Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Analysis of variance ▷ F-test for grand mean in two-way ANOVA

Theorem: Assume the two-way analysis of variance model

\[\label{eq:anova2} \begin{split} y_{ijk} &= \mu + \alpha_i + \beta_j + \gamma_{ij} + \varepsilon_{ijk} \\ \varepsilon_{ijk} &\overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2), \; i = 1, \ldots, a, \; j = 1, \ldots, b, \; k = 1, \dots, n_{ij} \; . \end{split}\]

Then, the test statistic

\[\label{eq:anova2-fgm} F_M = \frac{n (\bar{y}_{\bullet \bullet \bullet})^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2}\]

follows an F-distribution

\[\label{eq:anova2-fgm-h0} F_M \sim \mathrm{F}\left( 1, n-ab \right)\]

under the null hypothesis for the grand mean

\[\label{eq:anova2-h0} \begin{split} H_0: &\; \mu = 0 \\ H_1: &\; \mu \neq 0 \; . \end{split}\]

Proof: Applying Cochran’s theorem for two-analysis of variance, we find that the following squared sums

\[\label{eq:anova2-ss-dist} \begin{split} \frac{\mathrm{SS}_M}{\sigma^2} &= \frac{1}{\sigma^2} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (\bar{y}_{\bullet \bullet \bullet} - \mu)^2 = \frac{1}{\sigma^2} n (\bar{y}_{\bullet \bullet \bullet} - \mu)^2 \\ \frac{\mathrm{SS}_\mathrm{res}}{\sigma^2} &= \frac{1}{\sigma^2} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2 = \frac{1}{\sigma^2} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2 \end{split}\]

are independent and chi-squared distributed:

\[\label{eq:anova2-cochran-s1} \begin{split} \frac{\mathrm{SS}_M}{\sigma^2} &\sim \chi^2(1) \\ \frac{\mathrm{SS}_\mathrm{res}}{\sigma^2} &\sim \chi^2(n-ab) \; . \end{split}\]

Thus, the F-statistic from \eqref{eq:anova2-fgm} is equal to the ratio of two independent chi-squared distributed random variables divided by their degrees of freedom

\[\label{eq:anova2-fgm-ess-tss} \begin{split} F_M &= \frac{(\mathrm{SS}_M/\sigma^2)/(1)}{(\mathrm{SS}_\mathrm{res}/\sigma^2)/(n-ab)} \\ &= \frac{\mathrm{SS}_M/(1)}{\mathrm{SS}_\mathrm{res}/(n-ab)} \\ &\overset{\eqref{eq:anova2-ss-dist}}{=} \frac{n (\bar{y}_{\bullet \bullet \bullet} - \mu)^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2} \\ &\overset{\eqref{eq:anova2-h0}}{=} \frac{n (\bar{y}_{\bullet \bullet \bullet})^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2} \end{split}\]

which, by definition of the F-distribution, is distributed as

\[\label{eq:anova2-fia-qed} F_M \sim \mathrm{F}(1, n-ab)\]

under the null hypothesis for the grand mean.

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Metadata: ID: P374 | shortcut: anova2-fgm | author: JoramSoch | date: 2022-11-11, 16:54.