Week 9: RM ANOVA Presentation Instructor Notes/Transcript and Slide Handout

Slide Presenter Notes/Transcription

1 This week we are going to examine the Repeated Measures Analysis of Variance. We

will look at a simple one-way repeated measure and then a factorial repeated measure

ANOVA.

2 When you read Dr. Fields account of Repeated Measures (RM) ANOVA, you

probably got a knot in your stomach. I don’t know why he started out on a such a note

of doom but I rather like RM ANOVA – It is one of the tests that I use quite often and

don’t think it is so bad. Really there is just a little quirk with that issue of sphericity,

which can be overcome quite efficiently. Wait and see.

3 There are many wonderful things about a RM design; first, you will probably need far

few subjects to achieve power and second, it is an efficient test for change subsequent

to an intervention. Something we need quite frequently in nursing.

4 Now this is where you really need to hold on to your cookies. I don’t watch reality TV

because I can’t stand the thought of making another human being do such awful

things. But, this is the example that he gives us and despite the nauseating thought of

eating such things it is a rather good example.

5 Here we go – 8 celebrities appearing on a reality TV show are to eat four gross items.

The outcome measure is seconds until retching aka emesis, which we have all

measured throughout our careers in a variety of ways. Note here that the mean is

calculated by columns. This is every celebrity contributes one score for each entree.

These are the scores we would use if we did a regular ANOVA. We also see the grand

mean of 5.56. Then the mean is calculated for each entree by celebrity along with the

variance and df – which is 4 entrees – 1 = 3. Before we go any further, retrieve the

data Bushtucker.sav and follow the instructions provided by Fields for setting up the

analysis (pg. 468 – 473).

6 Here we see the variance partitioned out like we are used to seeing except there are

some different SS. The SS total is the same as what you have been seeing but instead

of going directly to the model and error, we see the variance partitioned out for that

which is within subjects and between. For now we will focus on the within subject

variance called SS within. That variance is further partitioned into SS for the model or

experiment and SS residual.

7 With the same participants in each condition, we expect to see correlation between

each of the conditions. Therefore we violate the assumption of independence – this is a

good thing. We adjust for this in our degrees of freedom as you will see in a minute.

8 We test this assumption using Mauchly’s test. We want a nonsignificant result.

Slide Presenter Notes/Transcription

9 What do we do if we violate that assumption and Mauchly’s test is significant? We

correct using one of these tests. Primarily the Greenhouse-Geisser Estimate and/or the

Huynh-Feldt Estimate or an average of the two. Hang in there as this will all make

sense shortly.

10 Using the data bushtucker gross foods data – we get a significant Mauchly’s test –

does that mean we get to quit for the week and go play. No way. We can correct this.

The next slide gives us some corrected RM ANOVA results for within subjects

difference.

11 Notice the first line for differences in time to retch by entrée is with the assumption of

sphericity – if Mauchly’s test were non-significant we would report these results. On

the next line we have the very conservative GG corrected results with a non-significant

F and the less conservative HF corrected results with a significant F. This is where

your judgment will be critical. You will see. Read page 476 & 477 of the text to

discover that you can actually take the average of the two scores and if it is significant

– you can assume significance. You can also use MANOVA – we will cover it in

detail in a few weeks but for now look at the results of the multivariate tests.

12 We can see here that the multivariate tests are significant so if we were vasilating on

our decision seeing this might help.

13 This is a simple presentation of means for each entree. You can easily visualize that

the stick insect and witchetty grub were more palatable than the kangaroo testicle and

fish eye.

14 In this slide we see the results of our contrasts: Level I to II, II to III, and III to IV.

Here we see that our only significant difference lies between the time it took to retch

when eating the stick insect and the time it took to retch when eating the kangaroo

testicle.

15 As we have done for the other ANOVA tests, we need to determine exactly where any

differences are and to do so the most conservative test is the Bonferroni test.

16 Here we see the results of our post hoc comparisons using the Bonferroni adjustment.

Stick to testicle .012

Stick to eye .006

Stick to wichetty

Testicle to stick

Testicle to eye

Testicle to wichetty

Eye to stick

Eye to Testicle

Eye to wichetty

Wichetty to stick

Slide Presenter Notes/Transcription

Wichetty to testicle

Wichetty to eye

17 Here is the issue when you are reporting. We would want to always be sure to report

the correct value. Our main finding is using the values of the Greenhouse geyser

correction. Our degrees of freedom model is equal to 1.60. If you need to go back to

the outcome slide, go find those numbers. Our degrees of freedom residual was 11.19,

our s-value was 3.79, our p was less than .05. We also should report Mockley’s test

and here the chi-square, 5 degrees of freedom, equals 11.41, our p-value is less than

.05, our Greenhouse geyser epsilon correction is .53 and we also have reported our

Pillay’s trace. Our f is 26.96; we have 3 and 5 degrees of freedom. Our p-value is

.002. These are the differences that we could and should report for our repeated

measures one-way ANOVA when we have violated this assumption of sphericity and

had to correct for it.

18 Now we start to really have fun with RM. We add additional IVs and look at the main

effects and the interaction effects. This will look very similar to the factorial ANOVA.

19 Stop now, retrieve the Attitude.sav data from the Field’s companion website. Run the

analysis described on pages 484 – 491.

20 You see the model breakdown here – you have the addition of the second IV and the

interaction effect.

21 Retrieve the Attitude.sav data from the Field’s companion website

Analyze - General Linear Model – Repeated Measures

Define Factors – Within subjects Factor Name – Drink (3 Levels) and Imagery

(3 Levels)

22 Click on all of the variables in order and move them to the within subjects variables

23 Make sure that your variables are listed in this order

24 Click on change contrast polynomial – select simple and change. Click on Continue

25 From the main dialog box choose Plots. Select drink and move to the Horizontal axis

and imagery to separate lines then click on Add.

26 From the main dialog box – choose options select Drink, Imagery, and Drink X

Imagery – Check Compare main effects.

27 The assumption of sphericity is violated for Drink and for imagery but not for Drink X

Imagery

28 Use the values for G G for Drink and Imagery

Use the values for Spherecity Assumed – for Drink X Imagery

Slide Presenter Notes/Transcription

29 Here we see our main effects for f for the main effective drink. We will be using the

Greenhouse geyser correction which makes us have unusual looking degrees of

freedom. 1.15 and 21.93 are our corrected degrees of freedom. It’s equal to 5.11 and

that p-value is less than .05. So our main effect for drink using the Greenhouse geyser

correction is a significant finding. You can see the significance between beer and

wine looks pretty good and looks pretty strong. Wine and water look pretty strong and

water and beer look pretty strong. Everything looks fairly different. We will look at

our specific comparisons here.

30 Look at the comparison between drink 2 and drink 3 – Wine is stronger than water.

31 Here is our main effect of imagery. Remember, we had to use our Greenhouse Geyser

correction on this one too. We have degrees of freedom at .150 and 28.40 and an f-

value of 122.57; highly significant at the .001 being less than .001.

32 Positive to Neutral

Positive to Negative

Neutral to Negative

Are all significantly different

33 Blue is positive

Red is Neutral

Green Negative

Negative imagery has a greater effect on wine and water than on beer

34 Beer to water is significant F(1,19) = 6.22, p < .05

Wine to water is significant F (1, 19) = 18.61, p < .05

For Positive to Neutral Imagery F(1, 19) = 142.19 < .05

For Negative to Neutral Imagery F (1, 19) = 47.07 < .05

Beer to water by positive to neutral imagery not significant

Beer to Water by negative to neutral imagery is significant F (1, 19) = 6.75, p < .05

Wine to water by positive to neutral imagery is nonsignificant

Wine to water by negative to neutral imagery is significant F(1, 19) = 26.91, p < .001

35 Work through these examples until you feel comfortable with the RM ANOVA

Complete Smart Alex’s task 1 part 2 SPSS only and compare your results with

Dr. Field’s

Complete the database assignment and submit via the assignment link

Discuss any problems, trials, frustrations with your classmates on the DB for

week 9

Before the end of the week, Take Week 9 Quiz

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Adapted from slides by Dr. Fields to accompany Discovering Statistics Using SPSS 3rd edition

Presented by Danita Alfred

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Rationale of Repeated Measures ANOVAOne‐ and two‐wayBenefits

Partitioning VarianceStatistical Problems with Repeated Measures DesignsSphericitySphericityOvercoming these problems

Interpretation

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SensitivitySensitivityUnsystematic variance is reduced.More sensitive to experimental effects.

EconomyLess participants are needed.But, be careful of fatigue.

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Are certain Bushtucker foods more revoltingthan others?than others?Four Foods tasted by 8 celebrities:

Stick InsectKangarooTesticleFish EyeballWitchetty GrubWitchetty Grub

Outcome:Time to retch (seconds).

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CelebrityStick Insect

Testicle Fish EyeWitchetty

GrubMean Variance Df

Insect Grub

1 8 7 1 6 5.50 9.67 3

2 9 5 2 5 5.25 8.25 3

3 6 2 3 8 4.75 7.58 3

4 5 3 1 9 4.50 11.67 3

5 8 4 5 8 6.25 4.25 3

6 7 5 6 7 6.25 0.92 3

7 10 2 7 2 5.25 15.58 3

8 12 6 8 1 6.75 20.92 3

Mean 8.13 4.25 4.13 5.75 24

Grand Mean = 5.56

SSTVariance between all scores

SSWVariance Within Individuals

SSBetween

SSMEffect of Experiment

SSRError

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Same participants in all conditionsSame participants in all conditions.Scores across conditions correlate.Violates assumption of independence (lecture 2).

Assumption of SphericitySphericity.Crudely put: the correlation across Crudely put: the correlation across conditions should be the same.Adjust Degrees of Freedom.

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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 ViolatedP < .05, Sphericity is Violated.P > .05, Sphericity is met.

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Three measures:Three measures:Greenhouse‐Geisser EstimateHuynh‐Feldt EstimateLower‐bound Estimate

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

ε̂ε~

p yG‐G is conservative, and H‐F liberal.

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Tests 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 13.707

Sphericity AssumedGreenhouse-GeisserHuynh-FeldtLower-boundSphericity AssumedGreenhouse-Geisser

SourceAnimal

Error(Animal)

Type III Sumof Squares df Mean Square F Sig.

153.375 11.190 13.707153.375 13.981 10.970153.375 7.000 21.911

Huynh-FeldtLower-bound

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Multivariate Testsb

Effect Value F Hypothesis df Error df Sig.animal Pillai's Trace .942 26.955a 3.000 5.000 .002

Wilks' Lambda .058 26.955a 3.000 5.000 .002

Hotelling's Trace 16.173 26.955a 3.000 5.000 .002

Roy's Largest Root 16.173 26.955a 3.000 5.000 .002

a. Exact statistic

b. Design: Intercept Within Subjects Design: animal

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Bushtucker Trials

to

Retc

h (

s)

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5

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Animal

Stick Insect Kangaroo Testicle Fish Eye Witchetty Grub

Tim

e

0

1

2

3

4

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Tests of Within-Subjects ContrastsTests of Within-Subjects Contrasts

Measure:MEASURE_1

Source animal Type III Sum of

Squares df Mean Square F Sig.

animal Level 1 vs. Level 2 120.125 1 120.125 22.803 .002

Level 2 vs. Level 3 .125 1 .125 .011 .920

Level 3 vs. Level 4 21.125 1 21.125 .796 .402

Error(animal) Level 1 vs Level 2 36 875 7 5 268 Error(animal) Level 1 vs. Level 2 36.875 7 5.268

Level 2 vs. Level 3 80.875 7 11.554

Level 3 vs. Level 4 185.875 7 26.554

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Compare each mean against all others (t‐p g (tests).

In general terms they use a stricter criterion toaccept an effect as significant.

Hence, control the familywise error rate.

Si l l i h B f i h dSimplest example is the Bonferroni method:

TestsofNumber αα =Bonferroni

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Using the values of the Greenhouse‐GeisserUsing the values of the Greenhouse Geisserdf model = 1.60df residual = 11.19F (1.60, 11.19) = 3.79, p> .05   OR

Mauchly’s test X2 (5) = 11.41, p < .05  GG ε = .53

Pillai’s traceV = 0.94, F(3, 5) = 26.96, p = .002

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Two Independent VariablespTwo‐way = 2 IVsThree‐Way = 3 IVs

The same participants in all conditions.Repeated Measures = ‘same participants’A k  ‘ ithi bj t ’A.k.a. ‘within‐subjects’

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Field (2009): Effects of advertising onField (2009): Effects of advertising onevaluations of different drink types.IV 1 (Drink): Beer,Wine,Water

IV 2 (Imagery): Positive, negative, neutral

Dependent Variable (DV): Evaluation ofDependent Variable (DV): Evaluation ofproduct from ‐100 dislike very much to+100 like very much)

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SSTVariance between all participants

SSSSMWithin‐ParticpantVariance Variance explained by the 

experimental manipulations

SSRBetween‐Participant Variance

SSAEffect of 

SSBEffect of 

SSA × BEffect of Effect of 

Drink Imagery Interaction

SSRAError for Drink

SSRBError for Imagery

SSRA × BError for 

Interaction

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F(1.15, 21.93) = 5.11, p < .0529

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F(1.50, 28.40) = 122.57, p < .00131

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F(4, 76) = 17.16, p < .00133

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Work through these examples until you feel g p ycomfortable with the RM ANOVAComplete Smart Alex’s task 1 part 2 SPSS only and compare your results with Dr. Field’sComplete the database assignment and submit via the assignment linksubmit via the assignment linkDiscuss any problems, trials, frustrations with your classmates on the DB for week 9Before the end of the week, Take Week 9 Quiz 

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