Mean-ratio preservation under a multiplicative random effect ($\mathbb{E}{Y}_j/\mathbb{E}{Y}_k=e^{{\alpha }_j-{\alpha }_k}$)

If $\mathbb{E}(Y_j\mid B,x)=B\exp ({\beta }_0+{\beta }^{T}x+{\alpha }_j)$ and the distribution of $B$ is the same at each observation time with finite positive mean, integrating out $B$ adds $\log \mathbb{E}B$ 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.