Example 11.4.4

In the balanced two-factor ANOVA model (ie, nijn0), the estimates are much simpler when one uses the following notations

$\begin{array}{ll}\hfill {\stackrel{-}{Y}}_{\cdot \cdot \cdot }& ={\left({n}_{0}ab\right)}^{-1}\sum _{i}\sum _{j}\sum _{k}{Y}_{ijk},\hfill \\ \hfill {\stackrel{-}{Y}}_{i\cdot \cdot }& ={\left({n}_{0}b\right)}^{-1}\sum _{j}\sum _{k}{Y}_{ijk},{\stackrel{-}{Y}}_{\cdot j\cdot }={\left({n}_{0}a\right)}^{-1}\sum _{i}\sum _{k}{Y}_{ijk},\text{and}\hfill \\ \hfill {\stackrel{-}{Y}}_{ij\cdot }& ={n}_{0}^{-1}\sum _{k}{Y}_{ijk}.\hfill \end{array}$ With these notations

$\begin{array}{l}\stackrel{^}{\mu }={\stackrel{-}{Y}}_{\cdot \cdot \cdot },{\stackrel{^}{\alpha }}_{i}={\stackrel{-}{Y}}_{i\cdot \cdot }-{\stackrel{-}{Y}}_{\cdot \cdot \cdot },{\stackrel{^}{\beta }}_{j}={\stackrel{-}{Y}}_{\cdot j\cdot }-{\stackrel{-}{Y}}_{\cdot \cdot \cdot },\hfill \\ {\stackrel{^}{\left(\alpha \beta \right)}}_{ij}={\stackrel{-}{Y}}_{ij\cdot }-{\stackrel{-}{Y}}_{i\cdot \cdot }-{\stackrel{-}{Y}}_{\cdot j\cdot }+{\stackrel{-}{Y}}_{\cdot \cdot \cdot },\text{and}\hfill \\ {Y}_{ijk}-\stackrel{^}{\mu }={\stackrel{^}{\alpha }}_{i}+{\stackrel{^}{\beta }}_{j}+{\stackrel{^}{\left(\alpha \beta \right)}}_{ij}+{\stackrel{^}{\epsilon }}_{ijk},\hfill \end{array}$ where $\left\{{\stackrel{^}{\epsilon }}_{ijk}={Y}_{ijk}-{\stackrel{-}{Y}}_{ij\cdot }\right\}$ are the residuals. If both sides are squared and summed over i, j, and

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