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

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())

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)

• Bingo, I knew I was missing something simple! Funny how the shorthand explanations we get from introductory material can end up confusing us way down the road. The "what we really mean is" you gave is very clear. Jun 15, 2018 at 13:53