Factorial Analysis of Variance

One-Way Analysis of Variance

Factorial Analysis of VarianceOne dependent variable, more than one independent variable (factor)12

KNR 445FACTORIALANOVASlide 2Two factors, more realityImagine you want to describe what makes GPA, body fat, a teams winning %, the outcome of an electoral poll varyDo they depend on just one thing?Of course notMore IVs simply get closer to the truth (to explaining all of the DV - increase overall R2)Factorial ANOVA & one-way ANOVAMultiple and simple regressionANOVA categorical IVs12

KNR 445FACTORIALANOVASlide 3Two factors, more realityHow factorial designs workConsider this experiment:Take 2 sets of golfers: 1 set (A1) is high anxious, 1 set (A2) is low anxiousAssign 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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KNR 445FACTORIALANOVASlide 4Two factors, more realitySo for assignment to groups we get:

SituationLow PressureModerate pressureHigh PressureAnxiety LevelLow Anxietyncell = 2ncell = 2ncell = 2n(A1) =6High Anxietyncell = 2ncell = 2ncell = 2n(A2) =6

n(B1) = 4n(B2) = 4n(B3) = 4ntotal = 12123

Vocabulary Factor = Independent variableTwo-factor ANOVA / Two-way ANOVA: an experiment with 2 independent variablesLevels: number of treatment conditions (groups) for a specific IV Notation3 X 2 ANOVA = experiment w/2 IVs: one w/3 levels, one w/2 levels2 X 2 ANOVA = experiment w/2 IVs: both w/2 levels3 X 2 X 2 = ????KNR 445FACTORIALANOVASlide 51234

KNR 445FACTORIALANOVASlide 6Two factors, more realitySuppose that the performance scores are

SituationLow PressureModerate pressureHigh PressureAnxiety LevelLow AnxietyM = 5(3, 7)M = 8(7, 9)M = 11(10, 12)MA1 = 8High AnxietyM = 4(3, 5)M = 6(5, 7)M = 2(2, 2)MA2 = 4

MB1 = 4.5MB2 = 7MB3 = 6.5Mtotal = 6123

KNR 445FACTORIALANOVASlide 7Introducing MAIN EFFECTSSuppose that the performance scores are

SituationLow PressureModerate pressureHigh PressureAnxiety LevelLow AnxietyM = 5(3, 7)M = 8(7, 9)M = 11(10, 12)MA1 = 8High AnxietyM = 4(3, 5)M = 6(5, 7)M = 2(2, 2)MA2 = 4

MB1 = 4.5MB2 = 7MB3 = 6.5Mtotal = 612

KNR 445FACTORIALANOVASlide 8MAIN EFFECTSWhat do we find?We can consider the overall effect of anxiety (Factor A) on performanceThe 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)

NB: if you were to do a 1-way ANOVA, youd ignore the effect of pressure (IVB) completely1

KNR 445FACTORIALANOVASlide 9MAIN EFFECTSThis overall effect of anxiety is called the main effect of anxiety1

KNR 445FACTORIALANOVASlide 10MAIN EFFECTSWhat do we find?We can also consider the overall effect of situation (Factor B) on performanceThe 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)

NB: here, youd ignore the effect of anxiety (IVA) completely1

KNR 445FACTORIALANOVASlide 11MAIN EFFECTSThis overall effect of situation is called the main effect of situationIn 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 effectThis is how it is ignored, statistically. Its effects are, quite literally, averaged outWHENEVER YOU INTERPRET A MAIN EFFECT, YOU SHOULD PAY ATTENTION TO THE FACT THAT IT AVERAGES ACROSS LEVELS OF THE OTHER FACTOR ESPECIALLY WHEN YOU GET12

KNR 445FACTORIALANOVASlide 12INTERACTIONSNote the difference between each pair of means in our original table of dataSituationLow PressureModerate pressureHigh PressureAnxiety LevelLow AnxietyM = 5(3, 7)M = 8(7, 9)M = 11(10, 12)MA1 = 8High AnxietyM = 4(3, 5)M = 6(5, 7)M = 2(2, 2)MA2 = 4

MB1 = 4.5MB2 = 7MB3 = 6.5Mtotal = 65-4 = 18-6 = 211-2 = 9123

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KNR 445FACTORIALANOVASlide 13INTERACTIONSThe magnitude of the difference changes depending on the pressure levelIn other wordsIn other words, the effect of anxiety on performance depends on the pressure level in which the participants are asked to performIn other words, the pressure level moderates the effect of anxiety on performanceIn other words, the anxiety-performance relationship differs depending on the pressure level12

KNR 445FACTORIALANOVASlide 14INTERACTIONSYou might find it easier to see in a graph:123Ordinal interaction = lines do not cross

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KNR 445FACTORIALANOVASlide 15INTERACTIONSThe 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 consideredNon-parallelism is a necessary but not sufficient condition for an interaction to be present12

KNR 445FACTORIALANOVASlide 16INTERACTIONSSo, is this an interaction?1

KNR 445FACTORIALANOVASlide 17INTERACTIONSHow about this?1

Disordinal interaction = lines crossKNR 445FACTORIALANOVASlide 18Interactions and (spurious) main effectsWith figure B, it seems we have a main effect of anxiety levelThat 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 mistakeA main effect would fail to acknowledge that the effect of anxiety changes across situationsIn which figure, A or C, would the main effect of anxiety be more likely?1234

SituationLow PressureModerate pressureHigh PressureAnxiety LevelLow AnxietyM = 8M = 9M = 7MA1 = 8High AnxietyM = 4M = 5M = 3MA2 = 4

MB1 = 6MB2 = 7MB3 = 5Mtotal = 6KNR 445FACTORIALANOVASlide 19Interactions and (spurious) main effects1

SituationLow PressureModerate pressureHigh PressureAnxiety LevelLow AnxietyM =5(3, 7)M = 8(7, 9)M = 11(10, 12)MA1 = 8High AnxietyM = 4(3, 5)M = 6(5, 7)M = 2(2, 2)MA2 = 4

MB1 = 4.5MB2 = 7MB3 = 6.5Mtotal = 6KNR 445FACTORIALANOVASlide 20Interactions and (spurious) main effects1234

SituationLow PressureModerate pressureHigh PressureAnxiety LevelLow AnxietyM = 10M = 8M = 3MA1 = 7High AnxietyM = 6M = 7M = 8MA2 = 7

MB1 = 8MB2 = 7.5MB3 = 5.5Mtotal = 7KNR 445FACTORIALANOVASlide 21Interactions and (spurious) main effects123

KNR 445FACTORIALANOVASlide 22Note on ordinal/disordinal interactionsNote: 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. pressure1

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