{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $g \\ge 2$ and $p \\ge 1$ be integers.\nFor $i = 1, \\ldots, g$ and $j = 1, \\ldots, n_i$, let $X_{ij}$ be independent random vectors in $\\R^p$ with $X_{ij} \\sim N_p(\\mu_i, \\Sigma)$, a [[multivariate normal distribution|0XN3K7RSZY]] with unknown $\\mu_i$ and unknown common $\\Sigma$.\nLet $N = n_1 + \\cdots + n_g$, let $\\bar X_i$ be the mean of group $i$ and $\\bar X$ the mean of all $N$ observations.\nThe between-group and within-group matrices are\n$$B = \\sum_{i=1}^g n_i (\\bar X_i - \\bar X)(\\bar X_i - \\bar X)^T, \\qquad W = \\sum_{i=1}^g \\sum_{j=1}^{n_i} (X_{ij} - \\bar X_i)(X_{ij} - \\bar X_i)^T.$$\n*One-way multivariate analysis of variance* is the [[test|087P03NEND]] of $H_0 \\colon \\mu_1 = \\cdots = \\mu_g$ by a statistic that depends on the eigenvalues of $W^{-1} B$.\nThe standard statistics are:\n\n1. Wilks' lambda $\\Lambda = \\det W / \\det(W + B)$, which rejects for small values.\n2. Pillai's trace $\\tr(B (B + W)^{-1})$.\n3. The Hotelling–Lawley trace $\\tr(W^{-1} B)$.\n4. Roy's largest root, the largest eigenvalue of $W^{-1} B$.\n\n## Remarks\n\nThe matrix $W$ is [[invertible|0EB9N0S33C]] with probability $1$ when $N - g \\ge p$ and $\\Sigma$ is positive definite.\nExample: for $p = 1$, $B$ and $W$ are the between and within sums of squares of the one-way [[analysis of variance|M9AQ5HKGP4]].\nThen $\\Lambda = W/(W + B)$, a decreasing function of the usual $F$ statistic, so MANOVA reduces to ANOVA.\nNon-example: with $p = 5$ responses and $g = 3$ groups of $2$ observations each, $N - g = 3 \u003c p$, so $W$ has rank at most $3$, is not invertible, and $\\Lambda = 0$ whatever the data.\nFor $g = 2$ the four statistics are equivalent to Hotelling's $T^2$ statistic.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "8J3B7N8CAT",
    "kind": "definition",
    "msc": [
      "62H15",
      "62J10"
    ],
    "name": "Multivariate analysis of variance",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance"
      }
    ],
    "tag": "849ASYBYJ6",
    "uses": [
      "M9AQ5HKGP4",
      "0XN3K7RSZY",
      "087P03NEND",
      "00KDZRW4C0",
      "0XC1RDBY7A",
      "0EB9N0S33C"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/849ASYBYJ6/",
  "links": [
    {
      "field": "defines",
      "target": "8J3B7N8CAT"
    },
    {
      "field": "uses",
      "target": "M9AQ5HKGP4"
    },
    {
      "field": "uses",
      "target": "0XN3K7RSZY"
    },
    {
      "field": "uses",
      "target": "087P03NEND"
    },
    {
      "field": "uses",
      "target": "00KDZRW4C0"
    },
    {
      "field": "uses",
      "target": "0XC1RDBY7A"
    },
    {
      "field": "uses",
      "target": "0EB9N0S33C"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/849ASYBYJ6.md",
  "msc": [
    "62H15",
    "62J10"
  ],
  "name": "Multivariate analysis of variance",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/849/849ASYBYJ6.md",
  "status": null,
  "tag": "849ASYBYJ6",
  "uri": "m2://849ASYBYJ6",
  "url": "https://tamnd.github.io/math2/wiki/multivariate-analysis-of-variance-definition/",
  "writing": {
    "tag": "849ASYBYJ6",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}