Logistic regression models the log odds of an event as some set of predictors. That is, $\log(p/(1-p))$ where p is the probability of some outcome. Thus, the interpretation of the raw logistic regression coefficients for some variable ($x$) has to be on the log odds scale. That is, if the coefficient for $x = 5$ then we know that a 1 unit change in x correspondents to 5 unit change on the log odds scale that an outcome will occur.
However, I often see people interpret exponentiated logistic regression coefficients as odds ratios. However, clearly $\exp(\log(p/(1-p))) = p/(1-p)$, which is an odds. As far as I understand it, an odds ratio is the odds of one event occurring (e.g., $p/(1-p)$ for event A) over the odds of another event occurring (e.g., $p/(1-p)$ for event B).
What am I missing here? Is seems like this common interpretation of exponentiated logistic regression coefficients is incorrect.
Best Answer
@Laconic's answer is great and complete, in my opinion. Something I wanted to add is that the original coefficients describe a difference in the log odds for two units who differ by 1 in the predictor. E.g., for a coefficient on $X$ of 5, we can say that the difference in log odds between two units who differ on $X$ by 1 is 5. Mathematically,
$$\beta = \log(\text{odds}(p|X=x_0+1))-\log(\text{odds}(p|X=x_0)) $$
When you exponentiate $\beta$, you get
$$\exp(\beta) = \exp(\log(\text{odds}(p|X=x_0+1))-\log(\text{odds}(p|X=x_0))) \\ = \frac{\exp(\log(\text{odds}(p|X=x_0+1)))}{\exp(\log(\text{odds}((p|X=x_0)))} \\ = \frac{\text{odds}(p|X=x_0+1)}{\text{odds}(p|X=x_0))}$$
which is a ratio of odds, an odds ratio.