. I read Poisson regression to estimate relative risk for binary outcomes and I am still uncertain as to the merits of poisson regression in this situation.

## 2 Answers

One solution to this problem is to assume that the number of events (like flare-ups) is proportional to time. If you denote the individual level of exposure (length of follow-up in your case) by $t$, then $\frac{E[y \vert x]}{t}=\exp\{x'\beta\}.$ Here a follow-up that is twice as long would double the expected count, all else equal. This can be algebraically equivalent to a model where $E[y \vert x]=\exp\{x'\beta+\log{t}\},$ which is just the Poisson model with the coefficient on $\log t$ constrained to $1$. You can also test the proportionality assumption by relaxing the constraint and testing the hypothesis that $\beta_{log(t)}=1$.

However, it does not sound like you observe the number of events, since your outcome is binary (or maybe it's not meaningful given your disease). This leads me to believe a logistic model with an logarithmic offset would be more appropriate here.

This dataset sounds like a person-years dataset, the outcome being an event (is this correct?) and uneven follow-up until the event. In that case, this is sounds like a cohort study of some sort (assuming I understood what is being researched), and thus, either poisson regression OR a survival analysis may be warranted (kaplan-meier & cox-proportional hazards regression).

• Wouldn't the response be more like binomial than Poisson? – Martijn Weterings Apr 23 '18 at 6:08
• True, but a 0/1 response (binomial) dataset can be turned into a count dataset. Effectively, you collapse into groups/strata by predictors, then sum the number of events and separately the number of person years. Time-to-event (survival data) can be analysed as survival or as count data, the simpler option often is the survival analysis. – Nicolas Smoll Apr 24 '18 at 6:27
• Isn't that like turning a 0/1 response (Bernouilli) dataset, into a count dataset. You only end up with a Poisson distribution/process by approximation of the Binomial distribution (for the finite cohort size). – Martijn Weterings Apr 24 '18 at 8:06
• @NicolasSmoll "True, but a 0/1 response (binomial) dataset can be turned into a count dataset." How to do that? – vasili111 Jul 9 at 14:45