Random intercept

A random intercept gives every group a latent additive shift, inducing dependence among observations from the same group.

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.

A random-intercept linear mixed model has the formwhere independent group effects $b_i$ induce positive covariance among observations from the same group.