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