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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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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
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 reality1
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)

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Two factors more reality2
Two factors, more reality

  • So for assignment to groups we get:

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Vocabulary
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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Two factors more reality3
Two factors, more reality

  • Suppose that the performance scores are…

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Introducing main effects
Introducing MAIN EFFECTS

  • Suppose that the performance scores are…

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Main effects
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 effects1
MAIN EFFECTS

  • This overall effect of anxiety is called the main effect of anxiety

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Main effects2
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 effects3
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…


Interactions

8-6 = 2

11-2 = 9

5-4 = 1

INTERACTIONS

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

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Interactions1
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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Interactions2
INTERACTIONS

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

Ordinalinteraction = lines do not cross

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Interactions3
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


Interactions4
INTERACTIONS

  • So, is this an interaction?

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Interactions5
INTERACTIONS

Disordinalinteraction = lines cross

  • How about this?

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Interactions and spurious main effects
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 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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