# Estimates diverging using continuous probabilities in logistic regression

When using a binomial family, logit link for GLM (or GEE in my case), I notice that my model estimates diverge when my response variables (which are continuous probabilities with range 0 to 1) include 0 or 1 (or 0 <= y <= 1) as observed values, but the models with response variables that don't include 0 or 1 (or 0 < y < 1) are able to converge just fine.

Question:

• Why does this happen?

When running a logistic regression model (with 0 < y < 1) the model runs fine, as does the model when the response variable is dichotomous 0/1.

I suspect the following: say I have observations 0 < y <= 1. In this case, the algorithm sees my ones but not any zeros, and then craps out saying "some groups have fewer than x observations," the aforementioned group being the ones that are supposed to have zeros.

Secondary question:

• If I exclude observations that are 0 or 1 in order to fit my models, am I biasing my results?

Here's an example: my response variable is graduation rate expressed as a percentage. For the logistic regression models, there are apparently schools that have 100% graduation rate (seen as a 1 in my dataset). Would it be a valid strategy to drop these schools from the model, and what are the implications in interpretation? Is this akin to dropping outliers willy-nilly?

• Reading this may help: stats.stackexchange.com/questions/9330 – GaBorgulya Apr 29 '11 at 21:17
• I am not sure I understand how are you using a binomial family with a response variable that is a percentage. Do you have the numerator and denominator counts? – Aniko Apr 29 '11 at 21:57

It shouldn't happen, if you do the taylor series well - I'd suggest starting it at different initial values. A good choice is to set the intercept equal to the logit of the total proportion in your sample, and all other betas to zero. So you have $p_{i}=\frac{y_{i}}{n_{i}}$ as the observed proportions for each unit. Just set $$\beta_0=logit\left(\frac{\overline{y}}{\overline{n}}\right)$$
$$\tilde{p}_{i}=\frac{y_{i}+1}{n_{i}+2}$$