How do I account for extra information I have about the distribution of the response variable in a logistic regression analysis? I've fitted a logistic regression, response is gender of teachers in a school leadership role (female=1), and the predictor is 'type' of school. School type is categorical with three values: co-educational (mixed gender); all girls; all boys.
Here's the model output:
Call:
glm(formula = cbind(data$q8a, data$q8b) ~ temp.coed, family = "binomial")

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-1.55935  -0.60008  -0.08811   0.61469   1.75349  

Coefficients:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)      0.6150     0.1076   5.714  1.1e-08 ***
temp.coedBoys   -1.0745     0.3841  -2.797  0.00515 ** 
temp.coedGirls   0.8956     0.2888   3.101  0.00193 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 53.301  on 48  degrees of freedom
Residual deviance: 31.759  on 46  degrees of freedom
  (7 observations deleted due to missingness)
AIC: 163.08

Number of Fisher Scoring iterations: 4

I can see that, at a girls school, a school leader is (exp(0.8956)) 2.4 times more likely to be a female at a girls school than at a co-ed school (the baseline), and about a third (exp(-1.074)) as likely to a female at a boys school than at a co-ed school.
I have some additional information about teachers in these schools: I know the gender distribution of all teachers (so, not just teachers in leadership roles as in the regression). From summary statistics, I can see that the change in leadership gender distribution is similar to the change in teacher gender distribution, across school type.
I think it could be that the results I'm seeing in this analysis are due to the overall change in teacher gender across school type, and are not because of some effect of school type on who is a school leader. How do I account for this extra information I have? I wondered if I should include the overall teacher gender proportions as a predictor in the model?
 A: Including the information about gender fraction of teachers would certainly make sense. I'd recommend thinking about this in reverse from the way you started (although the final analysis is the same): see how well just the proportion of female faculty works to predict fraction of female leaders. Then see whether adding the type of school to that proportion adds any more useful information. Do that comparison with a simple anova() or aov() test between the two models, as you care about school type overall rather than the individual reported school-type regression coefficients, which are the differences of the two non-reference school categories from the reference.
I'm a little worried about the possibility that you are overfitting. With the 2-column structure of your outcome model it's hard to see how many women total are in leadership positions. (It seems that for each school you have taken together all "leadership" positions rather than limiting say to the chief executive, with the response columns being female and male leader numbers for each school.) You typically need to limit yourself to about 1 predictor per 15 cases in the less-frequent class, which is presumably female leaders here.
Your current model already has 2 predictors (each level of the categorical predictor beyond the reference). Adding fraction of female faculty would add a third. So provided that you have on the order of 50 female leaders overall you should be OK.
It might be nice to examine an interaction between that proportion and the type of school in the model. That would mean an extra 2 predictors and a correspondingly greater number of female leaders in your sample to avoid overfitting.
With the known association between female teacher fraction and the school type there will be some danger of those correlations leading to high standard errors in the coefficient estimates, which could affect significance testing even in the simplest additive teacher fraction plus school type model. That could be even more of a problem if you include an interaction. That's why I recommend thinking about and presenting the results with teacher fraction first as a single predictor, and moving on up from there.
