Conditions for the Uniqueness, Finiteness, and Possible Location of the Maximum Likelihood Estimate with a Log Binomial Model

• Activity Number: 87 - Invited ePoster Session: a Statistical Smörgåsbord
• Type: Invited
• Date/Time: Sunday, July 29, 2018 : 8:30 PM to 10:30 PM
• Sponsor: Section on Statistics in Epidemiology
• Author(s): Gurbakhshash Singh* and Gordon Hilton Fick
• Companies: University of Calgary and University of Calgary
• Keywords: Generalized Linear Model; Log Binomial Model; Maximum Likelihood Estimate; Uniqueness; Finiteness; Location

Abstract

For binary outcomes, the Generalized Linear Model with the log link is sometimes called the Log-Binomial Model (LBM). For this model, the literature on the Maximum Likelihood Estimate (MLE) has been highly problematic. These problems are partially due to the implied constraints on the parameter space but there are other issues in play as well. This poster presents developments in the LBM dealing with conditions for 1) the uniqueness of the MLE, 2) the finiteness of the MLE and 3) the possible location of the MLE. In particular, for the uniqueness of the MLE, the full column rank of certain subsets of the covariate matrix is shown to be a sufficient condition. For the possibility of non-finite components of the MLE, a method is proposed based on determining directions of recession of the log-likelihood. Finally, the possible location of the MLE is discussed in terms of the counts of groupings/stratification of the data based on unique covariate vectors.