A note on A. Albert and J. A. Anderson's conditions for the existence of maximum likelihood estimates in logistic regression models

Abstract

This note expands the paper by Albert & Anderson (1984) on the existence and uniqueness of maximum likelihood estimates in logistic regression models. Their three possible mutually exclusive data patterns: (i) overlap, (ii) complete separation, and (iii) quasiseparation are considered. The maximum likelihood estimate exists only in (i). Modifications of the statement and proofs of Albert & Anderson's results are given for (ii) and (iii) The identifiability for a more general model arising in the study of (iii) is discussed together with the maximization of the corresponding likelihood. A linear program is presented which determines whether data is of type (i), (ii) or (iii), and in the case of (iii) identifies Albert & Anderson's minimal set Qm.