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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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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? • 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
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
Two factors, more reality • So for assignment to groups we get: 2 1 3
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
Two factors, more reality • Suppose that the performance scores are… 1 2 3
Introducing MAIN EFFECTS • Suppose that the performance scores are… 1 2
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
MAIN EFFECTS • This overall effect of anxiety is called the main effect of anxiety 1
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
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…
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
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
INTERACTIONS • You might find it easier to see in a graph: Ordinalinteraction = lines do not cross 1 2 4 3
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
INTERACTIONS • So, is this an interaction? 1
INTERACTIONS Disordinalinteraction = lines cross • How about this? 1
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
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