Random-intercept attenuation of marginal logistic slopes ($\frac{d}{d\eta }\mathrm{logit}M(\eta )=1-\frac{\mathrm{Var}(p_B)}{M(1-M)}$)

Put $p_B={\mathrm{logit}}^{-1}(\eta +B)$ and $M(\eta )=\mathbb{E}(p_B)$. Differentiation gives $M^{\prime }=\mathbb{E}[p_B(1-p_B)]=M(1-M)-\mathrm{Var}(p_B)$. 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.