Analysis of variance

Definition

Let $k \ge 2$ and $n_1, \dots, n_k \ge 1$, with $N = n_1 + \cdots + n_k > k$. Let $y_{ij}$, for $i = 1, \dots, k$ and $j = 1, \dots, n_i$, be observations of $k$ groups. Let $\bar y_i = \frac{1}{n_i} \sum_j y_{ij}$ be the mean of group $i$ and $\bar y = \frac{1}{N} \sum_{i,j} y_{ij}$ the overall mean. The one-way analysis of variance is the decomposition $$\sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar y)^2 = \sum_{i=1}^k n_i (\bar y_i - \bar y)^2 + \sum_{i=1}^k \sum_{j=1}^{n_i} (y_{ij} - \bar y_i)^2,$$ written $SS_{\mathrm{tot}} = SS_{\mathrm{between}} + SS_{\mathrm{within}}$, together with the statistic $$F = \frac{SS_{\mathrm{between}} / (k - 1)}{SS_{\mathrm{within}} / (N - k)},$$ defined when $SS_{\mathrm{within}} > 0$. Consider the model $y_{ij} = \mu_i + \varepsilon_{ij}$ with independent errors $\varepsilon_{ij} \sim N(0, \sigma^2)$. The ANOVA $F$-test of level $\alpha$ rejects the hypothesis $\mu_1 = \cdots = \mu_k$ when $F$ is large. The threshold is the upper $\alpha$-quantile of the F-distribution with $k - 1$ and $N - k$ degrees of freedom.