# Interpretation of Coefficient

In Poisson Regression, suppose we have the following regression equation: $\ln(E(Y_i|X_1)) = \beta_0+ \beta_{1}X_{1}$ where $Y$ is the number of heart attacks. Also suppose $X_1$ is a binary variable (e.g. smokers versus non-smokers). How would you interpret the coefficient $\beta_1$? Smokers, on average, have $\exp(\beta_1)$ more heart attacks than non-smokers?

• Related question: stats.stackexchange.com/questions/11096/… – user5594 Feb 26 '12 at 1:04
• You're right, Mike: the first part of this question and the one you reference are exactly the same. Normally that would be grounds to close this thread and merge the replies. But I notice that the second part of this question (the last line) does not appear in the duplicate and is addressed by a reply offered by @stask below. – whuber Feb 26 '12 at 19:53

## 1 Answer

It sort of depends on how your data are structured.

If you have counts in cross-sectional groups, then the Poisson model is typically used to answer the question of more frequently rather than simply more. The RHS is then interpreted as the rates of events, but to make the comparison of the rates meaningful, you need to offset (*) the number of cases by the population size on which this regression is based. If you have 1000 non-smokers and 100 smokers, then with heart attack rates of 5% and 20%, respectively, you would see about 50 events for non-smokers and about 20 for smoker, so smokers would have a lower coefficient if you don't standardize by the (vastly different) population size.

(*) An offset is a variable that enters the regression with a fixed coefficient of 1.

If you have longitudinal data on the same people with different number of heart attacks over the lifetime, then the interpretation you offered is valid, although you might still want to offset by the duration of observation -- having 5 heart attacks over 40 years is arguably not as bad as 3 over 10 years.

• Poisson regression can be used to measure counts, as well. In fact, I think that's the usual use. See e.g. Wikipedia. – Peter Flom Feb 25 '12 at 23:51
• @PeterFlom, this is true, of course. There are many context in which Poisson regression can be used, I responded to one of them, and should have clarified it. – StasK Feb 26 '12 at 0:12