Mixed model

Remarks

The model implies $y \sim N_n(X\beta, \, Z G Z^T + R)$. Without the term $Z u$ it is a general linear model with one response.

Example: the random intercept model. For groups $i = 1, \dots, a$ and members $j = 1, \dots, m$, let $y_{ij} = \beta_0 + b_i + \varepsilon_{ij}$ with independent $b_i \sim N(0, \tau^2)$ and $\varepsilon_{ij} \sim N(0, \sigma^2)$. This fits the definition with $p = 1$, $X$ the column of ones, $q = a$, $Z$ the matrix of group indicators, $G = \tau^2 I_a$ and $R = \sigma^2 I_{am}$. Two members of the same group have covariance $\tau^2$ and correlation $\tau^2 / (\tau^2 + \sigma^2)$, the intraclass correlation.

Non-example: if the group shifts $b_i$ are treated as unknown constants rather than random variables, clause 2 fails. The model is then a fixed effects model, a general linear model with the group indicators among the columns of $X$.

Given the variance parameters, $\beta$ is estimated by generalized least squares and $u$ is predicted by its best linear unbiased predictor. The variance parameters are usually estimated by restricted maximum likelihood.