F-Distribution and ANOVA

Let $V_1\sim\chi^2_{d_1}$ and $V_2\sim\chi^2_{d_2}$ be independent. Then $F = (V_1/d_1)/(V_2/d_2)\sim F_{d_1,d_2}$. Moreover, $1/F\sim F_{d_2,d_1}$, and the $\alpha$ quantile of $F_{d_1,d_2}$ equals the reciprocal of the $1 - \alpha$ quantile of $F_{d_2,d_1}$.

Each Chi-squared is normalized by its own degrees of freedom so that $V_i/d_i\to 1$ in probability as $d_i\to\infty$. The ratio approaches one under the null, with spread that shrinks as both degrees of freedom grow. The reciprocal-symmetry result lets you read lower-tail quantiles from upper-tail quantiles of the swapped distribution.

Every classical F-test has the form (numerator-variance estimate) / (denominator-variance estimate), where both estimates are scaled Chi-squareds when the underlying data are Normal. ANOVA, variance-comparison tests, and regression F-tests all fit this pattern. The asymptotic Chi-squared distribution of the LRT in regular models, multiplied by an appropriate factor, also gives a small-sample F approximation.

The construction requires independent Chi-squareds. ANOVA, regression, and variance-comparison tests use specific quadratic forms whose independence is guaranteed by Cochran's theorem applied to Normal samples. For non-Normal samples, the components are only approximately Chi-squared and only approximately independent, so the F-test is asymptotic and its accuracy depends on tail weight.

Let $\bar Y_{i\cdot}$ be the mean of group $i$ and $\bar Y_{\cdot\cdot}$ the grand mean. Then $\underbrace{\sum_{i=1}^g\sum_{j=1}^{n_i}(Y_{ij} - \bar Y_{\cdot\cdot})^2}_{\text{SST (total)}} = \underbrace{\sum_{i=1}^g n_i(\bar Y_{i\cdot} - \bar Y_{\cdot\cdot})^2}_{\text{SSB (between)}} + \underbrace{\sum_{i=1}^g\sum_{j=1}^{n_i}(Y_{ij} - \bar Y_{i\cdot})^2}_{\text{SSW (within)}}.$ The decomposition is algebraic and holds for every dataset, regardless of the model.

Total sum of squares is the squared distance from each point to the grand mean. Decompose each deviation $Y_{ij} - \bar Y_{\cdot\cdot}$ as $(Y_{ij} - \bar Y_{i\cdot}) + (\bar Y_{i\cdot} - \bar Y_{\cdot\cdot})$. The cross term vanishes after summing within each group (because deviations from a group mean sum to zero within the group), leaving SST = SSW + SSB.

The decomposition is the algebraic foundation for the F-test. Under the model $Y_{ij} = \mu + \tau_i + \epsilon_{ij}$ with $\epsilon_{ij}$ i.i.d. Normal and $\sum n_i\tau_i = 0$, the cross-product structure makes the between and within sums of squares independent. Each becomes a Chi-squared after dividing by $\sigma^2$, and their ratio (with appropriate degrees-of-freedom normalization) is the F-statistic.

The decomposition is purely algebraic and always holds, but the inferential interpretation of SSB and SSW as variance estimates of $\sigma^2$ requires the Normal-i.i.d.-equal-variance assumption. Heteroscedasticity (group-specific variances) breaks the F-test's exactness; non-Normal errors break it more mildly through the central limit theorem. For the heteroscedastic case, the Welch-style ANOVA replaces the pooled within-variance.

To test $H_0: \mu_1 = \cdots = \mu_g$ against $H_1$: at least two group means differ, compute the F-statistic $F = \frac{\text{SSB}/(g-1)}{\text{SSW}/(N-g)} = \frac{\text{MSB}}{\text{MSW}},$ where MSB and MSW are the mean squares between and within. Under $H_0$ and the Normal-equal-variance assumption, $F\sim F_{g-1, N-g}$ exactly. The test rejects at level $\alpha$ when $F$ exceeds the $1 - \alpha$ quantile of $F_{g-1, N-g}$.

Under the null, all group means equal a common $\mu$. SSB measures variability of the sample group means around the grand mean. SSW measures variability within groups around their own means. Both estimate $\sigma^2$ under the null: SSB/(g-1) and SSW/(N-g) are both unbiased estimators of $\sigma^2$. The ratio MSB/MSW concentrates near one under the null and is inflated when group means differ.

ANOVA is the standard test for comparing three or more group means in agricultural, industrial, clinical, and behavioral research. The test has high power against any deviation from equality among group means, not just specific contrast directions. When the null is rejected, follow up with post-hoc pairwise comparisons (Tukey's HSD, Scheffe, Bonferroni) to localize which means differ; see p-hacking and multiple testing for the multiple-comparison correction discussion.

The exact F distribution requires Normality, equal variances, and independence. Modest violations of Normality are tolerable in large samples by the central limit theorem; unequal variances are more damaging. Welch's ANOVA generalizes Welch's t-test to multiple groups and is the right alternative when equal variances are implausible. Independence violations (e.g., repeated measures on the same subject) require a mixed-effects model or a randomization-test approach.

To test $H_0:\sigma_X^2 = \sigma_Y^2$, compute the ratio of sample variances $F = \frac{S_X^2}{S_Y^2}.$ Under $H_0$ and the Normal assumption, $F\sim F_{m-1, n-1}$ exactly. A two-sided test at level $\alpha$ rejects when $F < F_{m-1, n-1, \alpha/2}$ or $F > F_{m-1, n-1, 1-\alpha/2}$.

Both $(m-1)S_X^2/\sigma_X^2$ and $(n-1)S_Y^2/\sigma_Y^2$ are Chi-squared with their respective degrees of freedom. Their ratio, normalized by degrees of freedom, is F-distributed. Under the null of equal variances the $\sigma^2$ cancels and the statistic is exactly $F_{m-1, n-1}$.

The variance-ratio F-test is used as a preliminary check before deciding whether to use the equal-variance pooled t-test or the Welch t-test. In practice, the test has low power for small samples and high sensitivity to non-Normality; Levene's test (which uses absolute deviations from group medians) is the standard alternative under non-Normality and is preferred in modern statistical practice.

The F-test for equal variances is notoriously sensitive to non-Normality, more so than the t-test for equal means. Even modest skewness or kurtosis substantially inflates the Type I error. Do not use this test as a "screening" decision for whether to pool variances; just use Welch's t-test directly, which avoids needing the equality test in the first place.