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Factorial Analysis of VariancePowerPoint Presentation

Factorial Analysis of Variance

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### Factorial Analysis of Variance

One dependent variable, more than one independent variable (“factor”)

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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

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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)

- Consider this experiment:

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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 = ????

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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)

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

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)

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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

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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…

11-2 = 9

5-4 = 1

INTERACTIONS1

- Note the difference between each pair of means in our original table of data

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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

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INTERACTIONS

- You might find it easier to see in a graph:

Ordinalinteraction = lines do not cross

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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

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

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?

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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

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