Random effect

A random effect is a latent variable used to describe variation between observational units or dependence among related observations. Its distribution supplements the fixed effects used in statistical modelling. Independent random effects produce extra variability; correlated random effects can produce serial dependence.

Let $Y\mid \Lambda \sim \mathrm{Poisson}(e^{\eta +\Lambda })$, 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 $\mathrm{Var}(Y)=m+m^2(e^{{\tau }^2}-1)$, 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.

Random effects are crossed when an observation can belong to levels of two grouping factors without one grouping factor being nested within the other. Independent location intercepts and incubator slopes, for example, contribute separate terms to the observation covariance matrix.

A random slope multiplies a predictor by a group-specific latent coefficient. With $b_j\sim N(0,{\tau }^2)$, a contribution $b_j{t}_i$ induces covariance ${\tau }^2{t}_i{t}_k$ for two observations in group $j$. A random slope does not necessarily include a random intercept.

When a person's latent slope has normal distribution $N({\beta }_1,{\tau }_1^2)$, its $u$th quantile is ${\beta }_1+{\tau }_1{\Phi }^{-1}(u)$. Multiply by an elapsed time to obtain the corresponding latent change quantile. This describes population variation between people; it is distinct from a confidence interval for the mean slope and from a prediction interval that includes measurement error.

The Gaussian model $Y=X\beta +Zb+\epsilon$ with $b\sim N(0,D)$ and independent $\epsilon \sim N(0,{\sigma }^{2}I)$ has marginal law $N(X\beta ,ZD{Z}^T+{\sigma }^{2}I)$. Its group coefficients are integrated out of the marginal likelihood function. The special random-slope model uses predictor values in the grouping columns of $Z$.