= Poisson-lognormal random-effect model
{c}
{title2=$\operatorname{Var}(Y)=m+m^2(e^{\tau^2}-1)$}
Let $Y\mid\Lambda\sim\operatorname{Poisson}(e^{\eta+\Lambda})$, with $\Lambda\sim N(0,\tau^2)$. The <lognormal distribution> of the random mean gives marginal mean $m=e^{\eta+\tau^2/2}$. The <law of total variance> gives $\operatorname{Var}(Y)=m+m^2(e^{\tau^2}-1)$, strictly larger than $m$ when $\tau>0$. Thus independent observation-level <random effects> produce <overdispersion> even though the conditional <Poisson distributions> have variance equal to their means.
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