Binary logistic regression: Interpreting odds ratio vs. comparing predictive probabilities

Question

Consider the following logistic regression:

set.seed(1839)
n <- 200
x <- rbinom(n, 1, .5)
y <- cut(x + rnorm(n), 2)
mod <- glm(y ~ x, family = binomial())
summary(mod)$coef

Which produces:

             Estimate Std. Error   z value     Pr(>|z|)
(Intercept) -1.172720  0.2244242 -5.225463 1.737202e-07
x            2.490961  0.3421728  7.279833 3.342331e-13

During my time in academia, I would report this as an OR of 12.07 (because exp(coef(mod)[[2]])) and say that "people with an x score of 1 are 12 times as likely then people with an x score of 0 to have a positive outcome on y" (where we assume that 1 is a positive outcome for y).

Now that I'm in industry and working with clients more, people like to see raw percentage point increases. Which we can approximate from comparing the predicted values for people where x = 0 and x = 1:

> inv_logit(coef(mod)[[1]])
[1] 0.2363636

> inv_logit(sum(coef(mod)))
[1] 0.7888889

So we could show predicted probabilities (usually shown as percentages for ease to the reader) as 78.9% when x = 1 and 23.6% when x = 0. This is a difference of 55.3 percentage points.

However, when we take those two predicted probabilities and divide them by one another, we don't get "12 times as likely." Instead, we get:

> inv_logit(sum(coef(mod))) / inv_logit(coef(mod)[[1]])
[1] 3.337607

Which is 3.34 times, not 12.07.

Now, I know that what I'm doing is:

$p(y = 1 | x = 1) \over p(y = 1 | x = 0)$

Whereas the odds ratio is doing:

$p(y = 1 | x = 1) \over 1 - p(y = 1 | x = 1)$

Which aren't the same, because inv_logit(coef(mod)[[1]]) + inv_logit(sum(coef(mod))) is greater than 1 (as it usually is).

I feel as if I am missing something basic here, but can't quite place it. Can anyone tell me where I'm going wrong in my interpretation here? Or perhaps I'm interpreting the odds ratio incorrectly? How do I reconcile these differences?

Answer 1

We often interpret the odds ratio like you did: $\newcommand{\Odds}{\text{Odds}}\newcommand{\event}{\text{event}}\newcommand{\noevent}{\text{no event}}$

people with an x score of 1 are 12 times as likely then people with an x score of 0 to have a positive outcome on y

But likely is purposefully vague here so as not to confuse non-statisticians.

What we really mean is

people with an x score of 1 have an odds 12 times greater than people with an x score of 0 of having a positive outcome on y

As ttnphns showed in the comments, this is expressed as $$ \frac{\Odds(\event|x=1)}{\Odds(\noevent|x=0)} = \frac{\frac{p(\event|x=1)}{p(\noevent|x=1)}}{\frac{p(\event|x=0)}{p(\noevent|x=0)}} $$

Basically, this is the difference between interpreting odds ratio and relative risk. (See here for more explanation)