The Efficiency of Logistic Regression Compared to Normal Discriminant Analysis

Abstract A random vector x arises from one of two multivariate normal distributions differing in mean but not covariance. A training set x 1, x 2, ··· x n of previous cases, along with their correct assignments, is known. These can be used to estimate Fisher's discriminant by maximum likelihood and then to assign x on the basis of the estimated discriminant, a method known as the normal discrimination procedure. Logistic regression does the same thing but with the estimation of Fisher's disriminant done conditionally on the observed values of x 1 x 2, ···, x n . This article computes the asymptotic relative efficiency of the two procedures. Typically, logistic regression is shown to be between one half and two thirds as effective as normal discrimination for statistically interesting values of the parameters.