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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?

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  • $\begingroup$ Does this answer your question? Converting log odds coefficients to probabilities $\endgroup$
    – Xi'an
    Commented Oct 6, 2021 at 11:27
  • $\begingroup$ 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%" $\endgroup$
    – Maarten Buis
    Commented Oct 6, 2021 at 20:03

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$\exp( \beta_j )$ 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.

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