= Random-intercept attenuation of marginal logistic slopes
{title2=$\frac{d}{d\eta}\operatorname{logit}M(\eta)=1-\frac{\operatorname{Var}(p_B)}{M(1-M)}$}
Put $p_B=\operatorname{logit}^{-1}(\eta+B)$ and $M(\eta)=\mathbb E(p_B)$. Differentiation gives $M'=\mathbb E[p_B(1-p_B)]=M(1-M)-\operatorname{Var}(p_B)$. Hence the displayed derivative lies between zero and one, strictly below one for a nondegenerate <random intercept>. A marginal <logit> is therefore generally nonlinear and its slope differs from the conditional slope even without <confounding>. This is a precise instance of <noncollapsibility of the odds ratio>.
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