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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Animal
Stick Insect Kangaroo Testicle Fish Eye Witchetty Grub
Tim
e
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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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