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**Definition:** Let there be an analysis of variance (ANOVA) model with [two](/D/anova2) or [more](/D/anovan) factors influencing the measured data $y$ (here, using the [standard formulation](/P/anova2-pss) of [two-way ANOVA](/D/anova2)):
Then, the interaction sum of squares is defined as the [explained sum of squares] (ESS) for each interaction, i.e. as the sum of squared deviations of the average for each cell from the average across all observations, controlling for the [treatment sums of squares](/D/trss) of the corresponding factors:
Here, $\bar{y} _{i j \bullet}$ is the mean for the $(i,j)$-th cell (out of $a \times b$ cells), computed from $n_{ij}$ values $y_{ijk}$, $\bar{y} _{i \bullet \bullet}$ and $\bar{y} _{\bullet j \bullet}$ are the level means for the two factors and and $\bar{y} _{\bullet \bullet \bullet}$ is the mean across all values $y_{ijk}$.
**Definition:** Let there be an analysis of variance (ANOVA) model with [one](/D/anova1), [two](/D/anova2) or [multiple](/D/anovan) factors influencing the measured data $y$ (here, using the [reparametrized version](/P/anova1-repara) of [one-way ANOVA](/D/anova1)):
Then, the treatment sum of squares is defined as the [explained sum of squares] (ESS) for each main effect, i.e. as the sum of squared deviations of the average for each level of the factor, from the average across all observations:
Here, $\bar{y}_i$ is the mean for the $i$-th level of the factor (out of $k$ levels), computed from $n_i$ values $y_{ij}$, and $\bar{y}$ is the mean across all values $y_{ij}$.
Assume that $\mu$ zero, according to $H_0$ given by \eqref{eq:anova2-h0}. Under this null hypothesis, we have:
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$$ \label{eq:yijk-h0}
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y_{ijk} \sim \mathcal{N}(\alpha_i + \beta_j + \gamma_{ij}, \sigma^2) \quad \text{for all} \quad i, j, k \; .
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$$
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Thus, the [random variable](/D/rvar) $U_{ijk} = (y_{ijk} - \alpha_i - \beta_j - \gamma_{ij})/\sigma$ [follows a standard normal distribution](/P/norm-snorm)
[Cochran's theorem](/P/snorm-cochran) states that, if a sum of squared [standard normal](/D/snorm)[random variables](/D/rvar) can be written as a sum of squared forms
where $J_n$ is an $n \times n$ matrix of ones, $J_{n,m}$ is an $n \times m$ matrix of ones and $\mathrm{diag}\left( A_1, \ldots, A_n \right)$ denotes a block-diagonal matrix composed of $A_1, \ldots, A_n$. We observe that those matrices satisfy
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$$ \label{eq:U-Q-B}
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\sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} U_{ijk]^2 = Q_1 + Q_2 + Q_3 = U^\mathrm{T} B^{(1)} U + U^\mathrm{T} B^{(2)} U + U^\mathrm{T} B^{(3)} U
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$$
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as well as
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$$ \label{eq:B-In}
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B^{(1)} + B^{(2)} + B^{(3)} = I_n
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$$
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and their ranks are
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$$ \label{eq:B-rk}
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\begin{split}
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\mathrm{rank}\left( B^{(1)} \right) &= n - a \cdot b \\
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\mathrm{rank}\left( B^{(2)} \right) &= 1 \\
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\mathrm{rank}\left( B^{(3)} \right) &= n - (n-ab) - 1 = a \cdot b - 1 \; .
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\end{split}
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$$
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Let's write down the [explained sum of squares](/D/ess) and the [residual sum of squares](/D/rss) for [two-way analysis of variance](/D/anova2) as
Because $\mathrm{ESS}/\sigma^2$ and $\mathrm{RSS}/\sigma^2$ are also independent by \eqref{eq:cochran-p2}, the F-statistic from \eqref{eq:anova2-fgm} is equal to the ratio of two independent [chi-squared distributed](/D/chi2)[random variables](/D/rvar) divided by their degrees of freedom
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Thus, the F-statistic from \eqref{eq:anova2-fgm} is equal to the ratio of two [independent](/D/ind)[chi-squared distributed](/D/chi2)[random variables](/D/rvar) divided by their degrees of freedom
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