# Interpretation of coefficients in logit model

We know that in a logit model, the coefficient $$\beta_j$$ for the variable $$x_j$$, measures the impact of the variable on the log(odds)

In order to measure the impact on the Odds, we have to consider the $$\exp(\beta_j)$$

Specifically, if we have that $$\exp(\beta_j)=1.16$$, we say that, for one unit increase of $$x_j$$, the odds increases 16%.

but in this formula $$[ ( \exp(\beta_j)-1 ) * 100 ]$$ , where does the $$-1$$ come from? why do we subtract $$1$$ from the exponential?

• Does this answer your question? Converting log odds coefficients to probabilities Oct 6, 2021 at 11:27
• I don't think so (though she may very well benefit from that question). I read Jenny's question as she wants to remain in the odds metric, and wonders how one goes from "the odds ratio is 1.16" to "the odds increase by 16%" Oct 6, 2021 at 20:03

$$\exp( \beta_j )$$ is the odds ratio, so for a unit change in $$x_j$$ we expect the odds to increase by a factor $$\exp( \beta_j )$$, i.e. we need to multiply the odds by that number. In your example you have an odds ratio of $$1.16$$. If you multiply a number with $$1$$ it remains the same, so anything on top of $$1$$ is a growth and anything below $$1$$ is a decrease. So if we want to focus on the change we subtract $$1$$. This is where the $$-1$$ comes from. Notice that this formula does not add any information. We can say that an odds doubles or that it has grown by 100%, and we would be saying the exact same thing. This is just about finding a representation of your results that is easiest for your audience.