Logit: why aren't my exponentiated coefficients equaling my odds ratio (for nominal predictors)?

Question

I'm running logit in JMP. My understanding is that if you exponentiate the coefficients, you get the change in odds ratios from a 1 unit change in X.

This works perfectly for continuous independent variables, but not for nominal (0,1) independent variables. Example below:

For parameter estimates I'm getting:

Term Estimate Intercept -1.516621
Gender[Female] 0.22189387

Level1 /Level2 Odds Ratio
Male Female 0.6416016
Female Male 1.5585996

Shouldn't my odds ratio for males be exp(-1.516621)=0.220486 and for females be exp(0.20461503)=1.2270525?

Befuddled by this. Would really appreciate some help. Thanks!

Answer 1

You're mistaking odds for odds ratio here. The odds - not the odds ratio! - for males is odds(male)=exp(-1.516621), whereas for females it's odds(female)=exp(-1.516621+0.22189387). Now in the next step you get odds ratio for males, calculated as OR(male)=odds(male)/odds(female) - hence you can compare the two genders. Odds ratio for females is calculated the same way: OR(female)=odds(male)/odds(female)=1/OR(male). Your results seem perfectly fine :)

See e.g. http://www.ats.ucla.edu/stat/stata/faq/oratio.htm for more detailed explanation and a similar example.

Answer 2

Per the SAS JMP forum apparently you have to make the gender variable ordinal and not nominal to get the right output. I find this confusing, especially since the odds ratio shows as the same either way, but the coefficients seem to be "wrong" when using nominal.