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In the logisitic regression chapter of "Data Analysis Using Regression and
Multilevel/Hierarchical Models" by Gelman and Hill, The "Divide by 4" rule is presented to approximate average marginal effects.

Essentially, dividing the estimated log-odds ratio gives the maximum slope (or the maximum change in probabilities) of the logistic function.

Since the text above states that the "divide by 4 rule" gives the maximum change in $P(y=1)$ with a unit change in x, why is the estimated 8% less than the 13% calculated from actually taking the derivative of the logistic function in the example given?

Does the "divide by 4 rule" actually give the upper bound marginal effect?

Other "divide by 4" resources:

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


• Divide by 4 Rule for Marginal Effects - Econometric Sense


• http://vulstats.ucsd.edu/pdf/Gelman.ch-05.logistic-regression.pdf

I think it's a typo.

The derivative of the logistic curve with respect to $x$ is:
$$
fracbetamathrme^alpha + beta xleft(1 + mathrme^alpha + beta xright)^2
$$

So for their example where $alpha = -1.40, beta = 0.33$ it is:
$$
frac0.33mathrme^-1.40 + 0.33 xleft(1 + mathrme^-1.40 + 0.33 xright)^2
$$
Evaluated at the mean $barx=3.1$ gives:
$$
frac0.33mathrme^-1.40 + 0.33 cdot 3.1left(1 + mathrme^-1.40 + 0.33cdot 3.1right)^2 = 0.0796367
$$
This result is very close to the maximum slope of $0.33/4 = 0.0825$ which is attained at $x=-fracalphabeta=4.24$, supporting their claim.

On page 82, they write

But $0.33mathrme^-0.39/left(1+mathrme^-0.39right)^2neq 0.13$. Instead, it's around $0.08$, as shown above.

For a continuous variable $x$, the marginal effect of $x$ in a logit model is

$$Lambda(alpha + beta x)cdot left[1-Lambda(alpha + beta x)right]cdotbeta = p cdot (1 - p) cdot beta,$$ where the inverse logit function is
$$Lambda(z)=fracexpz1+expz.$$

Here $p$ is a probability, so the factor $pcdot (1-p)$ is maximized when $p=0.5$ at $0.25$, which is where the $frac14$ comes from. Multiplying by the coefficient gives you the upper bound on the marginal effect. Here it is

$$0.25cdot0.33 =0.0825.$$

Calculating the marginal effect at the mean income yields,

$$mathbfinvlogit(-1.40 + 0.33 cdot 3.1)cdot left(1-mathbfinvlogit(-1.40 + 0.33 cdot3.1)right)cdot 0.33 = 0.07963666$$

These are pretty close, with the approximate maximum marginal effect bounding the marginal effect at the mean.