6.8: Complexity Happens
- Page ID
- 33665
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)From what we have discussed so far, we see that even for the simplest multi-factor studies (i.e. those involving only two factors), there are many possibilities of treatment designs generated by either factor being fixed or random, and factors being crossed or nested.
For any of these possibilities, we can carry out the hypothesis tests using the EMS expressions to identify the correct denominator for the relevant \(F\)-statistics.
| Crossed | ||||
|---|---|---|---|---|
| Source | d.f. | A fixed, B fixed | A fixed, B random | A random, B random |
| A | \(a-1\) | \(\sigma^{2} + nb \frac{\sum \alpha_{i}^{2}}{a-1}\) | \(\sigma^{2} + nb \frac{\sum \alpha_{i}^{2}}{a-1} + n \sigma_{\alpha \beta}^{2}\) | \(\sigma^{2} + nb \sigma_{\alpha}^{2} + n \sigma_{\alpha \beta}^{2}\) |
| B | \(b-1\) | \(\sigma^{2} + na \frac{\sum \beta_{i}^{2}}{b-1}\) | \(\sigma^{2} + na \sigma_{\beta}^{2}\) | \(\sigma^{2} + na \sigma_{\beta}^{2} + n \sigma_{\alpha \beta}^{2}\) |
| A×B | \((a-1)(b-1)\) | \(\sigma^{2} + n \frac{\sum \sum (\alpha \beta)_{ij}^{2}}{(a-1)(b-1)}\) | \(\sigma^{2} + n \sigma_{\alpha \beta}^{2}\) | \(\sigma^{2} + n \sigma_{\alpha \beta}^{2}\) |
| \(\sigma^{2}\) | \(\sigma^{2}\) | \(\sigma^{2}\) | ||
| Nested | ||||
|---|---|---|---|---|
| Source | d.f. | A fixed, B fixed | A fixed, B random | A random, B random |
| A | \(a-1\) | \(\sigma^{2} + bn \frac{\sum \alpha_{i}^{2}}{a-1}\) | \(\sigma^{2} + bn \frac{\sum \alpha_{i}^{2}}{a-1} + n \sigma_{\beta(\alpha)}^{2}\) | \(\sigma^{2} + bn \sigma_{\alpha}^{2} + n \sigma_{\beta(\alpha)}^{2}\) |
| B(A) | \(a(b-1)\) | \(\sigma^{2} + n \frac{\sum \sum \beta_{j(i)}^{2}}{a(b-1)}\) | \(\sigma^{2} + n \sigma_{\beta(\alpha)}^{2}\) | \(\sigma^{2} + n \sigma_{\beta(\alpha)}^{2}\) |
| Error | \(ab(n-1)\) | \(\sigma^{2}\) | \(\sigma^{2}\) | \(\sigma^{2}\) |