One-way ANOVA

Sums of squares for $y$

With subscript $i$ for participants/observations and $j$ for groups, the total sum of squares of $y$ can be expressed as

$$\begin{eqnarray*} \sum_{j=1}^g\limits \sum_{i=1}^{n_g}\limits (y_{ij}-\overline{y})^2 = & \sum_{j=1}^g\limits n_j(\overline{y}_j-\overline{y})^2 & + \sum_{j=1}^g\limits \sum_{i=1}^{n_g} \limits (y_{ij}-\overline{y}_j)^2 \\ SS_{tot} = & SS_{between} & + \text{ } SS_{within}, \end{eqnarray*}$$ where $SS_{between}$ is the sum of squares of the differences between groups, $SS_{within}$ the sum of squares of the differences within the groups, $n_j$ the number of observations in group $j$ and $g$ the number of groups.

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$F$-test ANOVA

$$\begin{equation*} F=\frac{MS_{between}}{MS_{within}}=\frac{SS_{between}/(g-1)}{SS_{within}/(n-g)}=\frac{\Big(\sum_{j=1}^g\limits n_j(\overline{y}_j-\overline{y})^2\Big) / (g-1)}{\Big(\sum_{j=1}^g\limits \sum_{i=1}^{n_g} \limits (y_{ij}-\overline{y}_j)^2\Big)/(n-g)}, \end{equation*}$$ with $df_1=g-1$ and $df_2=n-g$, where $n$ the total number of observations, $g$ the number of groups, and $n_j$ the number of observations in group $j$.

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$F$-test ANOVA - equal group sizes

$$\begin{equation*} F=\frac{MS_{between}}{MS_{within}}=\frac{\Big(n_c\sum_{j=1}^g\limits (\overline{y}_j-\overline{y})^2\Big) / (g-1)}{\Big(\sum_{j=1}^g \limits {s_j^2}\Big)/g} \end{equation*}$$ with $df_1=g-1$ (between) and $df_2=n-g$ (within) degrees of freedom, where $n_c$ denotes the number of observations in each group ($n_1=n_2=\cdots=n_g=n_c$).

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Proportion explained variation (eta$^2$)

$$\begin{equation*} \eta^2=\frac{SS_{between}}{SS_{tot}}=1-\frac{SS_{within}}{SS_{tot}} \end{equation*}$$

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Effect sizes ANOVA

$$\begin{equation*} f = \sqrt{\frac{SS_{between}}{SS_{within}}}=\sqrt{\frac{\eta^2}{1-\eta^2}} \end{equation*}$$

• $f=0.10$ small effect
• $f=0.25$ medium effect
• $f=0.40$ large effect

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$100(1-\alpha)\%$ - confidence interval post hoc comparisons groups $j$ and $k$

$$\begin{equation*} (\overline{y}_j-\overline{y}_k)-t_{\alpha/2}\cdot se_{\overline{y}_j-\overline{y}_k}\le (\mu_j-\mu_k) \le(\overline{y}_j-\overline{y}_k)+t_{\alpha/2}\cdot se_{\overline{y}_j-\overline{y}_k}, \end{equation*}$$ with $$\begin{equation*} se_{\overline{y}_j-\overline{y}_k}= \sqrt{\frac{SS_{within}}{n-g}\Big(\frac{1}{n_j}+ \frac{1}{n_k}\Big)},\end{equation*}$$ where $n$ is the ${\bf total}$ sample size, $g$ is the ${\bf total}$ number of groups and $df=(n-g)$ are the degrees of freedom associated with $MS_{within}$.

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Two-way ANOVA

Sums of squares for $y$

With factors $A$ and $B$ the total sum of squares of $y$ can be expressed as

$$\begin{eqnarray*} SS_T = SS_A+SS_B+SS_{AB}+SS_{within}. \end{eqnarray*}$$

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Mean sums of squares for $y$ (mean squares)

With $a$ levels for factor $A$ and $b$ levels for factor $B$ the mean squares are

$$\begin{eqnarray*} MS_A & = & SS_A/(a-1) \\ MS_B & = & SS_B/(b-1) \\ MS_{AB} & = & SS_{AB}/((a-1)(b-1)) \\ MS_{within} & = & SS_{within}/(n-ab) \end{eqnarray*}$$ $\textbf{NB}$ $MS_{within}$ is also referred to as $MS_{error}$.

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$F$-test interaction $A\times B$

$$\begin{equation*} F=MS_{AB}/MS_{within} \end{equation*}$$ with $df_1=(a-1)(b-1)$ and $df_2=n-ab$ (also: $df_{AB}$ and $df_{error}$)

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$F$-test main effect $A$

$$\begin{equation*} F=MS_{A}/MS_{within} \end{equation*}$$ with $df_1=a-1$ and $df_2=n-ab$ (also: $df_A$ and $df_{error}$).

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$F$-test main effect $B$

$$\begin{equation*} F=MS_{B}/MS_{within} \end{equation*}$$ with $df_1=b-1$ and $df_2=n-ab$ (also: $df_B$ and $df_{error}$).