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Transcript of Chapter 17 Factorial Analysis of Variance Fundamental Statistics for the Behavioral Sciences, 5th...
Chapter 17Factorial Analysis of
VarianceFundamental Statistics for the Fundamental Statistics for the
Behavioral Sciences, 5th editionBehavioral Sciences, 5th edition
David C. HowellDavid C. Howell
©2003 Brooks/Cole Publishing Company/ITP
2Chapter 17 Factorial Analysis of Variance
Major PointsMajor Points
• What is a factorial design?What is a factorial design?
• An exampleAn example
• Main effectsMain effects
• InteractionsInteractions
• Simple effectsSimple effects
Cont.
3Chapter 17 Factorial Analysis of Variance
Major Points-cont.Major Points-cont.
• Unequal sample sizesUnequal sample sizes
• Magnitude of effectMagnitude of effect
• Review questionsReview questions
4Chapter 17 Factorial Analysis of Variance
What is a FactorialWhat is a Factorial
• At least two independent variablesAt least two independent variables
• All combinations of each variableAll combinations of each variable
• R X C factorialR X C factorial
• CellsCells
5Chapter 17 Factorial Analysis of Variance
Video ViolenceVideo Violence
• Bushman studyBushman study Two independent variablesTwo independent variables
• Two kinds of videosTwo kinds of videos
• Male and female subjectsMale and female subjects
• See following diagramSee following diagram
6Chapter 17 Factorial Analysis of Variance
2 X 2 Factorial2 X 2 Factorial
ViolentVideo
NonviolentVideo
Male
Female
7Chapter 17 Factorial Analysis of Variance
Bushman’s Study-cont.Bushman’s Study-cont.
• Dependent variable = number of Dependent variable = number of aggessive associatesaggessive associates
• 50 observations in each cell50 observations in each cell
• We will work with means and st. We will work with means and st. dev., instead of raw data.dev., instead of raw data. This illustrates important concepts.This illustrates important concepts.
8Chapter 17 Factorial Analysis of Variance
The Data The Data (cell means and standard (cell means and standard
deviations)deviations)ViolentVideo
NonviolentVideo Means
Male 7.7(4.6)
6.2(3.5)
6.95
Female 6.5(4.2)
5.1(2.8)
5.80
Means 7.1 5.65 6.375
9Chapter 17 Factorial Analysis of Variance
Plotting ResultsPlotting Results
0
2
4
6
8
10
Violent Video Nonviolent Video
Aggre
ssiv
e A
ssoci
ate
s
Male Female
10Chapter 17 Factorial Analysis of Variance
Effects to be estimatedEffects to be estimated• Differences due to videosDifferences due to videos
Violent appear greater than nonviolentViolent appear greater than nonviolent
• Differences due to genderDifferences due to gender Males appear higher than femalesMales appear higher than females
• Interaction of video and genderInteraction of video and gender What is an interaction?What is an interaction?
Does violence affect males and females equally?Does violence affect males and females equally?
Cont.
11Chapter 17 Factorial Analysis of Variance
Estimated Effects--cont.Estimated Effects--cont.
• ErrorError average within-cell varianceaverage within-cell variance
• Sum of squares and mean squaresSum of squares and mean squares Extension of the same concepts in the Extension of the same concepts in the
one-wayone-way
12Chapter 17 Factorial Analysis of Variance
CalculationsCalculations
• Total sum of squaresTotal sum of squares
• Main effect sum of squaresMain effect sum of squares
2..XXSStotal
2..XXngSS Vvideo
2..XXnvSS Ggender
Cont.
13Chapter 17 Factorial Analysis of Variance
Calculations--cont.Calculations--cont.
• Interaction sum of squaresInteraction sum of squares Calculate SSCalculate SScellscells and subtract SS and subtract SSVV and SS and SSGG
• SSSSerrorerror = SS = SStotaltotal - SS - SScellscells
or, or, MSMSerrorerror can be found as average of cell variances can be found as average of cell variances
2..)( XXnSS ijcells
14Chapter 17 Factorial Analysis of Variance
Degrees of FreedomDegrees of Freedom
• dfdf for main effects = number of for main effects = number of levels - 1levels - 1
• dfdf for interaction = product of for interaction = product of dfdfmain main
effectseffects
• dfdf errorerror = = NN - - abab = = NN - # cells - # cells
• dfdftotaltotal = = NN - 1 - 1
15Chapter 17 Factorial Analysis of Variance
Calculations for Bushman Calculations for Bushman DataData
• SSSStotaltotal requires raw data. requires raw data.
It is actually = 171.50It is actually = 171.50
• SSSSvideovideo
125.105
375.665.5375.61.7250
..22
2
XXngSS Vvideo
Cont.
16Chapter 17 Factorial Analysis of Variance
Calculations--cont.Calculations--cont.
• SSSSgendergender
125.66
375.680.5375.695.6)2(50
..22
2
XXnvSS Ggender
Cont.
17Chapter 17 Factorial Analysis of Variance
Calculations--cont.Calculations--cont.
• SSSScellscells
• SSSSVXGVXG = SS = SScellscells - SS - SSvideovideo - SS - SSgendergender
== 171.375 - 105.125 - 66.125 = 0.125 171.375 - 105.125 - 66.125 = 0.125
375.171)4275.3(50
)375.61.5()375.65.6(
)375.62.6()375.67.7(50
..)(
22
22
2
XXnSS cellcells
Cont.
18Chapter 17 Factorial Analysis of Variance
Calculations--cont.Calculations--cont.
• MSMSerrorerror = average of cell variances = = average of cell variances =(4.6(4.622 + 3.5 + 3.522 + 4.2 + 4.222 + 2.8 + 2.822)/4 )/4 =58.89/4 = 14.723 =58.89/4 = 14.723
• Note that this is MSNote that this is MSerrorerror and not SS and not SSerrorerror
19Chapter 17 Factorial Analysis of Variance
Summary TableSummary Table
Source df SS MS FVideo 1 105.125 105.125 7.14Gender 1 66.125 66.125 4.49VXG 1 0.125 0.125 .01Error 196 2885.610 14.723Total 199 3056.980
20Chapter 17 Factorial Analysis of Variance
ConclusionsConclusions
• Main effectsMain effects Significant difference due to videoSignificant difference due to video
• More aggressive associates following More aggressive associates following violent videoviolent video
Significant difference due to genderSignificant difference due to gender• Males have more aggressive associates Males have more aggressive associates
than females.than females.
Cont.
21Chapter 17 Factorial Analysis of Variance
Conclusions--cont.Conclusions--cont.
• InteractionInteraction No interaction between video and No interaction between video and
gendergender• Difference between violent and Difference between violent and
nonviolent video is the same for males nonviolent video is the same for males (1.5) as it is for females (1.4)(1.5) as it is for females (1.4)
• We could see this in the graph of the We could see this in the graph of the data.data.
22Chapter 17 Factorial Analysis of Variance
Elaborate on InteractionsElaborate on Interactions
• Diagrammed on next slide as line graphDiagrammed on next slide as line graph
• Note parallelism of linesNote parallelism of lines Means video differences did not depend on Means video differences did not depend on
gendergender
• A significant interaction would have A significant interaction would have nonparallel linesnonparallel lines Ordinal and disordinal interactionsOrdinal and disordinal interactions
23Chapter 17 Factorial Analysis of Variance
Line Graph of InteractionLine Graph of Interaction
0123456789
Violent Video Nonviolent Video
Aggre
ssiv
e A
ssoci
ate
s
MaleFemale
24Chapter 17 Factorial Analysis of Variance
Simple EffectsSimple Effects
• Effect of one independent variable Effect of one independent variable at one level of the other.at one level of the other.
• e.g. Difference between males and e.g. Difference between males and females for only violent videofemales for only violent video
• Difference between males and Difference between males and females for only nonviolent videofemales for only nonviolent video
25Chapter 17 Factorial Analysis of Variance
Unequal Sample SizesUnequal Sample Sizes
• A serious problem for hand A serious problem for hand calculationscalculations
• Can be computed easily using Can be computed easily using computer softwarecomputer software
• Can make the interpretation difficultCan make the interpretation difficult Depends, in part, on why the data are Depends, in part, on why the data are
missing.missing.
26Chapter 17 Factorial Analysis of Variance
Magnitude of EffectMagnitude of Effect
• Eta SquaredEta Squared
InterpretationInterpretation
• Omega squaredOmega squared Less biased estimateLess biased estimate
total
effect
SS
SS2
errortotal
erroreffect
MSSS
MSkSS
)1(2
k = number of levels for the effectin question
Cont.
27Chapter 17 Factorial Analysis of Variance
Effect Size—cont.Effect Size—cont.
• As with one-way, we can calculate As with one-way, we can calculate effect size for each kind of effect effect size for each kind of effect separately.separately.
• Most sensible to stick to Most sensible to stick to comparisons of two groups.comparisons of two groups.
• Same formulae as for Same formulae as for tt tests. tests.
28Chapter 17 Factorial Analysis of Variance
Minitab ExampleMinitab Example
• Analysis of Variance for AGGASSOCAnalysis of Variance for AGGASSOC• Source DF SS MS F PSource DF SS MS F P
• GENDER 1 66.1 66.1 4.49 0.035GENDER 1 66.1 66.1 4.49 0.035
• VIDEO 1 105.1 105.1 7.14 0.008VIDEO 1 105.1 105.1 7.14 0.008
• Interaction 1 0.1 0.1 0.01 0.927Interaction 1 0.1 0.1 0.01 0.927
• Error 196 2885.6 14.7Error 196 2885.6 14.7
• Total 199 3057.0Total 199 3057.0
Cont.
29Chapter 17 Factorial Analysis of Variance
Minitab--cont.Minitab--cont.
Individual 95% CIGENDER Mean --------+---------+---------+---------+---1 6.95 (----------*----------)2 5.80 (----------*----------) --------+---------+---------+---------+--- 5.60 6.30 7.00 7.70
Individual 95% CIVIDEO Mean ---------+---------+---------+---------+--1 7.10 (---------*--------)2 5.65 (---------*--------) ---------+---------+---------+---------+-- 5.60 6.40 7.20 8.00
30Chapter 17 Factorial Analysis of Variance
Review QuestionsReview Questions
• What is the definition of a factorial What is the definition of a factorial design?design?
• How many independent variables How many independent variables can you have in a factorial design?can you have in a factorial design? How many levels of an independent How many levels of an independent
variable can you have?variable can you have?
Cont.
31Chapter 17 Factorial Analysis of Variance
Review Questions--cont.Review Questions--cont.
• What do all of the calculations for What do all of the calculations for sums of squares have in common?sums of squares have in common?
• How does a main effect differ from How does a main effect differ from an interaction?an interaction?
• How does a main effect differ from a How does a main effect differ from a simple effect?simple effect?
Cont.
32Chapter 17 Factorial Analysis of Variance
Review Questions--cont.Review Questions--cont.
• Give an example of a situation where Give an example of a situation where you would commonly expect an you would commonly expect an interaction.interaction.
• What happens to What happens to FF values when values when MSMSerrorerror decreases? decreases?
• How do eta-squared and omega-How do eta-squared and omega-squared differ?squared differ?