I am reading the second edition of Categorical Data Analysis by Alan Agresti, and somehow stuck in the following second paragraph:
I don't quite understand why $\beta\pi(\hat{x})(1 - \pi(\hat{x}))$ will give the probability when $x = 26.3$, can anyone enlighten me? Thanks.
The answer is near the bottom of p166. It's using a linear approximation (what social scientists would call a 'marginal effect'). A small change $\delta x$ in $x$ gives a change in probability of: $$\delta\pi \approx \frac{\partial \pi(x)}{\partial x} \delta x.$$ With $\operatorname{logit}(\pi(x)) = \alpha + \beta x$, it's straightforward to show that $ \partial \pi(x) / \partial x = \beta \pi(x)(1-\pi(x))$.
