Using the "Divide by 4 Rule" to Interpret Logistic Regression Coefficients

I was recently reading a bit about logistic regression in Gelman and Hill's book on hierarchical/multilevel modeling when I first learned about the "divide by 4 rule" for quickly interpreting coefficients in a logistic regression model in terms of the predicted probabilities of the outcome. The idea is pretty simple. The logistic curve (predicted probabilities) is steepest at the center where a+ßx=0, where logit^-1(x+ßx)=0.5. See the plot below (or use the R code to plot it yourself).

The slope of this curve (1st derivative of the logistic curve) is maximized at a+ßx=0, where it takes on the value:

ße⁰/(1+e⁰)²

=ß(1)/(1+1)²

=ß/4

So you can take the logistic regression coefficients (not including the intercept) and divide them by 4 to get an upper bound of the predictive difference in probability of the outcome y=1 per unit increase in x. This approximation the best at the midpoint of x where predicted probabilities are close to 0.5, which is where most of the data will lie anyhow.

So if your regression coefficient is 0.8, a rough approximation using the ß/4 rule is that a 1 unit increase in x results in about a 0.8/4=0.2, or 20% increase in the probability of y=1.