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