# Logistic Binary or Poisson?

My outcome had a prevalence of 51% (yes, disease). I need to know which is the best regression to use Binary or Poisson Logistics in the identification of the associated factors. All my data is categorical (male, female / yes, not / younger, older) .... What are the implications on my results of using one regression instead of another?

• I never heard of "Poisson Logistics"? Any reference? Fitting Poisson with logistic function? Commented Apr 24, 2017 at 1:32
• @SmallChess My best guess is actually the opposite - you can use Poisson regression to approximate a binomial distribution. Commented Apr 24, 2017 at 1:36

## 2 Answers

What the "best" regression to use is will depend on many factors. And, as @SmallChess has noted, I've never heard of "Poisson Logistics" despite doing an awful lot of disease related research, so I'm going to guess and give you three options for what I think you mean:

1. Logistic Regression: Conventionally, in clinical and epidemiological studies, logistic regression is used to study case-control data with rare outcomes, as in that case the odds ratio obtained from logistic regression approximate the relative risk you would obtain from a cohort study. You don't specify if your data is a case-control study or not. If it is, and you use logistic regression, in your case the odds ratio does not approximate the relative risk.
2. Binomial Regression: Binomial regression can be used in a cohort of patients with binary outcomes to direct estimate the relative risk. If you can use binomial regression, it's good to try it. Unfortunately, unlike logistic regression, binomial regression models are often sometimes poorly behaved, and have trouble converging.
3. Approximation of a Binomial model using Poisson regression: Poisson regression with a robust variance estimate can be used to approximate what you'd get from a binomial regression model, and is likely better behaved. See Zou G. A modified Poisson regression approach to prospective studies with binary data. Am J Epidemiol. 2004;159(7):702–706. for a theoretical treatment of the topic, and Rivers, C., M.S. Majumder and E.T. Lofgren. Risks of Death and Severe Disease in Patients With Middle East Respiratory Syndrome Coronavirus, 2012–2015. Am J Epidemiol. 2016;184(6): 460-4. for an example of using this in practice.
• Sorry for the language, I'm not native. In a journal submission, the reviewer suggested that I use Poisson regression instead of binary logistical because of the high prevalence found (51%). I do not know to what extent this is really important given that the study design is cross-sectional. Is the reviewer correct? Commented Apr 24, 2017 at 1:53
• @AndréAraújo The reviewer is suggesting #3 in my list, which is appropriate for cross-sectional data. Commented Apr 24, 2017 at 1:54
• @DanChaltiel Your link leads to a login screen. Commented Oct 6, 2018 at 0:28
• @DanChaltiel The Zou paper is mentioned as #3 in my answer... Commented Oct 7, 2018 at 18:36
• @arielhasidim Yes, that's correct. The "log" link function is what I'm referring to. Commented Jun 9, 2023 at 7:03

Your reviewer suggested the right thing which is to choose Poisson regression. And I disagree with Formite in that the reasoning to choose Poisson regression over logistic is not to approximate binomial model. This is a very poor justification. The only reason to choose Poisson regression is because you are doing a large cross-sectional study, which means the total sample including all cases and controls is a random variable following Poisson distribution, as opposed to the binomial (number of either exposed or diseased fixed) or multinomial model (total sample size fixed).

Whether to choose logistic or Poisson regression depends on your sampling scheme. Logistic regression enjoys certain attractive properties such as the unbiasedness of the regression coefficient estimators under different sampling scheme including case-control sampling. That said, you could still use a logistic regression in your case, however, the resulting estimator will yield a higher variance than the one from a Poisson regression so why to use it? Additionally, odds ratio in logistic regression only gives information about the odds but not the risk or rate for each exposure group (unless the disease is rare) and in some research, the risk/rate for each exposure can be of interests.

However, I'm not sure why your reviewer suggested Poisson regression based on the crude prevalence of the disease. This does not sound conventional and perhaps, you could as him/her why? Does this mean if the disease prevalence is low, you are to use logistic regression? This is a quite strange justification. Please keep me posted if you find out.