Repeated Measures ANOVA

Anthony Greene 1

Repeated Measures ANOVA

I. Between Subjects Variance Is Partitioned From the Error Variance

II. Conceptual Approach

III. Using The Source Table

Anthony Greene 2

The Source Table• Keeps track of all data in complex ANOVA

designs

• Source of SS, df, and Variance (MS)– Partitioning the

SS, df and MS– All variability is attributable to

effect differences or error (all unexplained differences)

Total Variability

Effect Variability

(MS Between)

ErrorVariability

(MS Within)

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Source Table for 1-Way ANOVA

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Source Table for 1-Way ANOVA

Effect VariabilityError Variability

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Measurement 1 Measurement 2 Measurement 3

Treatment 1 Treatment 2 Treatment 3

M1 M2 M3

An Example of a Repeated Measures Design

S Base Perf

Pf After 2 Cups Coffee



A 4 6 6 3

B 5 8 6 2

C 9 11 9 7

D 6 7 8 9

E 5 6 5 2

F 7 10 8 7

Avg 6 8 7 5

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Repeated Measures ANOVA• Recall the differences between a t-test a

tD-test:1. The same group is used in the experimental

and control group via the before-after design2. Thereby individual

differences are removed from the error term

• Identical concepts to repeated measures ANOVA

• Individual Variation is computed and subtracted from the error term

n x1 x2 D

A 2 6 4

B 13 17 4

C 24 28 4

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Partitioning Variability For Repeated Measures ANOVA

Total Variability

Effect Variability

(MS Between)

MS Within

Individual Variability

ErrorVariability

Stage 1 {{Stage 2

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Partitioning Variability For Repeated Measures ANOVA

Total Variability

Effect Variability

(MS Between)

MS Within


ErrorVariability

Stage 1 {{Stage 2

Numerator of F-ratio

Denominator of F-ratio

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Source Table For Repeated Measures ANOVA

Tests of Within-Subjects Effects

Source SS df MS F p

FACTOR1 30.0 3 10.0 6.3 .005

Error 23.5 15 1.5

Subj. 72.5 5

Within 96.0 20

Total 126.0 23

Source SS df MS F p

FACTOR1 30.0 3 10.0 2.1 n.s.

Error 23.5 15

Subj. 72.5 5

Within 96.0 20 4.8Total 126.0 23

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Source SS df MS F Sig.

FACTOR1 30.0 3 10.0 6.3 .005

Error 23.5 15 1.5

Subj. 72.5 5

Within 96.0 20Total 126.0 23

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FACTOR1 30.0 3 10.0 6.3 .005

Error 23.5 15 1.5

Subj. 72.5 5

Within 96.0 20Total 126.0 23

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FACTOR1 30.0 3 10.0 6.3 .005

Error 23.5 15 1.5

Subj. 72.5 5

Within 96.0 20Total 126.0 23

0

1

2

3

4

5

6

7

8

9

Base 2 4 6

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Tests of Within-Subjects Contrasts


Linear 4.8 1 4.8 1.5 .263

Quadratic 24.0 1 24.0 18.4 .008

Cubic 1.2 1 1.2 3.1 .136



FACTOR1 30.0 3 10.0 6.3 .005

Error 23.5 15 1.5

Subj 72.5 5

Within 96.0 20Total 126.0 23

0

1

2

3

4

5

6

7

8

9

Base 2 4 6

Doing the ComputationsS Base

PerfPerf After 2 Cups Coffee

Perf After 4 Cups Coffee


Person Totals P

A 4 6 6 3 19

B 5 8 6 2 21

C 9 11 9 7 36

D 6 7 8 9 30

E 5 6 5 2 18

F 7 10 8 7 32Avg 6 8 7 5 26

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S Base Perf




Person Totals P

A 4 6 6 3 19

B 5 8 6 2 21

C 9 11 9 7 36

D 6 7 8 9 30

E 5 6 5 2 18

F 7 10 8 7 32

Avg 6 8 7 5 26

1..

22

nfd

N

G

k

P

Total Variability

Effect Variability

(MS Between)

MS Within


ErrorVariability

Stage 1 {

{Stage 2

New computation: 1. Compute SS for P and deg freedom2. Subtract those from SS w/in and d.f. w/in3. This leaves SS error and d.f. error

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Tests of Within-Subjects EffectsSource SS df MS F Sig. FACTOR1 30.0 3

ErrorSubj. STAGE 1

Within 96.0 20Total 126.0 23

Source SS df MS F Sig. FACTOR1 30.0 3

ErrorSubj. 72.5 5 STAGE 2

Within 96.0 20Total 126.0 23

Source SS df MS F Sig. FACTOR1 30.0 3 10.0 6.3 .005

Error 23.5 15 1.5Subj. 72.5 5

Within 96.0 20Total 126.0 23

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Source d.f. SS MS F

Between 11.67

Within 12Subject 3 3

Residual

Total 15 51

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Source d.f. SS MS F

Between 3 11.67


Residual

Total 15 51

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Source d.f. SS MS F

Between 3 35 11.67


Residual

Total 15 51

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Source d.f. SS MS F

Between 3 35 11.67

Within 12 16Subject 3 3

Residual

Total 15 51

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Source d.f. SS MS F

Between 3 35 11.67


Residual 9

Total 15 51

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Source d.f. SS MS F

Between 3 35 11.67


Residual 9 13

Total 15 51

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Source d.f. SS MS F

Between 3 35 11.67


Residual 9 13 1.44

Total 15 51

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Source d.f. SS MS F

Between 3 35 11.67 9.0


Residual 9 13 1.44

Total 15 51

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Repeated-Measures ANOVA

Before Therapy

After Therapy

6-month follow-up

4 3 1

2 5 4

4 7 6

1 2 6

An experimenter wishes to examine whether a certain therapy for Attention Deficit Disorder is effective. Four ADD children, 12 years old, are selected. The dependent measure is a standard test of attention wherein the child hears a narrative and is then asked questions about its content

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Before Therapy

After Therapy

6-month follow-up

5 3 1

2 5 4 G=48

4 6 9

1 2 6

T1=12 T2=16 T3=20

Compute the treatment totals, T, and the grand total, G

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Before Therapy

After Therapy

6-month follow-up

5 3 1

2 5 4 G=48

4 6 9 N=12

1 2 6 k=3

T1=12 T2=16 T3=20

n1=4 n2=4 n3=4

Count n for each treatment, the total N, and k

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Before Therapy

After Therapy

6-month follow-up

5 3 1

2 5 4 G=48

4 6 9 N=12

1 2 6 k=3

T1=12 T2=16 T3=20

n1=4 n2=4 n3=4

MX1=3 MX2 =4 MX3 =5

Compute the treatment means

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Before Therapy

After Therapy

6-month follow-up

5 3 1

2 5 4 G=48

4 6 9 N=12

1 2 6 k=3

T1=12 T2=16 T3=20

n1=4 n2=4 n3=4

MX1=3 MX2 =4 MX3 =5

SS=10 SS=10 SS=34

Compute the treatment SSs

(5-3)2=4

(2-3)2=1

(4-3)2=1

(1-3)2=4

sum

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Before Therapy

After Therapy

6-month follow-up

5 3 1

2 5 4

4 6 9

1 2 6

T1=12 T2=16 T3=20 G=48

n1=4 n2=4 n3=4 N=12

MX1=3 MX2 =4 MX3 =5 k=3

SS=10 SS=10 SS=34 X2= 254

Compute all X2s and sum them

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Add in Person totals column

Before Therapy

After Therapy

6-month follow-up

P totals

5 3 1 9

2 5 4 11

4 6 9 19

1 2 6 9

T1=12 T2=16 T3=20 G=48

n1=4 n2=4 n3=4 N=12

MX1=3 MX2 =4 MX3 =5 k=3

SS=10 SS=10 SS=34 X2= 254

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Before Therapy

After Therapy

6-month follow-up

P totals

5 3 1 9

2 5 4 11

4 6 9 19

1 2 6 9

T1=12 T2=16 T3=20 G=48

n1=4 n2=4 n3=4 N=12

MX1=3 MX2 =4 MX3 =5 k=3

SS=10 SS=10 SS=34 X2= 254

Compute SSTotal

SSTotal= X2 – G2/N

SSTotal=62

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Before Therapy

After Therapy

6-month follow-up

P totals

5 3 1 9

2 5 4 11

4 6 9 19

1 2 6 9

T1=12 T2=16 T3=20 G=48

n1=4 n2=4 n3=4 N=12

MX1=3 MX2 =4 MX3 =5 k=3

SS1=10 SS2=10 SS3=34 X2= 254

T2/n=36 T2/n=64 T2/n=100

Compute SSbetween

SSbetween= T2/n – G2/N

SSbetween=8

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Before Therapy

After Therapy

6-month follow-up

P totals

5 3 1 9

2 5 4 11

4 6 9 19

1 2 6 9

T1=12 T2=16 T3=20 G=48

n1=4 n2=4 n3=4 N=12

MX1=3 MX2 =4 MX3 =5 k=3

SS1=10 SS2=10 SS3=34 X2= 254

Compute SSWithin

SSWithin= SSi

SSWithin = 54

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Before Therapy

After Therapy

6-month follow-up

P

5 3 1 9

2 5 4 11

4 6 9 19

1 2 6 9

T1=12 T2=16 T3=20 G=48

n1=4 n2=4 n3=4 N=12

MX1=3 MX2 =4 MX3 =5 k=3

SS1=10 SS2=10 SS3=34 X2= 254

Compute SSBtw sub

SSBtw Sub=

P2/k – G2/N

SSBtw Sub =22.66

SSerror = SSwithin-SSbtw sub = 54-22.66 = 31.34

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Before Therapy

After Therapy

6-month follow-up

5 3 1 G=48

2 5 4 N=12

4 6 9 k=3

1 2 6 X2= 254

T1=12 T2=16 T3=20

n1=4 n2=4 n3=4 d.f. Within=9

MX1=3 MX2 =4 MX3 =5 d.f. Between=2

SS1=10 SS2=10 SS3=34 d.f.bet sub= 3

d.f.error=6

d.f. Total=11

Determine d.f.s

d.f. Within=N-k

d.f. Between=k-1

d.f. Total=N-1

d.f. error = (N-k) – (n-1)

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Before Therapy

After Therapy

6-month follow-up

5 3 1 G=48

2 5 4 N=12

4 6 9 k=3

1 2 6 X2= 254

T1=12 T2=16 T3=20 SSWithin=54

n1=4 n2=4 n3=4 d.f. Within=9

MX1=3 MX2 =4 MX3 =5 d.f. Between=2

SS1=10 SS2=10 SS3=34 d.f.bet sub= 3

d.f.error=6

d.f. Total=11

Ready to move it to a source table

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Source SS d.f. MS F

Between8 2 4 0.766

Within 54 9Subject 22.66 3

Error 31.34 6 5.22

Total 62 11

Compare to critical value of F(2,6) = 5.14

Fail to reject (no significant difference)

Repeated Measures ANOVA

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