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  "body": "\n## Definition\n\nLet $n, p, m \\ge 1$.\nA *general linear model* is a statistical model for a random $n \\times m$ response matrix $Y$ of the form\n$$Y = X B + U,$$\nwhere:\n\n1. $X$ is a known real $n \\times p$ matrix, the *design matrix*;\n2. $B$ is an unknown real $p \\times m$ matrix of coefficients;\n3. $U$ is a random $n \\times m$ error matrix whose rows are independent with [[expected value|0WQNRCNY3M]] $0$ and a common [[covariance matrix|0V7G6WMFK0]] $\\Sigma$.\n\nOften the rows of $U$ are assumed to follow the [[multivariate normal distribution|0XN3K7RSZY]] $N_m(0, \\Sigma)$.\n\n## Remarks\n\nWith $m = 1$, the model is $y = X\\beta + \\varepsilon$, multiple [[linear regression|41FVC4P5JB]], and $\\Sigma = \\sigma^2$ is a single variance.\n\nExample: the comparison of two group means.\nLet observations $1, \\dots, n_1$ belong to group $1$ and the others to group $2$, and let $X$ have two columns: the indicator of group $1$ and the indicator of group $2$.\nThen $\\beta = (\\mu_1, \\mu_2)^T$, and the model says that $y_i$ has mean $\\mu_1$ or $\\mu_2$ according to its group, with a common error variance.\nThis is the model of the two-sample $t$-test.\n\nNon-example: the model $E[y_i] = \\exp(x_i^T \\beta)$ is not a general linear model.\nIts mean is not a linear function of $\\beta$, so it cannot be written as the $i$-th entry of $X B$.\n\nWhen $X$ has rank $p$, the least squares estimate is $\\hat B = (X^T X)^{-1} X^T Y$, computed column by column.\n",
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