= Marginal and conditional slopes agree for an independent log-link random intercept
{title2=$\log\mathbb E(Y\mid x)=\beta_0+\log\mathbb E(e^b)+\beta^{\mathsf T}x$}
For a conditional count mean $\mathbb E(Y\mid x,b)=\exp(b+\beta_0+\beta^{\mathsf T}x)$, suppose the distribution of $b$ is <independent> of the covariates and has finite $\mathbb E e^b$. Integrating over $b$ gives the displayed <logarithmic link function> mean: every slope stays unchanged, and only the intercept shifts. This is an exact property of the log link; it does not hold in this form for the <logit>.
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