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I run a simple binary logit regression with a single dummy as IV. The output gives log odds and is:

Intercept: 0.3045
Female: 0.4521

Calculating the odds ratio for female is $exp(0.4521) = 1.572$ and I usually interpret it like this: "If the person is female, this multiplies the odds of Y = 1 by 1.572."

I was now interested in the probability and couldn't get my head around it. First I calculated the probability from the OR for female, which is $\frac{1.572}{1+1.572} = 0.61$. I thought it translates into something like: "If the person is female, the probability of Y = 1 increases by 61%". However, when calculating the fitted probability of Y = 1 | female and Y = 1 | male I get:

$\frac{exp(0.3045+0.4521)}{1+exp(0.3045+0.4521)} = 0.681 $ for female and $\frac{exp(0.3045)}{1+exp(0.3045)} = 0.576 $ for male.

This is clearly no increase by 61% from male to female.

My question: How does the probability of 0.61, calculated above, relate to the fitted probabilities? Did I do anything wrong?

Cheers!

mire
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  • Doesn't https://stats.stackexchange.com/questions/133623 answer your question? – whuber Dec 12 '19 at 16:36
  • Thanks for the quick reply. I am not sure but I don't think so. I understand how to get from the fitted probabilities to odds (e.g. 0.681 translates to odds of 2.131) and OR and back. But I do not understand where the probability of 0.61 comes into play, what it means and how it is interpreted. – mire Dec 12 '19 at 17:08
  • I believe my answer in that thread discusses every one of those points explicitly. – whuber Dec 12 '19 at 17:08
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    Alright. I think my mistake is, that I incorrectly calculated the probability for female. I just took the OR for female instead of taking the intercept (i.e. the whole model) into account. So, it would never be exp(0.4521)/1+exp(0.4521) = 0.61 but rather exp(0.3045+0.4521)/1+exp(0.3045+0.4521) = 0.681 in the first place (vs. the 0.576 for the empty model). Then, the OR also fits and nothing is "left over". Does this make sense? Thanks for the help @whuber – mire Dec 12 '19 at 17:46

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