I am running logistic models with data where I typically have proportions for the dependent variable that looks like the following:
0 1 Grp1 0.18 0.82 Grp2 0.24 0.76
Now I run a logistic regression with independent variables Grp (0 if Grp1 and 1 if Grp2) and control variables A, B and C (including a number of interactions) that gives me an estimate of 0.5 for Grp. I then calculate the predicted values for both groups where A, B and C are taken at their mean values, which gives me a probability of 0.82 for Grp2 and 0.88 for Grp1.
I am not sure I understand why it is possible for both groups to have a predicted probability above the actual proportions. Assuming that the calculations are correct (admittedly an ambitious assumption, although I have tried to verify with different ways of calculating the predicted values), is it theoretically possible to have higher predictions for both groups?