On the Existence of an Analytical Solution in Multiple Logistic Regression

Fitting a logistic regression model to a given data starts from the likelihood function. Typically, the regression parameters are solved by maximizing the likelihood function. In general, there is no analytical solution since these regression parameters fall into a set of nonlinear equations. So far, only two cases have been known to have an analytical closed-form solution for univariate logistic regression: a dichotomous variable and a categorical variable. In this paper we study the existence of an analytical solution for multiple logistic regression. We discover 3 types of data that yields an analytical solution: Partially balanced data; Perfectly balanced data; Quasi-saturated data. We show that the 2 known cases for univariate logistic regression are special cases to ours with quasi-saturated data.