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Biometrika 1986 73(3):755-758; doi:10.1093/biomet/73.3.755
© 1986 by Biometrika Trust
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A note on A. Albert and J. A. Anderson's conditions for the existence of maximum likelihood estimates in logistic regression models

THOMAS J. SANTNER and DIANE E. DUFFY

School of Operations Research and Industrial Engineering, Cornell University Ithaca, New York 14853, U.S.A.

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.

Key Words: Complete separation • Log linear model • Logistic discrimination • Mixed linear program • Overlap • Quasiseparation


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