Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Analysis of variance ▷ Interaction sum of squares

Definition: Let there be an analysis of variance (ANOVA) model with two or more factors influencing the measured data $y$ (here, using the standard formulation of two-way ANOVA):

\[\label{eq:anova} y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + \varepsilon_{ijk}, \; \varepsilon_{ijk} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .\]

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 of the corresponding factors:

\[\label{eq:iass} \begin{split} \mathrm{SS}_\mathrm{A \times B} &= \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} ([\bar{y}_{i j \bullet} - \bar{y}_{\bullet \bullet \bullet}] - [\bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet \bullet \bullet}] - [\bar{y}_{\bullet j \bullet} - \bar{y}_{\bullet \bullet \bullet}])^2 \\ &= \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (\bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} + \bar{y}_{\bullet \bullet \bullet})^2 \; . \end{split}\]

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 $y_{ijk}$.

 
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Metadata: ID: D184 | shortcut: iass | author: JoramSoch | date: 2022-12-14, 13:14.