The Sign of the Logistic Regression Coefficient

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.

• LD50
• Log-concavity
• Probit model
• Sufficiency