The Sign of the Logistic Regression Coefficient

A. B. Owen and P. A. Roediger

The American Statistician, 2014, vol. 68, issue 4, 297-301

Abstract: Let Y be a binary random variable and X a scalar. Let be the maximum likelihood estimate of the slope in a logistic regression of Y on X with intercept. Further let and 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 . More generally, if the x i are vector valued, then we show that if and only if . This holds for logistic regression and also for more general binary regressions with inverse link functions satisfying a log-concavity condition. Finally, when then the angle between and is less than 90° in binary regressions satisfying the log-concavity condition and the separation condition, when the design matrix has full rank.

Date: 2014
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DOI: 10.1080/00031305.2014.951128

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