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  "body": "\n## Definition\n\nA *linear mixed model* is a statistical model for a random vector $y \\in \\R^n$ of the form\n$$y = X\\beta + Z u + \\varepsilon,$$\nwhere:\n\n1. $X$ is a known real $n \\times p$ matrix and $\\beta \\in \\R^p$ is an unknown vector of *fixed effects*;\n2. $Z$ is a known real $n \\times q$ matrix and $u$ is a random vector of *random effects* with $u \\sim N_q(0, G)$;\n3. $\\varepsilon \\sim N_n(0, R)$ is a random error vector, independent of $u$;\n4. the [[covariance matrices|0V7G6WMFK0]] $G$ and $R$ are [[positive definite|0KX66CZMBB]] and depend on unknown variance parameters.\n\nHere $N_k(m, S)$ is the [[multivariate normal distribution|0XN3K7RSZY]].\n\n## Remarks\n\nThe model implies $y \\sim N_n(X\\beta, \\, Z G Z^T + R)$.\nWithout the term $Z u$ it is a [[general linear model|A06SGFCSCE]] with one response.\n\nExample: the random intercept model.\nFor 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 $\\varepsilon_{ij} \\sim N(0, \\sigma^2)$.\nThis 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 $R = \\sigma^2 I_{am}$.\nTwo members of the same group have covariance $\\tau^2$ and correlation $\\tau^2 / (\\tau^2 + \\sigma^2)$, the intraclass correlation.\n\nNon-example: if the group shifts $b_i$ are treated as unknown constants rather than random variables, clause 2 fails.\nThe model is then a fixed effects model, a general linear model with the group indicators among the columns of $X$.\n\nGiven the variance parameters, $\\beta$ is estimated by generalized least squares and $u$ is predicted by its best linear unbiased predictor.\nThe variance parameters are usually estimated by restricted maximum likelihood.\n",
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