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  "body": "\n## Definition\n\nLet $k \\ge 2$ and $n_1, \\dots, n_k \\ge 1$, with $N = n_1 + \\cdots + n_k \u003e k$.\nLet $y_{ij}$, for $i = 1, \\dots, k$ and $j = 1, \\dots, n_i$, be observations of $k$ groups.\nLet $\\bar y_i = \\frac{1}{n_i} \\sum_j y_{ij}$ be the mean of group $i$ and $\\bar y = \\frac{1}{N} \\sum_{i,j} y_{ij}$ the overall mean.\nThe *one-way analysis of variance* is the decomposition\n$$\\sum_{i=1}^k \\sum_{j=1}^{n_i} (y_{ij} - \\bar y)^2 = \\sum_{i=1}^k n_i (\\bar y_i - \\bar y)^2 + \\sum_{i=1}^k \\sum_{j=1}^{n_i} (y_{ij} - \\bar y_i)^2,$$\nwritten $SS_{\\mathrm{tot}} = SS_{\\mathrm{between}} + SS_{\\mathrm{within}}$, together with the statistic\n$$F = \\frac{SS_{\\mathrm{between}} / (k - 1)}{SS_{\\mathrm{within}} / (N - k)},$$\ndefined when $SS_{\\mathrm{within}} \u003e 0$.\nConsider the model $y_{ij} = \\mu_i + \\varepsilon_{ij}$ with independent errors $\\varepsilon_{ij} \\sim N(0, \\sigma^2)$.\nThe *ANOVA $F$-test* of level $\\alpha$ rejects the hypothesis $\\mu_1 = \\cdots = \\mu_k$ when $F$ is large.\nThe threshold is the upper $\\alpha$-quantile of the [[F-distribution|PBT8VRJADK]] with $k - 1$ and $N - k$ degrees of freedom.\n\n## Remarks\n\nThe decomposition holds because, inside each group, the deviations $y_{ij} - \\bar y_i$ sum to zero, so the cross term vanishes.\nUnder the hypothesis of equal means and [[normal|05AQE0QR0D]] errors, $F$ has the $F$-distribution with $k - 1$ and $N - k$ degrees of freedom.\n\nExample: groups $\\{1, 2, 3\\}$ and $\\{4, 5, 6\\}$, so $k = 2$, $N = 6$, $\\bar y_1 = 2$, $\\bar y_2 = 5$, $\\bar y = 3.5$.\nThen $SS_{\\mathrm{between}} = 3 \\cdot 1.5^2 + 3 \\cdot 1.5^2 = 13.5$ and $SS_{\\mathrm{within}} = 2 + 2 = 4$.\nTheir sum $17.5$ equals $\\sum (y - 3.5)^2 = 2(6.25 + 2.25 + 0.25)$.\nSo $F = 13.5 / (4/4) = 13.5$ on $1$ and $4$ degrees of freedom.\n\nNon-example: if the groups have very different error variances, the hypothesis of a common $\\sigma^2$ fails.\nThen $F$ no longer has the $F$-distribution under equal means, and Welch's test is used instead.\n\nThe same idea extends to several factors, with one sum of squares for each main effect and interaction.\nAll these fixed-effects analyses are special cases of the [[general linear model|A06SGFCSCE]].\nWith $k = 2$, the $F$ statistic is the square of the two-sample $t$ statistic with pooled variance.\n",
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