Repeated ANOVARepeated ANOVA
OutlineOutline
When to use a repeated ANOVAWhen to use a repeated ANOVA
How variability is partitionedHow variability is partitioned
Interpretation of the F-ratioInterpretation of the F-ratio
How to compute & interpret one-way How to compute & interpret one-way ANOVAANOVA
Post Hoc TestsPost Hoc Tests
HypothesesHypotheses
Null hypothesis:
Ho: 1 = 2 = 3 = …= k
Researcher Hypothesis:
H1: At least one population mean is different from the others
F-ratioF-ratio
For Repeated Measures ANOVA:For Repeated Measures ANOVA:
The repeated measures design removes The repeated measures design removes individual differences fromindividual differences from both because the both because the same people are in all treatments.same people are in all treatments.
error alexperiment
error alexperiment +effect treatment F
Partitioning VariabilityPartitioning Variability
First Stage: First Stage: – Compute SSCompute SSTotalTotal
– SSSSTotalTotal partitioned into between- and within- partitioned into between- and within-
treatmentstreatments
Second Stage:Second Stage:– Compute variability between subjects Compute variability between subjects
(SS(SSBetween Subjects)Between Subjects)
Total variability
(SStotal)
Between-groups
variability(SSbetween)
Within-groups
variability(SSwithin)
Between-subjects
variability(SSbtwn Ss)
Error variability
(Sserror/residual)
Used to calculate numerator of F
Used to calculate denominator of F
Formulas for Repeated-Formulas for Repeated-Measures ANOVAMeasures ANOVA
error
treatmentsbetween
MS
MSF .
Critical Value & F notationCritical Value & F notation
Use the F distribution table Use the F distribution table
DF of Numerator: DF of Numerator: dfdfbetween between (k-1)(k-1)
DF of Denominator: DF of Denominator: dfdferror/residual error/residual (N-k)-(n-1)(N-k)-(n-1)
F notationF notationFF(df(dfbetweenbetween, df, dferrorerror) = F score) = F score, p, p < 0.05 < 0.05
Calculating ANOVA by handCalculating ANOVA by hand
General Notation:General Notation:– n = number of scores in the group/treatment
condition– T = sum of the scores in a group/treatment
condition– k = number of treatments; number of levels of
the IV/factor– N = total number of scores in the study (n x k)
Calculating ANOVA by handCalculating ANOVA by hand
General Notation:General Notation:– G = sum of all the scores in the total study
(ΣT)– Σx2 = sum of the squared values of each of
the scores in the total study
One new notational symbol for person total/ participant total: – S = sum of the scores across treatments for
each person
Calculating ANOVA by handCalculating ANOVA by hand
Example: Effects of label information on Example: Effects of label information on perceived quality of wineperceived quality of wine– FrenchFrench– ItalianItalian– CanadianCanadian
DV: perceived quality of wine (1 to 20, with DV: perceived quality of wine (1 to 20, with higher scores indicating better taste)higher scores indicating better taste)
The Data….The Data….
Participant French Italian Canadian
1 14 10 9
2 16 12 12
3 17 13 14
4 16 14 16
5 15 12 10
6 12 11 8
Step 1: State HypothesesStep 1: State Hypotheses
Ho:
H1:
Step 2: Compute dfStep 2: Compute df
dftotal=N-1
dfbtwn=k-1
dfwithin=N-K
dferror = (N-K) – (n-1)
Step 3: Determine F-criticalStep 3: Determine F-critical
=.05=.05dfdferrorerror==
dfdfbetweenbetween==
F critical =F critical =
Step 4: Calculate SSStep 4: Calculate SS
SSSSTOTALTOTAL = = 22 – – GG22
NN
Step 4: Calculate SSStep 4: Calculate SSSSSSBETWEENBETWEEN = = TT22 – – GG22
nn NN
Step 4: Calculate SSStep 4: Calculate SSSSSSwithinwithin = = SS SS inside each treatmentinside each treatment
Step 4: Calculate SSStep 4: Calculate SS
N
G
k
PSS jectsbetweenSub
22
Step 4: Calculate SSStep 4: Calculate SS
jectsBetweenSubWithin SSSSSSerror jectsBetweenSubWithin SSSSSSerror
Step 5: Calculate MS Step 5: Calculate MS
MSMSBETWEENBETWEEN = = SSSSBETWEENBETWEEN
dfdfBETWEENBETWEEN
MSMSerrorerror = = SSSSerrorerror
dfdferrorerror
Error
BetweenTx
MS
MSF
Error
BetweenTx
MS
MSF
Step 7: Summary Table Step 7: Summary Table
Source SS df MS F
Between
Within
Subjects
Error
Total
Step 9: Statistical Decision Step 9: Statistical Decision
F notation:
F (dfbetween, dferror) = F score, p < 0.05
Strength of relationshipStrength of relationship
Proportion of variance accounted for Proportion of variance accounted for ((22))
jectsBetweenSubSSSS
SS
Total
atmentBetweenTre2jectsBetweenSubSSSS
SS
Total
atmentBetweenTre2
Tukey’s HSD: Honestly Tukey’s HSD: Honestly Significant Difference Significant Difference
q= studentized range statisticq= studentized range statistic
MSMSerrorerror= error term= error term
n=number of scores in each treatmentn=number of scores in each treatment
n
MSqHSD Error
n
MSqHSD Error We use MSerror
instead of MSwithin
Assumptions for One-Way Assumptions for One-Way ANOVAANOVA
1. The observations within each treatment must be independent.
2. The population distribution within each treatment must be normally distributed.
3. The variances of the population distributions for each treatment should be equivalent.
4. Homogeneity of Covariance
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