The sign of the logistic regression coefficient

Let Y be a binary random variable and X a scalar. Let β̂ be the maximum likelihood estimate of the slope in a logistic regression of Y on X with intercept. Further let x̄0 and x̄1 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(β̂) = sign(x̄1− x̄0). More generally, if xi are vector valued then we show that β̂ = 0 if and only if x̄1 = x̄0. This holds for logistic regression and also for more general binary regressions with inverse link functions satisfying a log-concavity condition. Finally, when x̄1 6= x̄0 then the angle between β̂ and x̄1− x̄0 is less than ninety degrees in binary regressions satisfying the log-concavity condition and the separation condition, when the design matrix has full rank.