More sophisticated ANOVA More sophisticated ANOVA applicationsapplications
Repeated measures and Repeated measures and factorialfactorial
PSY295-001 SP2003PSY295-001 SP2003
Major TopicsMajor Topics
What are repeated-measures?What are repeated-measures?
An exampleAn example
AssumptionsAssumptions
Advantages and disadvantagesAdvantages and disadvantages
Review questionsReview questions
Effects of Counseling For Effects of Counseling For Post-Traumatic Stress Post-Traumatic Stress
DisorderDisorderFoa, Foa, et al.et al. (1991) (1991)– Provided supportive counseling (and Provided supportive counseling (and
other therapies) to victims of rapeother therapies) to victims of rape– Do number of symptoms change with Do number of symptoms change with
time?time?Point out lack of control groupPoint out lack of control group
– Not a test of effectiveness of supportive counselingNot a test of effectiveness of supportive counseling
Foa actually had controls.Foa actually had controls.
Cont.
Effect of Counseling--cont.Effect of Counseling--cont.
– 9 subjects measured before therapy, after 9 subjects measured before therapy, after therapy, and 3 months latertherapy, and 3 months later
We are ignoring Foa’s other treatment We are ignoring Foa’s other treatment conditions.conditions.
Therapy for PTSDTherapy for PTSD
Dependent variable = number of reported Dependent variable = number of reported symptoms.symptoms.
Question--Do number of symptoms Question--Do number of symptoms decrease over therapy and remain low?decrease over therapy and remain low?
Data on next slideData on next slide
The DataThe Data
Patient Pre PostFollow-up
SubjectMean
1 21 15 15 17.002 24 15 8 15.673 21 17 22 20.004 26 20 15 20.335 32 17 16 21.676 27 20 17 21.337 21 8 8 6.338 25 19 15 19.679 18 10 3 10.33
Mean 23.89 15.67 13.22 17.59s.d. 4.20 4.24 5.78 12.51
Plot of the DataPlot of the Data
0
5
10
15
20
25
30
PreTest PostTest FollowUp
Report
ed S
ympto
ms
Preliminary ObservationsPreliminary Observations
Notice that subjects differ from each other.Notice that subjects differ from each other.– Between-subjects variabilityBetween-subjects variability
Notice that means decrease over timeNotice that means decrease over time– Faster at first, and then slowerFaster at first, and then slower– Within-subjects variabilityWithin-subjects variability
Partitioning VariabilityPartitioning Variability
Total Variability
Between-subj. variability
Within-subj. variability
Time Error
This partitioning is reflected in the summary table.
Summary TableSummary Table
Source df SS MS FBet-subj 8 397.85
W/in-subj 18 716.66
Time 2 562.07 281.04 29.09 Error 16 154.59 9.66
Total 26 1114.51
InterpretationInterpretation
Note parallel with diagramNote parallel with diagram
Note subject differences not in error termNote subject differences not in error term
Note MSNote MSerrorerror is denominator for is denominator for FF on Time on Time
Note SSNote SStimetime measures what we are measures what we are
interested in studyinginterested in studying
AssumptionsAssumptions
Correlations between trials are all equalCorrelations between trials are all equal– Actually more than necessary, but closeActually more than necessary, but close– Matrix shown belowMatrix shown below
Pre Post FollowupPre 1.00 .637 .434
Post 1.00 .742
Followup 1.00
Cont.
Assumptions--cont.Assumptions--cont.
Previous matrix might look like we violated Previous matrix might look like we violated assumptionsassumptions– Only 9 subjectsOnly 9 subjects– Minor violations are not too serious.Minor violations are not too serious.
Greenhouse and Geisser (1959) Greenhouse and Geisser (1959) correctioncorrection– Adjusts degrees of freedomAdjusts degrees of freedom
Multiple ComparisonsMultiple Comparisons
With few means:With few means:– tt test with Bonferroni corrections test with Bonferroni corrections– Limit to important comparisonsLimit to important comparisons
With more means:With more means:– Require specialized techniquesRequire specialized techniques
Trend analysisTrend analysis
Advantages of Repeated-Advantages of Repeated-Measures DesignsMeasures Designs
Eliminate subject differences from Eliminate subject differences from error termerror term– Greater powerGreater power
Fewer subjects neededFewer subjects needed
Often only way to address the problemOften only way to address the problem– This example illustrates that case.This example illustrates that case.
DisadvantagesDisadvantages
Carry-over effectsCarry-over effects– Counter-balancingCounter-balancing
May tip off subjectsMay tip off subjects
Major PointsMajor Points
What is a factorial design?What is a factorial design?
An exampleAn example
Main effectsMain effects
InteractionsInteractions
Simple effectsSimple effects
Cont.
Major Points-cont.Major Points-cont.
Unequal sample sizesUnequal sample sizes
Magnitude of effectMagnitude of effect
Review questionsReview questions
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
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
2 X 2 Factorial2 X 2 Factorial
ViolentVideo
NonviolentVideo
Male
Female
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. dev., We will work with means and st. dev., instead of raw data.instead of raw data.– This illustrates important concepts.This illustrates important concepts.
The Data The Data (cell means and standard deviations)(cell means and standard 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
Plotting ResultsPlotting Results
0
2
4
6
8
10
Violent Video Nonviolent Video
Aggre
ssiv
e A
ssoci
ate
s
Male Female
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 Does violence affect males and females
equally?equally?
Cont.
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
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
ConclusionsConclusions
Main effectsMain effects– Significant difference due to videoSignificant difference due to video
More aggressive associates following violent videoMore aggressive associates following violent video
– Significant difference due to genderSignificant difference due to genderMales have more aggressive associates than Males have more aggressive associates than females.females.
Cont.
Conclusions--cont.Conclusions--cont.
InteractionInteraction– No interaction between video and genderNo interaction between video and gender
Difference between violent and nonviolent video is Difference between violent and nonviolent video is the same for males (1.5) as it is for females (1.4)the same for males (1.5) as it is for females (1.4)
We could see this in the graph of the data.We could see this in the graph of the data.
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
Line Graph of InteractionLine Graph of Interaction
0123456789
Violent Video Nonviolent Video
Aggre
ssiv
e A
ssoci
ate
s
MaleFemale
Simple EffectsSimple Effects
Effect of one independent variable at one Effect of one independent variable at one level of the other.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 females for Difference between males and females for only nonviolent videoonly nonviolent video
Unequal Sample SizesUnequal Sample Sizes
A serious problem for hand calculationsA serious problem for hand calculations
Can be computed easily using computer Can be computed easily using computer softwaresoftware
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.
Minitab ExampleMinitab Example
Analysis of Variance for AGGASSOCAnalysis of Variance for AGGASSOCSource 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 14.7Total 199 3057.0Total 199 3057.0
Cont.
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