The sign of the logistic regression coefficient

Let Y be a binary random variable and X a scalar. Let $\widehat{\beta }$ be the maximum likelihood estimate of the slope in a logistic regression of Y on X with intercept. Further let ${\bar{x}}_0$ and ${\bar{x}}_1$ 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($\widehat{\beta }$) = sign($\bar x_1-\bar x_0$)\mathrm{.}Moregenerally,if$x_i$arevectorvaluedthenweshowthat$\hat\beta=0$ifandonlyif$\bar x_1=\bar x_0$. This holds for logistic regression and also for more general binary regressions with inverse link functions satisfying a log-concavity condition. Finally, when $\bar x_1\ne \bar x_0$thentheanglebetween$\hat\beta$and$\bar x_1-\bar x_0$ is less than ninety degrees in binary regressions satisfying the log-concavity condition and the separation condition, when the design matrix has full rank.