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  "body": "\n## Definition\n\nLet $d_1, d_2 \\ge 1$ be integers and let $\\alpha \\in (0, 1)$.\nAn *F-test* is a [[hypothesis test|087P03NEND]] of a null hypothesis $H_0$ based on a [[statistic|036756NFD6]] $F \\ge 0$ with the following property.\nUnder $H_0$, $F$ has the [[F-distribution|PBT8VRJADK]] with $d_1$ and $d_2$ degrees of freedom.\nThe test at level $\\alpha$ rejects $H_0$ when $F \u003e F_{1 - \\alpha}(d_1, d_2)$, the $1 - \\alpha$ quantile of that law.\n\n## Remarks\n\nExample: in one-way analysis of variance, there are $k$ groups and $N$ observations in total, with independent normal errors of common variance.\nLet $\\mathrm{SSB}$ be the sum of squares between groups and $\\mathrm{SSW}$ the sum of squares within groups.\nUnder $H_0$ that all group means are equal, $F = \\frac{\\mathrm{SSB}/(k - 1)}{\\mathrm{SSW}/(N - k)}$ has the F-distribution with $k - 1$ and $N - k$ degrees of freedom.\nThe check: $\\mathrm{SSB}/\\sigma^2$ and $\\mathrm{SSW}/\\sigma^2$ are independent chi-squared variables with $k - 1$ and $N - k$ degrees of freedom, by Cochran's theorem.\nNon-example: for two non-normal samples, the variance ratio $S_1^2/S_2^2$ does not have an F-distribution under equal variances in general.\nSo comparing it with F quantiles is not an F-test in this sense, and its level can be far from $\\alpha$.\n",
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