1. Chapter 14Prepared by Samantha Gaies, M.A.1 Chapter 14: Two-Way ANOVA Let’s begin by reviewing one-way ANOVA. Try this example… Does motivation level affect performance?

2. Chapter 14Prepared by Samantha Gaies, M.A.2 The answer to the question posed on the previous slide appears to be “no,” because F calc = 30/11.11 = 2.7 < F.05 (2, 27) = 3.35. Now suppose that we add a second, preexisting, factor: “the need to achieve”… The new error term is MS within-cell, which is the average of all the cell variances. Squaring the six s’s in the above table and averaging, we get MS w-cell = 5.0, less than half of the error term for the one- way ANOVA. The F for Motivation is now 30/5.0 = 6.0, p <.05. The reduced error term that is typical when adding an individual- difference variable that also affects your DV is just one of the possible advantages of the two-way ANOVA.

3. Chapter 14Prepared by Samantha Gaies, M.A.3 Completing the Two-Way ANOVA We recalculated the F ratio for Motivation using the new (within-cell) error term; the numerator of that F ratio did not change, because the column means aren’t changed by adding the second variable. In a two-way ANOVA, this is referred to as testing the main effect of Motivation. Next, we test the main effect of need for achievement (NAch) using the same error term as the other main effect. The numerator is based on the means for high (M = 6.0) and low (M = 2.0) achievement. F calc for the main effect of NAch = 120 / 5.0 = 24, which is easily significant. There is a third F ratio to test, which is the interaction of the two independent variables (called factors). Note that the pattern of Motivation means for High NAch is different from the pattern for the Low NAch cells. The more discrepant the patterns, the greater the F ratio for the interaction. We will describe the calculation of the two-way ANOVA in greater detail soon.

4. Chapter 14Prepared by Samantha Gaies, M.A.4 Completely Crossed Factorial Design This is the simplest way to combine two or more factors (called a factorial design, for short). –With two factors, every level of one factor is combined with every level of the other factor, such that every possible combina- tion of levels, called a cell, corresponds to a group of subjects in the experiment – that is, none of the cells is empty. –In a balanced design, all cells have the same number of subjects The Motivation/Need for Achievement example is a balanced, factorial design. It is common to describe a factorial design in terms of the number of levels of each factor–e.g., the Motivation/NAch experiment is an example of a 2 X 3 factorial design.

5. Chapter 14Prepared by Samantha Gaies, M.A.5 Calculating the Two-Way ANOVA Calculate MS within-cell (MS W, for short) –In a balanced design, MS w is the simple average of all the cell variances (otherwise it is a weighted average) Calculate SS Bet for each main effect (in the formulas below, c stands for the number of columns, and r is the number of rows). –Apply the following formula to the column means in the Motivation/NAch example: –And again to the row means: Then, calculate SS Bet for all of the cells (SS Bet-cell ), using the same formula on the cell means: Finally, find the SS for interaction by subtraction: SS inter = SS bet-cell – SS Mot – SS ach = 240 –60 –120 = 60

6. Chapter 14Prepared by Samantha Gaies, M.A.6 Calculating the Two-Way ANOVA (cont.) The degrees of freedom are as follows: df motiv = c – 1 = 2; df NAch = r – 1 = 1; df inter = (c – 1)(r – 1) = 2; df W = N T – rc = 30 – 6 = 24. The numerator MSs for the three F ratios are found by dividing the SSs by their dfs:

7. Chapter 14Prepared by Samantha Gaies, M.A.7 Calculating the Two-Way ANOVA (cont.) We have already calculated MS W directly by averaging the cell variances (the squared SDs) from the 2 X 3 table for this example. (We could have calculated SS total, directly from the data, and then subtracted SS bet-cell to obtain SS W.) MS W is used as the error term (i.e., denomi- nator) for all three of the F ratios in a two- way ANOVA, like so: The critical F for both the main effect of motivation and the interaction of the two factors is: F.05 (2, 24) = 3.40; and for need for achievement, F.05 (1, 24) = 4.26. Thus, all three F ratios are significant at the.05 level.

8. Chapter 14Prepared by Samantha Gaies, M.A.8 Interpreting the Results of a Two-Way ANOVA After obtaining one or more significant F ratios, it is likely that you will want to graph the cell means as an aid to interpreting your results and planning follow-up tests to further specify the observed effects, as we have done below for the motivation/need for achievement example. The source of the significant interaction can be seen where the lines sharply diverge from being parallel. You can also see that the High NAch line is generally higher than the line for Low NAch, and that both lines are lowest for Low Motivation.

9. Chapter 14Prepared by Samantha Gaies, M.A.9 Types of Interactions –As a rule, a significant interaction tells you not to take the main effects at face value. From a graph of the cell means, you can then see whether you are dealing with a(n): Ordinal interaction: The direction (order) of the effects is consistent—i.e., the lines on the graph slope in the same direction. Disordinal interaction: The direction of the effects for one factor reverses for different levels of the other factor—i.e., one line will slope upward, while the other slopes downward. When a disordinal interaction is most extreme, the lines on the graph will form a perfect X, and the main effects will be completely obliterated. (In general, disordinal interactions complicate the interpretation of the main effects more than do ordinal interactions. When a disordinal interaction is significant, it is common not to interpret the main effects, or conduct follow- up tests on them, even if they are significant. See next slide for examples.)

10. Chapter 14Prepared by Samantha Gaies, M.A.10 Ordinal Interaction Disordinal Interaction

11. Chapter 14Prepared by Samantha Gaies, M.A.11 Post Hoc Comparisons for Significant Main Effects –If the interaction is not significant and a significant main effect involves more than two levels Use appropriate post hoc tests Common to view each factor as a separate “family”, and therefore control familywise alpha (e.g., use the.05 level separately for both factors). For three levels, use LSD: where n level is the number of subjects contributing to each of the means being compared For more than three levels of a main effect, use Tukey’s test:

12. Chapter 14Prepared by Samantha Gaies, M.A.12 Post Hoc Comparisons for a Significant Interaction –Simple Main Effects The effect of one factor while holding the other factor fixed at one level The main effects tested in the two-way ANOVA are actually averages of the relevant simple main effects. Common approach: Test each simple main effect as a one-way ANOVA, using MS W from the two-way ANOVA as the error term in each case. –For significant simple effects, follow up with pairwise comparisons, using an appropriate post hoc test Alternative approaches: (1) Proceed directly to cell-to-cell comparisons. (2) Conduct only planned comparisons.

13. Chapter 14Prepared by Samantha Gaies, M.A.13 Effect Sizes in the Two-Way ANOVA –The ordinary eta squared (η 2 ord ), which equals SS effect / SS total in a two-way ANOVA, is misleading when both factors involve experimental manipulations. –In that case, calculate partial eta squared, η 2 p : –You can calculate η 2 p for a particular effect from the F ratio for testing that effect and the appropriate dfs:

14. Chapter 14Prepared by Samantha Gaies, M.A.14 Advantages of the Two-Way ANOVA When both factors are experimental: –Economy If interaction is not significant, you have performed two one-way ANOVAs in an efficient manner. –Exploration of interactions These are often more interesting and informative than the main effects. When one factor is a grouping (i.e., pre- existing, individual difference) factor: –Reduction in error term Removes differences on the grouping factor from the error term of the two-way ANOVA, thus increasing power for the experimental factor. –Exploration of interaction between the experimental factor and the grouping factor, which may act as a moderating variable.

15. Chapter 14Prepared by Samantha Gaies, M.A.15 A researcher is interested in whether caffeine affects learning on a memory task. She is also interested in whether there are gender differences. Here are the three null hypotheses: –H 0 : μ female = μ male –H 0 : μ large = μ moderate = μ small = μ zero –H 0 : The effects of the two factors will be additive (i.e., zero interaction) Try this example of a 2 X 4 ANOVA….

16. Chapter 14Prepared by Samantha Gaies, M.A.16 Try to interpret the graph below in light of the results in the box above. Results of the 2 X 4 ANOVA example F.05 (1,32) = 4.17 F.05 (3,32) = 2.92