PSYC 3031 INTERMEDIATE STATISTICS LABORATORY J. Elder FACTORIAL ANOVA
PSYC 3031 INTERMEDIATE STATISTICS LABORATORY J. Elder
FACTORIAL ANOVA
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Acknowledgements
¨ Some of these slides have been sourced or modified from slides created by A. Field for Discovering Statistics using R.
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Aims
¨ Rationale of factorial ANOVA ¨ Partitioning variance ¨ Interaction effects
¤ Interaction graphs ¤ Interpretation
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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What is Two-Way Independent ANOVA?
¨ Two independent variables ¤ Two-way = 2 Independent variables ¤ Three-way = 3 Independent variables
¨ Different participants in all conditions ¤ Independent = ‘different participants’
¨ Several independent variables is known as a factorial design.
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Benefit of Factorial Designs
¨ We can look at how variables interact. ¨ Interactions
¤ Show how the effects of one IV might depend on the effects of another
¤ Are often more interesting than main effects.
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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An Example
¨ Field (2009): Testing the effects of alcohol and gender on ‘the beer-goggles effect’:
¤ IV 1 (Alcohol): none, 2 pints, 4 pints
¤ IV 2 (Gender): male, female
¨ Dependent variable (DV) was an objective measure of the attractiveness of the partner selected at the end of the evening.
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Step 1: Calculate SST
65 50 70 45 55 30
50 55 65 60 65 30
70 80 60 85 70 30
45 65 70 65 55 55
55 70 65 70 55 35
30 75 60 70 60 20
70 75 60 80 50 45
55 65 50 60 50 40
Grand Mean = 58.33
66896614878190
12
.)(.)(SS grandT
=−=
−= Ns
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Step 2: Calculate SSM
( )∑ −= 2grandMSS xxn ii
16754791362412451125136258411121391362584136242
2883085458817485458829528
335883358833588
335883358833588
222222
222
222
.......
2.705)().().().().().(
).(35.625).(57.5).(66.875
).(62.5).(66.875).(60.625SSM
=+++++=
−+−++++=
−+−+−+
−+−+−=
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Step 2a: Calculate SSA
( )∑ −= 2grandASS xxn ii
A1: Female A2: Male
65 70 55 50 45 30
70 65 65 55 60 30
60 60 70 80 85 30
60 70 55 65 65 55
60 65 55 70 70 35
55 60 60 75 70 20
60 60 50 75 80 45
55 50 50 65 60 40
Mean Female = 60.21 Mean Male = 56.46
751689256838256842488124
33582433582422
22
...
1.87)().(
).(56.46).(60.21SSGender
=+=
−+=
−+−=
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Step 2b: Calculate SSB
Slide 12
( )∑ −= 2grandBSS xxn ii
B1: None B2: 2 Pints B3: 4 Pints
65 50 70 45 55 30
70 55 65 60 65 30
60 80 60 85 70 30
60 65 70 65 55 55
60 70 65 70 55 35
55 75 60 70 60 20
60 75 60 80 50 45
55 65 50 60 50 40
Mean None = 63.75 Mean 2 Pints = 64.6875
Mean 4 Pints = 46.5625
2 2 2Alcohol
2 2 2
SS 16(63.75 58.33) 16(64.6875 58.33) 16(46.5625 58.33)
16(5.42) 16(6.3575) 16( 11.7675)470.0224 646.6849 2215.58493332.292
= − + − + −
= + + −= + +=
Factorial ANOVA
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Step 2c: Calculate SSAxB
Slide 13
MSS SS SS SSA B A B× = − −
MSS SS SS5479.167 168.75 3332.2921978.125
A B A BSS× = − −= − −=
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Step 3: Calculate SSR
Slide 14
2 2 2 2R group1 1 group2 2 group3 3 group SS ( 1) ( 1) ( 1) ( 1)n ns n s n s n s n= − + − + − −K
5234878782135091096300974685171
717571747172
111
111
62
52
42
32
22
12
.....
)17.41()0()56.7()2.86()06.7()4.55(
)()()(
)()()(SS
group6group5group4
group3group2group1R
=+++++=×+×+×+×+×+×=
−+−+−+
−+−+−=
nsnsns
nsnsns
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Orthogonal contrasts for the alcohol variable
Factorial ANOVA
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Running a Factorial ANOVA in R
¨ Packages: ¤ car
n Levene’s test ¤ compute.es
n Effect sizes ¤ ggplot2
n Graphs ¤ multcomp
n Post-hoc tests ¤ pastecs
n Descriptive statistics ¤ reshape
n Reshaping the data
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Procedure
¨ Explore the data ¤ Tests of normality, homogeneity of variance, etc.
¨ Construct contrasts ¨ Compute ANOVA ¨ Compute contrasts or post-hoc tests
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Fitting a Factorial ANOVA Model
gogglesModel<-aov(attractiveness ~ gender + alcohol + gender:alcohol, data = gogglesData)
¨ Or: gogglesModel<-aov(attractiveness ~ alcohol*gender, data = gogglesData)
¨ If we want to look at the Type III sums of squares for the model, we need to also execute this command after we have created the model: ¤ Anova(gogglesModel, type="III")
Factorial ANOVA
J. Elder PSYC 3031 INTERMEDIATE STATISTICS LABORATORY
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Interpreting Factorial ANOVA
Factorial ANOVA
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Interpretation: Main Effect Alcohol
Factorial ANOVA
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Interpretation: Main Effect Gender
Factorial ANOVA
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Interpretation: Interaction
Factorial ANOVA
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What Is an Interaction?
6.25 6.25
5.625
−21.875
Is There Likely to Be a Significant Interaction Effect?
Is There Likely to Be a Significant Interaction Effect?
Factorial ANOVA
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Interpreting Contrasts
¨ To see the output for the contrasts that we specified, execute: summary.lm(gogglesModel)
Factorial ANOVA
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Interpreting Contrasts