2. Factorial Analysis of Variance One dependent variable, more than one independent variable (“factor”) 1 2

3. KNR 445 FACTORIAL ANOVA Slide 2 Two factors, more reality  Imagine you want to describe what makes GPA, body fat, a team’s winning %, the outcome of an electoral poll vary…  Do they depend on just one thing?  Of course not  More IVs simply get closer to the truth (to explaining all of the DV - increase overall R 2 )  Factorial ANOVA & one-way ANOVA  Multiple and simple regression  ANOVA – categorical IVs 1 2

4. KNR 445 FACTORIAL ANOVA Slide 3 Two factors, more reality  How factorial designs work  Consider this experiment:  Take 2 sets of golfers: 1 set (A 1 ) is high anxious, 1 set (A 2 ) is low anxious  Assign 1/3 of each set of golfers to a different performance scenario: Low pressure (B 1 ), Moderate pressure (B 2 ), High pressure (B 3 ) 1 2 3

5. KNR 445 FACTORIAL ANOVA Slide 4 Two factors, more reality  So for assignment to groups we get: Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety n cell = 2 n(A 1 ) =6 High Anxiety n cell = 2 n(A 2 ) =6 n(B 1 ) = 4n(B 2 ) = 4n(B 3 ) = 4n total = 12 1 2 3

6. Vocabulary  Factor = Independent variable  Two-factor ANOVA / Two-way ANOVA: an experiment with 2 independent variables  Levels: number of treatment conditions (groups) for a specific IV  N OTATION  3 X 2 ANOVA = experiment w/2 IVs: one w/3 levels, one w/2 levels  2 X 2 ANOVA = experiment w/2 IVs: both w/2 levels  3 X 2 X 2 = ???? KNR 445 FACTORIAL ANOVA Slide 5 1 2 3 4

7. KNR 445 FACTORIAL ANOVA Slide 6 Two factors, more reality  Suppose that the performance scores are… Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety M = 5 (3, 7) M = 8 (7, 9) M = 11 (10, 12) M A1 = 8 High Anxiety M = 4 (3, 5) M = 6 (5, 7) M = 2 (2, 2) M A2 = 4 M B1 = 4.5M B2 = 7M B3 = 6.5M total = 6 1 2 3

8. KNR 445 FACTORIAL ANOVA Slide 7 Introducing MAIN EFFECTS  Suppose that the performance scores are… Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety M = 5 (3, 7) M = 8 (7, 9) M = 11 (10, 12) M A1 = 8 High Anxiety M = 4 (3, 5) M = 6 (5, 7) M = 2 (2, 2) M A2 = 4 M B1 = 4.5M B2 = 7M B3 = 6.5M total = 6 1 2

9. KNR 445 FACTORIAL ANOVA Slide 8 MAIN EFFECTS  What do we find?  We can consider the overall effect of anxiety (Factor A) on performance  The null hypothesis here would be  This is analogous to doing a t-test or 1-way ANOVA on the row means of M A1 (8) and M A2 (4) NB: if you were to do a 1-way ANOVA, you’d ignore the effect of pressure (IV B ) completely 1

10. KNR 445 FACTORIAL ANOVA Slide 9 MAIN EFFECTS  This overall effect of anxiety is called the main effect of anxiety 1

11. KNR 445 FACTORIAL ANOVA Slide 10 MAIN EFFECTS  What do we find?  We can also consider the overall effect of situation (Factor B) on performance  The null hypothesis here would be  This is analogous to doing a 1-way ANOVA on the row means of M B1 (4.5), M B2 (7) and M B3 (6.5) NB: here, you’d ignore the effect of anxiety (IV A ) completely 1

12. KNR 445 FACTORIAL ANOVA Slide 11 MAIN EFFECTS  This overall effect of situation is called the main effect of situation  In each of the main effects, note that each mean within the main effect has been computed by averaging across levels of the factor not considered in the main effect  This is how it is ignored, statistically. Its effects are, quite literally, averaged out WHENEVER YOU INTERPRET A MAIN EFFECT, YOU SHOULD PAY ATTENTION TO THE FACT THAT IT AVERAGES ACROSS LEVELS OF THE OTHER FACTOR – ESPECIALLY WHEN YOU GET… 1 2

13. KNR 445 FACTORIAL ANOVA Slide 12 INTERACTIONS  Note the difference between each pair of means in our original table of data Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety M = 5 (3, 7) M = 8 (7, 9) M = 11 (10, 12) M A1 = 8 High Anxiety M = 4 (3, 5) M = 6 (5, 7) M = 2 (2, 2) M A2 = 4 M B1 = 4.5M B2 = 7M B3 = 6.5M total = 6 5-4 = 18-6 = 2 11-2 = 9 1 2 3 4

14. KNR 445 FACTORIAL ANOVA Slide 13 INTERACTIONS  The magnitude of the difference changes depending on the pressure level  In other words…  In other words, the effect of anxiety on performance depends on the pressure level in which the participants are asked to perform  In other words, the pressure level moderates the effect of anxiety on performance  In other words, the anxiety-performance relationship differs depending on the pressure level 1 2

15. KNR 445 FACTORIAL ANOVA Slide 14 INTERACTIONS  You might find it easier to see in a graph: 1 2 3 Ordinal interaction = lines do not cross 4

16. KNR 445 FACTORIAL ANOVA Slide 15 INTERACTIONS  The essential point is, when the lines are significantly non-parallel, you have an interaction, and the effect of one factor on the dependent variable depends on the level of other factor being considered Non-parallelism is a necessary but not sufficient condition for an interaction to be present 1 2

17. KNR 445 FACTORIAL ANOVA Slide 16 INTERACTIONS  So, is this an interaction? 1

18. KNR 445 FACTORIAL ANOVA Slide 17 INTERACTIONS  How about this? 1 Disordinal interaction = lines cross

19. KNR 445 FACTORIAL ANOVA Slide 18 Interactions and (spurious) main effects  With figure B, it seems we have a main effect of anxiety level  That implies that the effect of anxiety on performance can be generalized across different pressure conditions.  With figures A and C, generalization across situations would be a serious mistake  A main effect would fail to acknowledge that the effect of anxiety changes across situations  In which figure, A or C, would the main effect of anxiety be more likely? 1 2 3 4

20. Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety M = 8M = 9M = 7 M A1 = 8 High Anxiety M = 4M = 5M = 3 M A2 = 4 M B1 = 6M B2 = 7M B3 = 5M total = 6 KNR 445 FACTORIAL ANOVA Slide 19 Interactions and (spurious) main effects 1

21. Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety M =5 (3, 7) M = 8 (7, 9) M = 11 (10, 12) M A1 = 8 High Anxiety M = 4 (3, 5) M = 6 (5, 7) M = 2 (2, 2) M A2 = 4 M B1 = 4.5M B2 = 7M B3 = 6.5M total = 6 KNR 445 FACTORIAL ANOVA Slide 20 Interactions and (spurious) main effects 1 2 3 4

22. Situation Low Pressure Moderate pressure High Pressure Anxiety Level Low Anxiety M = 10M = 8M = 3 M A1 = 7 High Anxiety M = 6M = 7M = 8 M A2 = 7 M B1 = 8M B2 = 7.5M B3 = 5.5M total = 7 KNR 445 FACTORIAL ANOVA Slide 21 Interactions and (spurious) main effects 1 2 3

23. KNR 445 FACTORIAL ANOVA Slide 22 Note on ordinal/disordinal interactions  Note: whether an interaction is disordinal or not is often just a matter of how it is drawn. If you reversed the IVs for figure A, you would find a disordinal interaction. It was ordinal w.r.t. anxiety, but disordinal w.r.t. pressure 1 2