= Mean-ratio preservation under a multiplicative random effect
{title2=$\mathbb EY_j/\mathbb EY_k=e^{\alpha_j-\alpha_k}$}
If $\mathbb E(Y_j\mid B,x)=B\exp(\beta_0+\beta^Tx+\alpha_j)$ and the distribution of $B$ is the same at each observation time with finite positive mean, integrating out $B$ adds $\log\mathbb EB$ to the marginal log-mean intercept. The conditional positive-$B$ mean ratio and the marginal mean ratio both equal $e^{\alpha_j-\alpha_k}$. This is a property of the multiplicative <logarithmic link function>; it need not hold for nonlinear links such as the <logit link>.
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