# Interpreting logit results, what do coefficients mean?

I ran an experiment for which I computed personality scores of the Big Five of subjects, and subsequently asked to rank several video games franchises, from 1 (really wants to play it) to 7 (really do not want to play it).

To see which personality could predict the preference for a video game franchise, I ran a logit regression, with a DV scored 1 if the game was ranked 1, and 0 else. I also controlled for Gender (Man-Woman, binary), Age, and if the subject was a video game user or not.

Here's the result of the logit regression:

The thing is, I have trouble to interpret the coefficient. Naively, I would say that an increase in, for instance the EXT (for Extraversion) score, would increase the probability to play this specific type of game. Or is that what the odd ratio (so if I recall right, the exp of the coefficient) is telling us (in this case, it would be 1.10%, so an 10% increase for one-unit increase in the EXT score)?

I'm a bit lost, and any help would be appreciated! Thank you!

Logistic regression models the log odds as linear

$$\log\left( \dfrac{p}{1-p} \right) = \beta_0 + \beta_1x_1 + \cdots$$

The coefficients you see are the $$\beta$$ in the model above. If you do the algebra, a one unit increase in the predictor leads to the following change in the log odds

$$\log\left( \dfrac{p(x_1+1)}{1-p(x_1+1)} \right) - \log\left( \dfrac{p(x_1)}{1-p(x_1)} \right) = \log\left( \dfrac{p(x_1+1) (1-p(x_1))}{p(x_1)(1-p(x_1+1))} \right) = \beta_1$$

So the coefficients describe the log odds ratio; how much the log odds change when the predictor changes by one unit. If we exponentiation the coefficient, we get the odds ratio. The interpretation is how much the odds change (on a multiplicative scale) when we increase the predictor by a single unit.

It isn't as easy to make inferences about how the coefficient changes the probability because changes to the odds depend on the baseline risk. If you want to know how changing the EXT changes the probability, you will need to compute the risk for a reference case under each EXT measure.

• Thank you for your very detailed answer! Commented May 10, 2021 at 19:20