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Factorial Analysis of Variance

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

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Factorial Analysis of Variance

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

  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 R2) • Factorial ANOVA & one-way ANOVA • Multiple and simple regression • ANOVA – categorical IVs 1 2

  3. Two factors, more reality • How factorial designs work • Consider this experiment: • Take 2 sets of golfers: 1 set (A1) is high anxious, 1 set (A2) is low anxious • Assign 1/3 of each set of golfers to a different performance scenario: Low pressure (B1), Moderate pressure (B2), High pressure (B3) 1 2 3

  4. Two factors, more reality • So for assignment to groups we get: 2 1 3

  5. 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 • Notation • 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 = ???? 1 2 3 4

  6. Two factors, more reality • Suppose that the performance scores are… 1 2 3

  7. Introducing MAIN EFFECTS • Suppose that the performance scores are… 1 2

  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 MA1 (8) and MA2 (4) 1 NB: if you were to do a 1-way ANOVA, you’d ignore the effect of pressure (IVB) completely

  9. MAIN EFFECTS • This overall effect of anxiety is called the main effect of anxiety 1

  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 MB1 (4.5), MB2 (7) and MB3 (6.5) 1 NB: here, you’d ignore the effect of anxiety(IVA) completely

  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 1 2 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…

  12. 8-6 = 2 11-2 = 9 5-4 = 1 INTERACTIONS 1 • Note the difference between each pair of means in our original table of data 4 3 2

  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

  14. INTERACTIONS • You might find it easier to see in a graph: Ordinalinteraction = lines do not cross 1 2 4 3

  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 1 2 Non-parallelism is a necessary but not sufficient condition for an interaction to be present

  16. INTERACTIONS • So, is this an interaction? 1

  17. INTERACTIONS Disordinalinteraction = lines cross • How about this? 1

  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 3 2 4

  19. Interactions and (spurious) main effects 1

  20. Interactions and (spurious) main effects 1 2 3 4

  21. Interactions and (spurious) main effects 1 2 3

  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

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