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How do I interpret a regression coefficient in a logistic regression with two predictors?

$\hat{L} = -14.27+3.32(3)+0.88(7)$

My understanding is to take the anti-log of the coefficient, like $e^{3.32}$, and use that to interpret the percentage increase of odds ratio. But when I do $e^{3.32}$, I get $27.66$, so does that mean that the increase is by $2666\%$ or only $66\%$?

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    $\begingroup$ For what it's worth, "anti-log" is usually spelled "exponential". $\endgroup$
    – Matthew Drury
    Commented Dec 15, 2015 at 2:28

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Personally, I think it is very difficult for people to grasp what does it mean by certain amount of change in odds...

But it is more clear to directly compare the difference of interest: the p's In your case.

At the current value of your predictors, increase 1 unit (can be any change you specify) in the first predictor means increase of

$logit^{-1}(−14.27+3.32(4)+0.88(7)) - logit^{-1}(−14.27+3.32(3)+0.88(7))$

$\approx0.99-0.86 = 0.13$, i.e., it is a 0.13 increase in probability of being 1.

But of course, this varies as the current values of the predictor changes, a common practice (if you are not interested in this specific predictor value) is to let the current predictor value be at the average, which corresponds the biggest change in probability, then you can say, as the predictor values move towards the end, changes in probability are expected to slow down.

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  • $\begingroup$ Great answer! Just to add an extra layer of clarity for people who might see this in the future: the $logit^{-1}(x)$ function mentioned extracts the probability from a logit, which is done as $logit^{-1}(x) = p(x) = exp(x) / (1 + exp(x))$ $\endgroup$
    – hellter
    Commented Oct 20, 2022 at 7:38
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It would be 2666% if you insist using percentage for a large number like that. Basically when you exponentiate the coefficients of a logistic regression, you end up with the odds ratio of being in class 1. In your case, it means 1 unit increase of that variable would result in 2666% compare to 1 unit less.

For more details you could look at http://www.ats.ucla.edu/stat/mult_pkg/faq/general/odds_ratio.htm with details of the interpretation.

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