Random effect vs fixed effect in logistic regression I have a logistic regression model: $y \sim x^{\top}\beta $. One of the covariates represents the doctor ID, as in this example. I wondered if I would obtain very different results by considering the doctor ID as another covariate (fixed effect) instead of a random effect as in the link provided above. 
I thought of this since another covariate is also "male" or "female", among other categorical predictors, then, why wouldn't these variables be also considered as random effects?
 A: Depending on the ratio of observations of patients (assuming that your observations are patients) to observations of doctors, including very many fixed effects in a nonlinear model (like logit or probit) introduces bias into the estimates of the coefficients of interest.  This is because the number of parameters to estimate grows with the size of the data, as the number of doctors grows (in this particular case).  
The lecture notes linked below give a pretty good technical exposition of the problem and ways around it:
http://www.crest.fr/ckfinder/userfiles/files/Pageperso/raeberhardt/080408.Panel_sem.varqual.beamer.pdf
You can find more googling "incidental parameters probem"
Note that the problem diminishes if you have many patients per doctor -- if the number of nuisance parameters grows slowly.
Note also that random effects assumptions are justified -- and RE will be more efficient -- if doctor-specific effects are independent of your other covariates.  While I'm not in the medical field, it doesn't seem like this is an assumption that will commonly hold in practice.
