Why are exponentiated logistic regression coefficients considered “odds ratios”?

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.

@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.

• This is extremely clear to me. My question is resolved. – jack Aug 9 '18 at 18:51

Consider two set of conditions, the first described by the vector of independent variables $X$, and the second described by the vector $X'$, which differs only in the ith variable $x_i$, and by one unit. Let $\beta$ be the vector of model parameters as usual.

According to the logistic regression model, the probability of the event occurring in the first case is $p_1 = \frac{1}{1 + \exp(-X \beta)}$, so that the odds of the event occurring is $\frac{p_1}{1-p_1} = \exp(X \beta)$.

The probability of the event occurring in the second case is $p_2 = \frac{1}{1 + \exp(-X' \beta)}$, so that the odds of the event occurring is $\frac{p_2}{1-p_2} = \exp(X' \beta) = \exp(X \beta + \beta_i)$.

The ratio of the odds in the second case to the odds in the first case is therefore $\exp(\beta_i)$. Hence the interpretation of the exponential of the parameter as an odds ratio.