Chapter 13 Two-Way ANOVA

Usage: To check for a difference between three or more group means with two sets of factors.

Requirements: Three or more variables that are interval/ratio and two grouping variables.

Steps to conducting an two-way ANOVA:

Write out null and alternative hypotheses.
A. Main effects of Factor A.
B. Main effects of Factor B.
C. Interaction of Factors A and B.
Determine critical F-values for rejection of null hypotheses.
A. Use degrees of freedom (df) and alpha level . B. One CV for each hypothesis.
Calculate 3 F-values for your data.
A. One for each hypothesis.
Compare your calculated F-value to the critical F-value.
A. If your calculated value is greater than the critical value, you reject the null hypothesis.
B. If your calculated value is less than the critical value, you fail to reject the null hypothesis.
Calculate effect size (eta squared \(\eta^2\)).

13.1 Example Setup

For Factor A:

\(\Large H_0: \mu_{a_1} = \mu_{a_2}\)

\(\Large H_1: \mu_{a_1} \neq \mu_{a_2}\)

For Factor B:

\(\Large H_0: \mu_{b_1} = \mu_{b_2}\)

\(\Large H_1: \mu_{b_1} \neq \mu_{b_2}\)

For Interaction:

\(\Large H_0: No\ interaction\)

\(\Large H_1: Interaction\ present\)

\(\Large N = number\ of\ participants\)

\(\Large n_{group} = number\ of\ participants\ in\ group\)

\(\Large k = number\ of\ groups\)

\(\Large df_{between} = k - 1\)

\(\Large df_{within} = (levels\ of\ A)*(levels\ of\ B)(n - 1)\)

\(\Large df_{total} = N - 1\)

\(\Large df_{A} = (levels\ of\ A) - 1\)

\(\Large df_{B} = (levels\ of\ B) - 1\)

\(\Large df_{interaction} = (df_A)*(df_B)\)

\(\Large \alpha = .05\)

\(\Large CV = [check\ table]\)

\(\Large T_{group} = \sum X_{group}\)

\(\Large SS_{total} = \sum X^2 - \frac{(\sum X)^2}{N}\)

\(\Large SS_{within} = \sum X^2 - \frac{\sum T^2}{n}\)

\(\Large SS_{between} = \frac{\sum T^2}{n} - \frac{(\sum X)^2}{N}\)

\(\Large SS_{A} = \sum(\frac{T^2_{row}}{n_{row}}) - \frac{(\sum X)^2}{N}\)

\(\Large SS_{B} = \sum(\frac{T^2_{col}}{n_{col}}) - \frac{(\sum X)^2}{N}\)

\(\Large SS_{AB} = SS_{between} - SS_A - SS_B\)

\(\Large MS_{A} = \frac{SS_{A}}{df_{A}}\)

\(\Large MS_{B} = \frac{SS_{B}}{df_{B}}\)

\(\Large MS_{AB} = \frac{SS_{AB}}{df_{AB}}\)

\(\Large F_A = \frac{MS_{A}}{MS_{within}}\)

\(\Large F_B = \frac{MS_{B}}{MS_{within}}\)

\(\Large F_{AB} = \frac{MS_{AB}}{MS_{within}}\)

If F is greater than the CV, reject the null hypothesis.
If F is less than the CV, fail to reject the null hypothesis.

\(\Large \eta^2_A = \frac{SS_{A}}{SS_{A} + SS_{within}}\)

\(\Large \eta^2_B = \frac{SS_{B}}{SS_{B} + SS_{within}}\)

\(\Large \eta^2_{AB} = \frac{SS_{AB}}{SS_{AB} + SS_{within}}\)

Draw a table like the one below to organize your calculations as you go.

Source	SS	df	MS	F
Main Effect Factor A	40	1	40	2
Main Effect Factor B	40	1	40	2
Interaction	80	1	80	4
Within	140	7	20
Total	300	10

F is actually closely related to t: \(\Large F = t^2\)