# Poisson regression to estimate relative risk for binary outcomes

Brief Summary

Why is it more common for logistic regression (with odds ratios) to be used in cohort studies with binary outcomes, as opposed to Poisson regression (with relative risks)?

Background

Undergraduate and graduate statistics and epidemiology courses, in my experience, generally teach that logistic regression should be used for modelling data with binary outcomes, with risk estimates reported as odds ratios.

However, Poisson regression (and related: quasi-Poisson, negative binomial, etc.) can also be used to model data with binary outcomes and, with appropriate methods (e.g. robust sandwich variance estimator), it provides valid risk estimates and confidence levels. E.g.,

From Poisson regression, relative risks can be reported, which some have argued are easier to interpret compared with odds ratios, especially for frequent outcomes, and especially by individuals without a strong background in statistics. See Zhang J. and Yu K.F., What's the relative risk? A method of correcting the odds ratio in cohort studies of common outcomes, JAMA. 1998 Nov 18;280(19):1690-1.

From reading the medical literature, among cohort studies with binary outcomes it seems that it is still far more common to report odds ratios from logistic regressions rather than relative risks from Poisson regressions.

Questions

For cohort studies with binary outcomes:

1. Is there good reason to report odds ratios from logistic regressions rather than relative risks from Poisson regressions?
2. If not, can the infrequency of Poisson regressions with relative risks in the medical literature be attributed mostly to a lag between methodological theory and practice among scientists, clinicians, statisticians, and epidemiologists?
3. Should intermediate statistics and epidemiology courses include more discussion of Poisson regression for binary outcomes?
4. Should I be encouraging students and colleagues to consider Poisson regression over logistic regression when appropriate?
• If you want a relative risk, why would you not just use binomial regression with log (rather than logistic) link? The mean-variance relationship of the Poisson family is doesn't make a lot of sense if you have conditioned on the number of possible events per observation. Nov 17, 2016 at 16:43
• @AndrewM How would you apply a Binomial regression with log link? Positive values of the regressor would imply probability values larger than 1.
– Rufo
May 17, 2017 at 14:42
• @Rufo: If I understand you, I would call this the linear predictor, rather than regressor. And yes, the parameter space is now constrained so that the linear predictor is negative, unlike the unconstrained case for the logistic link. Your predicted response (on new data) can be outside $[0,1]$, though I believe a MLE will always exist (maybe on the boundary of the parameter space). These models are sometimes finicky to fit. May 18, 2017 at 20:09
• @AndrewM Yes, I ment linear predictor, thank you :). But even when you manage to implement the model, I am not sure it is adequate. As I indicate in a comment in the first answer, if you swap 0s for 1s and vice versa for the response variable, as the log link is not symmetric around 0.5, the estimates of the relative risks are different (exp(beta_M1) =/= 1/exp(beta_M2)). That disturbs me quite a bit.
– Rufo
May 19, 2017 at 10:24
• @Rufo: Of course it isn't reciprocal. You are calculating a relative risk: $P(Y|X)/P(Y|X^c)$ and $P(Y|X)/P(Y|X^c) \neq P(Y^c|X)/P(Y^c | X^c)$, in general, no matter what link function you use. May 19, 2017 at 22:04

An answer to all four of your questions, preceeded by a note:

It's not actually all that common for modern epidemiology studies to report an odds ratio from a logistic regression for a cohort study. It remains the regression technique of choice for case-control studies, but more sophisticated techniques are now the de facto standard for analysis in major epidemiology journals like Epidemiology, AJE or IJE. There will be a greater tendency for them to show up in clinical journals reporting the results of observational studies. There's also going to be some problems because Poisson regression can be used in two contexts: What you're referring to, wherein it's a substitute for a binomial regression model, and in a time-to-event context, which is extremely common for cohort studies. More details in the particular question answers:

1. For a cohort study, not really no. There are some extremely specific cases where say, a piecewise logistic model may have been used, but these are outliers. The whole point of a cohort study is that you can directly measure the relative risk, or many related measures, and don't have to rely on an odds ratio. I will however make two notes: A Poisson regression is estimating often a rate, not a risk, and thus the effect estimate from it will often be noted as a rate ratio (mainly, in my mind, so you can still abbreviate it RR) or an incidence density ratio (IRR or IDR). So make sure in your search you're actually looking for the right terms: there are many cohort studies using survival analysis methods. For these studies, Poisson regression makes some assumptions that are problematic, notably that the hazard is constant. As such it is much more common to analyze a cohort study using Cox proportional hazards models, rather than Poisson models, and report the ensuing hazard ratio (HR). If pressed to name a "default" method with which to analyze a cohort, I'd say epidemiology is actually dominated by the Cox model. This has its own problems, and some very good epidemiologists would like to change it, but there it is.

2. There are two things I might attribute the infrequency to - an infrequency I don't necessarily think exists to the extent you suggest. One is that yes - "epidemiology" as a field isn't exactly closed, and you get huge numbers of papers from clinicians, social scientists, etc. as well as epidemiologists of varying statistical backgrounds. The logistic model is commonly taught, and in my experience many researchers will turn to the familiar tool over the better tool.

The second is actually a question of what you mean by "cohort" study. Something like the Cox model, or a Poisson model, needs an actual estimate of person-time. It's possible to get a cohort study that follows a somewhat closed population for a particular period - especially in early "Intro to Epi" examples, where survival methods like Poisson or Cox models aren't so useful. The logistic model can be used to estimate an odds ratio that, with sufficiently low disease prevalence, approximates a relative risk. Other regression techniques that directly estimate it, like binomial regression, have convergence issues that can easily derail a new student. Keep in mind the Zou papers you cite are both using a Poisson regression technique to get around the convergence issues of binomial regression. But binomial-appropriate cohort studies are actually a small slice of the "cohort study pie".

3. Yes. Frankly, survival analysis methods should come up earlier than they often do. My pet theory is that the reason this isn't so is that methods like logistic regression are easier to code. Techniques that are easier to code, but come with much larger caveats about the validity of their effect estimates, are taught as the "basic" standard, which is a problem.

4. You should be encouraging students and colleagues to use the appropriate tool. Generally for the field, I think you'd probably be better off suggesting a consideration of the Cox model over a Poisson regression, as most reviewers would (and should) swiftly bring up concerns about the assumption of a constant hazard. But yes, the sooner you can get them away from "How do I shoehorn my question into a logistic regression model?" the better off we'll all be. But yes, if you're looking at a study without time, students should be introduced to both binomial regression, and alternative approaches, like Poisson regression, which can be used in case of convergence problems.

• When you say Other regression techniques that directly estimate it [relative risk, I presume], like binomial regression, have convergence issues[...], how would you apply a binomial regression so that it gives you a relative risk? @AndrewM suggests a log link, but I fail to see how would you avoid the problem of having estimations of the probability of success higher than 1.
– Rufo
May 17, 2017 at 14:53
• @Rufo A binomial model with a log-link, when run on a cohort, will estimate relative risk. That these models sometimes estimate probabilities greater than 1 is indeed one of the reasons binomial models are harder to implement than is ideal. But I have succeeded in using them - it's helpful that your data often has probabilities well below 1, so the model may never end up with the problem you're worried about. May 17, 2017 at 15:35
• Would not the log link function give different results deppending on your codification of your response variable? I mean, if you swap 0s for 1s and vice versa, as the log link is not symmetric around 0.5, the estimates for the parameter $p$ given certain values of the covariates and the predictive estimates are different. That disturbs me quite a bit.
– Rufo
May 22, 2017 at 9:09

I too speculate at the prevalence of logistic models in the literature when a relative risk model would be more appropriate. We as statisticians are all too familiar with adherence to convention or sticking to "drop-down-menu" analyses. These create far more problems than they solve. Logistic regression is taught as a "standard off the shelf tool" for analyzing binary outcomes, where an individual has a yes/no type of outcome like death or disability.

Poisson regression is frequently taught as a method for analyzing counts. It is somewhat under emphasized that such a probability model works exceptionally well for modeling 0/1 outcomes, especially when they are rare. However, a logistic model is also well applied with rare outcomes: the odds ratio is approximately a risk ratio, even with outcome dependent sampling as with case control studies. The same cannot be said of relative risk or Poisson models.

A poisson model is useful too when individuals may have an "outcome" more than once, and you might be interested in cumulative incidence, such as outbreaks of herpes, hospitalizations, or breast cancers. For this reason, exponentiated coefficients can be interpreted as relative rates. To belabor the difference between rates and risks: If there are 100 cases per 1,000 person-years, but all 100 cases happened in one individual, the incidence (rate) is still 1 case per 10 person-years. In a health care delivery setting, you still need to treat 100 cases, and vaccinating 80% of the people has an 80% incidence rate reduction (a priori). However the risk of at least one outcome is 1/1000. The nature of the outcome and the question, together, determine which model is appropriate.

I would be concerned with saying "we fit a Poisson regression model for incidence to estimate relative rates" because this may introduce some confusion as to the nature of the outcome and whether one person may experience it more than once. If you are interested in relative risks, you must say so, and be prepared to discuss the sensitivities of the inappropriate variance assumption where the mean is proportional to the outcome when binary events have the following mean variance relationship: $\mbox{var}(y) = E(y)(1-E(y))$

My understanding is that if the scientific interest lies in estimating relative rates, there is a hybrid model: relative risk regression which is a GLM using the logistic variance structure and the poisson mean structure. That is to say: $\log (E[Y|X])= \beta_0 + \beta_1 X$ and $\mbox{var}(Y) = E[Y](1-E[Y])$,

By the way, the Zhang article provides a biased estimate of inference based on the relative risk estimate which doesn't account for variability in the intercept term. You can correct the estimator by bootstrapping.

To answer the specific questions:

1. If the outcome is rare they are approximately the same. If the outcome is common, the variance of the relative rate estimator from the Poisson might be over inflated, and we might prefer the odds ratio as a biased but efficient estimate of association between a binary outcome and several exposures. I also think that case-control studies justify use of the odds ratio as a measure which does not vary with outcome dependent sampling. Scott and Wild 97 discuss methods around this. Of course, other journals might not have dedicated statistical reviewers.

2.3. I think you are blaming and assuming overmuch about what happens in medical review and academics.

1. You should always be encouraging your students to use appropriate models whenever possible.

http://biostats.bepress.com/cgi/viewcontent.cgi?article=1128&context=uwbiostat

• "My understanding is that if the scientific interest lies in estimating relative rates, there is a hybrid model: relative risk regression which is a GLM using the logistic variance structure and the poisson mean structure": Also known as binomial regression with a log link. Nov 17, 2016 at 16:45
• @AndrewM Indeed. In fact, I think that is the preferred language. Thanks for pointing that out. I've edited the question to include a reference to a working paper from Thomas Lumley which emphasizes that the Poisson model is a "working model" in that it is an incorrect assumed mean-variance relationship. Nov 17, 2016 at 19:32
• What you mean by "If the outcome is rare they are approximately the same"? What is the maximum percentage of "rare" outcome in order to use OR instead of RR for estimating prevalence? Jul 9, 2019 at 14:14
• @vasili111 this is a hotly debated topic with no clear answer. Nowadays you see lots of critiques of people making the "rare" assumption when the incidence wasn't that rare at all, such as more than 1/30. And with multivariate models, anything goes! Jul 9, 2019 at 14:52