Repeated-measures designs (GLM 4)

Thanks to Andy Fieldhttps://

www.youtube.com/watch?v=wkMwW_2_TzY&feature=youtu.be

Aims• Rationale of Repeated Measures ANOVA

– One- and two-way– Benefits

• Partitioning Variance• Statistical Problems with Repeated

Measures Designs– Sphericity– Overcoming these problems

• Interpretation

Slide 2

Experimental rationale• Cause and effect.• Manipulate variables in a systematic

way to infer things about causality. • Previously different groups.• Now same group repeatedly.• Same patient, 2 different machines.

Benefits of Repeated Measures Designs

• Sensitivity–Unsystematic variance is reduced.–More sensitive to experimental

effects.• Economy

–Less participants are needed.–But, be careful of fatigue.

Slide 4

Counterbalancing Divide the participants into groups and have each group perform a slightly different task, so that any imbalance in the task is canceled out.

An Example• Are certain Bushtucker foods more

revolting than others?• Four Foods tasted by 8 celebrities:

– Stick Insect– Kangaroo Testicle– Fish Eyeball– Witchetty Grub

• Outcome:– Time to retch(seconds).

Slide 6

The Data

Theory of one-way repeated-measures

ANOVA

Problems with Analyzing Repeated Measures Designs• Same participants in all

conditions.– Scores across conditions correlate.– Violates assumption of

independence.• Assumption of Sphericity.

– Crudely put: the correlation across conditions should be the same.

– Adjust Degrees of Freedom.Slide 9

The Assumption of Sphericity

• Basically means that the correlation between treatment levels is the same.

• Actually, it assumes that variances in the differences between conditions is equal.

• Measured using Mauchly’s test.– P < .05, Sphericity is Violated. (boo)– P > .05, Sphericity is met. (Yay!)

Slide 10

What is Sphericity?

Slide 11

Estimates of Sphericity

• Three measures:– Greenhouse-Geisser Estimate– Huynh-Feldt Estimate– Lower-bound Estimate

• Multiply df by these estimates to correct for the effect of Sphericity.

• G-G is conservative, and H-F liberal.

Slide 12

~

Correcting for Sphericity

Mauchly's Test of Sphericity

Measure: MEASURE_1

.136 11.406 5 .047 .533 .666 .333Within Subjects EffectAnimal

Mauchly's WApprox.

Chi-Square df Sig.Greenhouse

-Geisser Huynh-Feldt Lower-bound

Epsilon

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables isproportional to an identity matrix.

Slide 13

df = 3, 21

Slide 14

OutputTests of Within-Subjects Effects

Measure: MEASURE_1

83.125 3 27.708 3.794 .02683.125 1.599 52.001 3.794 .06383.125 1.997 41.619 3.794 .04883.125 1.000 83.125 3.794 .092

153.375 21 7.304153.375 11.190 13.707153.375 13.981 10.970153.375 7.000 21.911

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-bound

SourceAnimal

Error(Animal)

Type III Sumof Squares df Mean Square F Sig.

Interpretation: Main Effect

Slide 15

Bushtucker Trials

Animal

Stick Insect Kangaroo Testicle Fish Eye Witchetty Grub

Tim

e to

Ret

ch (

s)

0

1

2

3

4

5

6

7

8

9

10

Post Hoc Tests• Compare each mean against all

others (t-tests).• In general terms they use a stricter

criterion to accept an effect as significant.– Hence, control the familywise error rate.– Simplest example is the Bonferroni

method:

Slide 16

TestsofNumber Bonferroni

TWO WAY ANOVA

19/03/16

What is Two-Way Repeated Measures

ANOVA?• Two Independent Variables

– Two-way = 2 IVs– Three-Way = 3 IVs

• The same participants in all conditions.– Repeated Measures = ‘same

participants’– A.k.a. ‘within-subjects’

Slide 18

An Example• Field (2013): Effects of

advertising on evaluations of different drink types.– IV 1 (Drink): Beer, Wine, Water– IV 2 (Imagery): Positive, negative,

neutral– Dependent Variable (DV):

Evaluation of product from -100 dislike very much to +100 like very much)

Slide 19

Slide 20

SST

Variance between all participants

SSMWithin-Particpant Variance Variance explained by the

experimental manipulations

SSRBetween-

Participant Variance

SSAEffect of

Drink

SSBEffect of Imagery

SSA BEffect of

Interaction

SSRAError for

Drink

SSRBError for Imagery

SSRA BError for

Interaction

Running the Analysis: Naming factors

Slide 21

Defining Variables

Slide 22

Defining Variables

Slide 23

Contrasts

Slide 24

Plots

Slide 25

Getting Means

Slide 26

Output: Sphericity

Slide 27

Output: Main ANOVA

Slide 28

Main Effect of Drink

Slide 29

F(1.15, 21.93) = 5.11, p = .01

Main Effect of Imagery

Slide 30

F(1.50, 28.40) = 122.57, p < .001

Drink by Dose Interaction p581 text

Slide 31

F(4, 76) = 17.16, p < .001

Contrasts

Summary• Repeated measures designs are very similar

to other ANOVAs– Interpret F ration for each effect

• Watch out for Sphericity– Mauchley’s test

• P <.05 sphericity can’t be assumed– Greehouse-Geisser correction– Huynh-Feldt correction

• Follow up tests– In SPSS can do built in contrasts only– Limited post-hoc tests (bonferroni)

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