The Sign of the Logistic Regression Coefficient

• Activity Number: 108
• Type: Invited
• Date/Time: Monday, August 4, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistics in Defense and National Security
• Author(s): Paul Roediger*+ and Art Owen
• Companies: UTRS and Stanford University

Abstract

The conjecture posed in our 3pod commentary (Ray, et. al.) has been settled in the affirmative. Specifically, let Y be a binary random variable and X a scalar. Let B be the maximum likelihood estimate of the slope in a logistic regression of Y on X with intercept. Further let m0 and m1 be the average of sample X values for cases with Y=0 and Y=1, respectively. Then under a condition that rules out separable predictors, we show that sign(B)=sign (m1-m0). More generally, if X are vector valued then we show that B=0 if and only if m1=m0. This holds for logistic regression and also for more general binary regressions with inverse link functions satisfying a log-concavity condition. Finally, when m1-m0 is not zero then the angle between B and m1-m0 is less than ninety degrees in binary regressions satisfying the log-concavity condition and the separation condition, when the matrix has full rank. For more information see http://arxiv.org/pdf/1402.0845.pdf.