Estimate just the constant coefficient in logistic regression How do I calculate the constant coefficient in logistic regression manually, i.e without having to use a calculator?
My model is
$g(Y) = X \beta + \alpha$
is it possible to calculate just the constant parameter $\alpha$ without performing the full regression fit?
 A: If all your predictors $(X_1, \dots, X_4)$ are zero centered then the constant term is what accounts for the class imbalance.
For example, if your sample has 100 1s and 400 0s then the overall background proportion of 1s is 20%, and alpha will be such that $g(0.2) = \alpha$.  If you are using logit link function then $logit(0.2) = \alpha = -1.386$.
The constant is zero when the classes are balanced, negative when 1s are "rare" and positive when 0s are "rare".
As for not using a calculator, unless you have log tables to hand, you will need a calculator to take the logit (log-odds).
Non-centered predictors
If the predictors are not centered in the sample, then their offset acts like another constant term.  You would have to know the $\beta$ parameters to know how much constant each predictor is providing.  In this case however, the $\alpha$ parameter doesn't have a useful interpretation.  It is simply the log odds when all predictors are zero (which is a non-interesting point if they are not centered, and in some cases is an impossible point, since some predictors may not even naturally go through zero).
