Rationale of Repeated Measures ANOVA Partitioning Variance

13 Pages·2012·1.38 MB·English

11/12/2012
PSY 511: Advanced Statistics for
Psychological and Behavioral Research 1
(cid:1)Rationale of Repeated Measures ANOVA
•One-way and two-way
•Benefits
(cid:1)Partitioning Variance
(cid:1)Statistical Problems with Repeated-
Measures Designs
•Sphericity
•Overcoming these problems
(cid:1)Interpretation
Repeated-Measures ANOVA
One or more factors
1. Situation/hypotheses dependent samples
Ho:µ1=µ2=…=µJ
2. Test statistic F = (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)
(cid:1)(cid:2)(cid:10)(cid:4)(cid:4)(cid:11)(cid:4)
3 .Distribution
FJ-1, (J-1)*(n-1)
4. Assumptions 1. Populations are normal
2. Equal population variances for
each cell
3. Sphericity(if more than three
conditions)
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(cid:1)Sensitivity
•Unsystematic variance is reduced
•More sensitive to experimental effects
(cid:1)Economy
•Fewer participants are needed
•…but participants may become fatigued
or experience practice effects
(cid:1)Are certain bushtuckerfoods(eaten by Australian
Aborigines)morerevoltingthanothers?
(cid:1)Fourfoodstastedby8celebrities:
• StickInsect
• KangarooTesticle
• FishEyeball
• WitchettyGrub
(cid:1)Outcome:
• Timetoretch(seconds)
Stick Witchetty
Celebrity Insect Testicle Fish Eye Grub Mean Variance
1 8 7 1 6 5.50 9.67
2 9 5 2 5 5.25 8.25
3 6 2 3 8 4.75 7.58
4 5 3 1 9 4.50 11.67
5 8 4 5 8 6.25 4.25
6 7 5 6 7 6.25 0.92
7 10 2 7 2 5.25 15.58
8 12 6 8 1 6.75 20.92
Mean 8.13 4.25 4.13 5.75
Grand Mean = 5.56
Grand Variance = 8.19
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(cid:1)The effect of the experiment is found in the within-
participant variance rather than in the between-
group variance
• Within-participant variance is composed of (1) treatment
effect and (2) individual differences in performance (which is
“error”)
(cid:1)The sources of variance for the repeated-measures
ANOVA are the same as those for the one-way
ANOVA…but the variances reflecting the treatment
effect and error are both part of the within-
participant variance for repeated-measures ANOVA
SS
Total
Variance between all scores
SS
Within SS
Variance Within Between
Individuals
This captures differences in
the outcome variable that
are not due to treatment
SESxETpfrefeeracimttm oefen ntt SESrErrorror weinfifdleli vcntio dotu rb aeelr duroisfrfee (dre et.ogn .cc, easlc).u Tlhaties
repeated-measures ANOVA
8 7 1 6
9 5 2 5
6 2 3 8
5 3 1 9
8 4 5 8
7 5 6 7
10 2 7 2
12 6 8 1
Grand Mean = 5.56
Grand Variance = 8.19
This is the total amount
of variability that can
be explained
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This is the amount of
variability that is
explained by within-
participant features (i.e.,
both treatment and
error)
This is the amount of variability that
is explained by the foods
This is the amount of variability that is explained by error
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(cid:1)MS = (cid:2)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)= (cid:14)(cid:15).(cid:17)(cid:15)= 27.71
Treatment (cid:12)(cid:13)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7) (cid:15)
(cid:1)MS = (cid:2)(cid:2)(cid:10)(cid:4)(cid:4)(cid:11)(cid:4)= (cid:17)(cid:18)(cid:15).(cid:15)(cid:19)= 7.30
Error (cid:12)(cid:13)(cid:10)(cid:4)(cid:4)(cid:11)(cid:4) (cid:20)(cid:17)
(cid:1)df = n*(J –1) = 8*3 = 24
Within
(cid:1)df = J –1 = 4 –1 = 3
Treatment
(cid:1)df = (J-1)*(n-1) = 3*7 = 21
Error
(cid:1)F = (cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:8)(cid:5)(cid:9)(cid:7)= (cid:20)(cid:19).(cid:19)(cid:17)= 3.79
(cid:1)(cid:2)(cid:10)(cid:4)(cid:4)(cid:11)(cid:4) (cid:19).(cid:15)(cid:21)
(cid:1)This is conceptually similar to the one-way
ANOVA
(cid:1) Same participants in all conditions
• Scores across conditions correlate
• Violates the assumption of independence
(cid:1) Assumption of Sphericity
• Sphericityrefers to the condition where the variances of the differences between all
possible pairs of groups (i.e., levels of the factor) are equal
• The violation of sphericityoccurs when the variances of the differences between all
combinations of the groups are not equal
• If sphericityis violated, then the variance calculations may be distorted which would
result in an inflated F-ratio
• Sphericitycan be evaluated when there are three or more levels of a repeated-
measures factor and, with each additional repeated-measures factor, the risk for
violating sphericityincreases
• If sphericityis violated, the degrees of freedom for the repeated-measures ANOVA
should be adjusted to correct for this violation
• Sphericityis measured using Mauchly’stest
(cid:2)p < .05 means that the sphericityassumption is violated
(cid:2)p> .05 means that the sphericityassumption is met
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Testicle - Eye – Witchetty Eye – Witchetty Witchetty
Stick Stick –Stick Testicle –Testicle –Eye
1 -1 -7 -2 -6 -1 5
2 -4 -7 -4 -3 0 3
3 -4 -3 2 1 6 5
4 -2 -4 4 -2 6 8
5 -4 -3 0 1 4 3
6 -2 -1 0 1 2 1
7 -8 -3 -8 5 0 -5
8 -6 -4 -11 2 -5 -7
Variance 5.27 4.29 25.70 11.55 14.29 26.55
(cid:1)Three corrections:
• Greenhouse-GeisserCorrection
(cid:2)This conservative correction should be used when sphericityestimates
are less than .75 or nothing is known about sphericity
• Huynh-FeldtCorrection
(cid:2)This liberal correction should be used when sphericityestimates
are.75 or greater
• Lower-Bound Correction
(cid:2)Not used very often because it is extremely conservative
(cid:1)Multiply degrees of freedom by these estimates to
correct for the effect of sphericity
(cid:1)Another way to deal with violations of the sphericity
assumption is to use a multivariate ANOVA (MANOVA)
because it does rely on a sphericityassumption
Df= 3, 21
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This is the amount of variability If we use the Greenhouse-Geissercorrection,
that is explained by the then we would retain the null hypothesis.
However, if we had used the Huynh-Feldt, then we
treatment effect would have rejected the null
This is the F-ratio…notice that it is the
This is the amount of same whether we assume the
variability that is explained sphericityassumption is met and or
by error violated (because the corrections only
adjust degrees of freedom)
(cid:1)Compareeachmeanagainstallothers(t-tests)
(cid:1)In general terms they use a stricter criterion to
acceptaneffectassignificant
• Theycontrolthefamilywiseerrorrate
• BonferronimethodforcontrollingTypeIerror:
(cid:1)Two factors
•Two-way = 2 factors
•Three-Way = 3 factors
(cid:1)The same participants in allconditions
•Repeated Measures = ‘same participants’
•This should be used to analyzedata from a within-
subjects design
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(cid:1)Effects of advertising on evaluations of
different drink types
•Factor A (Drink): Beer, Wine, Water
•Factor B (Imagery): Positive, Negative, Neutral
•Outcome Variable: Evaluation of product from -100
(dislike very much) to +100 (like very much)
SS
Total
Variance between all
participants
SS SS
Within Between
Variance explained by the Between-
experimental Participant
manipulations Variance
SSA SSB SSA ××××B
Effect of Effect of Effect of
Drink Imagery Interaction
SS SS SS
ErrorA ErrorB ErrorA××××B
Error for Error for Error for
Drink Imagery Interaction
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The sphericityassumption is
violated for the main effect of
drink…so we should use the
Greenhouse-Geissercorrection
The sphericityassumption is
violated for the main effect of
imagery…so we should use the
Greenhouse-Geissercorrection
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