Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Analysis of variance ▷ F-statistics in terms of OLS estimates

Theorem: Given the two-way analysis of variance assumption

\[\label{eq:anova2} y_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + \varepsilon_{ijk}, \; \varepsilon_{ijk} \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; ,\]

the F-statistics for the grand mean, the main effects and the interaction can be expressed in terms of ordinary least squares parameter estimates as

\[\label{eq:anova2-fols} \begin{split} F_M &= \frac{n \hat{\mu}^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \\ F_A &= \frac{\frac{1}{a-1} \sum_{i=1}^{a} n_{i \bullet} \hat{\alpha}_i^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \\ F_B &= \frac{\frac{1}{b-1} \sum_{j=1}^{b} n_{\bullet j} \hat{\beta}_j^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \\ F_{A \times B} &= \frac{\frac{1}{(a-1)(b-1)} \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij} \hat{\gamma}_{ij}^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \end{split}\]

where the predicted values $\hat{y}_{ijk}$ are given by

\[\label{eq:y-est} \hat{y}_{ijk} = \hat{\mu} + \hat{\alpha}_i + \hat{\beta}_j + \hat{\gamma}_{ij} \; .\]

Theorem: The F-statistics for the grand mean, the main effects and the interaction in two-way ANOVA are calculated as

\[\label{eq:anova2-f} \begin{split} F_M &= \frac{n (\bar{y}_{\bullet \bullet \bullet})^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2} \\ F_A &= \frac{\frac{1}{a-1} \sum_{i=1}^{a} n_{i \bullet} (\bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet \bullet \bullet})^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2} \\ F_B &= \frac{\frac{1}{b-1} \sum_{j=1}^{b} n_{\bullet j} (\bar{y}_{\bullet j \bullet} - \bar{y}_{\bullet \bullet \bullet})^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2} \\ F_{A \times B} &= \frac{\frac{1}{(a-1)(b-1)} \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij} (\bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} + \bar{y}_{\bullet \bullet \bullet})^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \bar{y}_{i j \bullet})^2} \end{split}\]

and the ordinary least squares estimates for two-way ANOVA are

\[\label{eq:anova2-ols} \begin{split} \hat{\mu} &= \bar{y}_{\bullet \bullet \bullet} \\ \hat{\alpha}_i &= \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet \bullet \bullet} \\ \hat{\beta}_j &= \bar{y}_{\bullet j \bullet} - \bar{y}_{\bullet \bullet \bullet} \\ \hat{\gamma}_{ij} &= \bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} + \bar{y}_{\bullet \bullet \bullet} \end{split}\]

where the the sample means are given by

\[\label{eq:mean-samp} \begin{split} \bar{y}_{\bullet \bullet \bullet} &= \frac{1}{n} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\ \bar{y}_{i \bullet \bullet} &= \frac{1}{n_{i \bullet}} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} y_{ijk} \\ \bar{y}_{\bullet j \bullet} &= \frac{1}{n_{\bullet j}} \sum_{i=1}^{a} \sum_{k=1}^{n_{ij}} y_{ijk} \\ \bar{y}_{i j \bullet} &= \frac{1}{n_{ij}} \sum_{k=1}^{n_{ij}} y_{ijk} \; . \end{split}\]

We first note that the predicted values can be evaluated as

\[\label{eq:y-est-qed} \begin{split} \hat{y}_{ijk} &= \hat{\mu} + \hat{\alpha}_i + \hat{\beta}_j + \hat{\gamma}_{ij} \\ &= \bar{y}_{\bullet \bullet \bullet} + (\bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet \bullet \bullet}) + (\bar{y}_{\bullet j \bullet} - \bar{y}_{\bullet \bullet \bullet}) + (\bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} + \bar{y}_{\bullet \bullet \bullet}) \\ &= \bar{y}_{i \bullet \bullet} + \bar{y}_{\bullet j \bullet} + \bar{y}_{i j \bullet} - \bar{y}_{i \bullet \bullet} - \bar{y}_{\bullet j \bullet} \\ &= \bar{y}_{i j \bullet} \; . \end{split}\]

Substituting this \eqref{eq:y-est-qed} and the OLS estimates \eqref{eq:anova2-ols} into the F-formulas \eqref{eq:anova2-f}, we obtain:

\[\label{eq:anova2-fols-qed} \begin{split} F_M &= \frac{n \hat{\mu}^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \\ F_A &= \frac{\frac{1}{a-1} \sum_{i=1}^{a} n_{i \bullet} \hat{\alpha}_i^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \\ F_B &= \frac{\frac{1}{b-1} \sum_{j=1}^{b} n_{\bullet j} \hat{\beta}_j^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \\ F_{A \times B} &= \frac{\frac{1}{(a-1)(b-1)} \sum_{i=1}^{a} \sum_{j=1}^{b} n_{ij} \hat{\gamma}_{ij}^2}{\frac{1}{n-ab} \sum_{i=1}^{a} \sum_{j=1}^{b} \sum_{k=1}^{n_{ij}} (y_{ijk} - \hat{y}_{ijk})^2} \; . \end{split}\]
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Metadata: ID: P380 | shortcut: anova2-fols | author: JoramSoch | date: 2022-11-16, 16:34.