Repeated Measures ANOVA
I.
Between Subjects Variance Is Partitioned
From the Error Variance
II. Conceptual Approach
III. Using The Source Table
Anthony Greene
1
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)
Anthony Greene
Error
Variability
(MS Within)
2
Source Table for 1-Way ANOVA
Anthony Greene
3
Source Table for 1-Way ANOVA
Error Variability
Effect Variability
4
Repeated Measures ANOVA
Treatment 1
Measurement 1
M1
Treatment 2
Measurement 2
M2
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Treatment 3
Measurement 3
M3
5
An Example of a Repeated
Measures Design
S
Base
Perf
Pf After 2
Pf After 4
Pf After 6
Cups Coffee Cups Coffee 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
8
7
5
Avg 6
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 design
2. Thereby individual
n
x
x
D
differences are
A
2
6
4
removed from the
B
13
17
4
C
24
28
4
error term
1
•
•
2
Identical concepts to repeated measures
ANOVA
Individual Variation is computed and
subtracted from the error term
Anthony Greene
7
Partitioning Variability For
Repeated Measures ANOVA
Stage 1
{
Stage 2
Total
Variability
Effect
Variability
(MS Between)
{
MS Within
Individual
Variability
Anthony Greene
Error
Variability
8
Partitioning Variability For
Repeated Measures ANOVA
Stage 1
{
Numerator
of F-ratio
Stage 2
Total
Variability
Effect
Variability
(MS Between)
{
MS Within
Individual
Variability
Denominator
of F-ratio
Error
Variability
9
Source Table For Repeated
Measures ANOVA
Tests of Within-Subjects Effects
Source
SS
df
MS
FACTOR1
30.0
3
10.0
Error 23.5
15
1.5
Subj. 72.5
5
Within
96.0
20
Total
126.0 23
Source
FACTOR1
Error
Subj.
Within
Total
SS
30.0
23.5
72.5
96.0
126.0
df
3
15
5
20
23
MS
10.0
F
6.3
p
.005
F
2.1
p
n.s.
4.8
Anthony Greene
10
Source Table For Repeated
Measures ANOVA
Tests of Within-Subjects Effects
Source
SS
df
MS
FACTOR1
30.0
3
10.0
Error 23.5
15
1.5
Subj. 72.5
5
Within
96.0
20
Total
126.0 23
F
6.3
Anthony Greene
Sig.
.005
11
Source Table For Repeated
Measures ANOVA
Tests of Within-Subjects Effects
Source
SS
df
MS
FACTOR1
30.0
3
10.0
Error 23.5
15
1.5
Subj. 72.5
5
Within
96.0
20
Total
126.0 23
F
6.3
Anthony Greene
Sig.
.005
12
Source Table For Repeated
Measures ANOVA
Tests of Within-Subjects Effects
Source
SS
df
MS
FACTOR1
30.0
3
10.0
Error 23.5
15
1.5
Subj. 72.5
5
9
Within
96.0
20
8
Total
126.0 23
7
F
6.3
Sig.
.005
6
5
4
3
2
1
0
Base
Anthony Greene
2
4
6
13
Source Table For Repeated
Measures ANOVA
Tests of Within-Subjects Effects
Source
SS
df
MS
FACTOR1
30.0
3
10.0
Error 23.5
15
1.5
Subj
72.5
5
Within
96.0
20
Total
126.0 23
F
6.3
Sig.
.005
9
8
7
6
5
4
3
2
1
0
Base
Tests of Within-Subjects Contrasts
2
Source
SS
df
MS
F
Sig.
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
Anthony Greene
4
6
14
Doing the Computations
Base
Perf
Perf After 2
Cups Coffee
Perf After 4
Cups Coffee
Perf After 6
Cups Coffee
Person Totals
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
S
P
S
Base
Perf
Perf After 2
Cups
Coffee
Perf After 4
Cups
Coffee
Perf After
6 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
32
Avg
6
8
7
5
26
2
Stage 1
Total
Variability
{
Effect
Variability
(MS
Between)
Stage 2
{
MS Within
Individual
Variability
Error
Variability
2
P G
k N
d. f .  n 1
New computation:
1. Compute SS for P and deg freedom
2. Subtract those from SS w/in and d.f. w/in
3. This leaves SS error and d.f. error
Anthony Greene
16
Tests of Within-Subjects Effects
Source
SS
df
MS
FACTOR1
30.0
3
Error
Subj.
Within
96.0
20
Total
126.0 23
Source
FACTOR1
Error
Subj.
Within
Total
SS
30.0
df
3
72.5
96.0
126.0
5
20
23
Source
FACTOR1
Error
Subj.
Within
Total
SS
30.0
23.5
72.5
96.0
126.0
df
3
15
5
20
23
MS
F
Sig.
STAGE 1
F
Sig.
STAGE 2
MS
10.0
1.5
F
6.3
Anthony Greene
Sig.
.005
17
Repeated Measures ANOVA
Source
d.f. SS
Between
Within
12
Subject 3
Residual
Total
15
MS
11.67
F
3
51
Anthony Greene
18
Repeated Measures ANOVA
Source
d.f. SS
Between 3
Within
12
Subject 3
Residual
Total
15
MS
11.67
F
3
51
Anthony Greene
19
Repeated Measures ANOVA
Source
d.f. SS
Between 3
35
Within
12
Subject 3
Residual
Total
15
MS
11.67
F
3
51
Anthony Greene
20
Repeated Measures ANOVA
Source
d.f. SS
Between 3
35
Within
12 16
Subject 3
Residual
Total
15
MS
11.67
F
3
51
Anthony Greene
21
Repeated Measures ANOVA
Source
d.f. SS
Between 3
35
Within
12 16
Subject 3
Residual 9
Total
15
MS
11.67
F
3
51
Anthony Greene
22
Repeated Measures ANOVA
Source
d.f. SS
Between 3
35
Within
12 16
Subject 3
Residual 9
Total
15
MS
11.67
F
3
13
51
Anthony Greene
23
Repeated Measures ANOVA
Source
d.f. SS
Between 3
35
Within
12 16
Subject 3
Residual 9
Total
15
3
13
MS
11.67
F
1.44
51
Anthony Greene
24
Repeated Measures ANOVA
Source
d.f. SS
Between 3
35
Within
12 16
Subject 3
Residual 9
Total
15
3
13
MS
F
11.679.0
1.44
51
Anthony Greene
25
RepeatedMeasures
ANOVA
Before
Therapy
4
2
4
1
After
Therapy
3
5
7
2
6-month
follow-up
1
4
6
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
Anthony Greene
26
RepeatedMeasures
ANOVA
Before
Therapy
5
2
4
1
Compute the
treatment totals, T,
and the grand total,
T1=12
G
After
Therapy
3
5
6
2
T2=16
Anthony Greene
6-month
follow-up
1
4
G=48
9
6
T3=20
27
Before
Repeated- Therapy
5
Measures
2
ANOVA
4
Count n for each
1
After
Therapy
3
5
6
2
6-month
follow-up
1
4
G=48
9
N=12
6
k=3
treatment, the
total N, and k
T1=12
n1=4
T2=16
n2=4
Anthony Greene
T3=20
n3=4
28
RepeatedMeasures
ANOVA
Compute the
treatment means
Before
Therapy
5
2
4
1
After
Therapy
3
5
6
2
T1=12
n1=4
MX1=3
T2=16
n2=4
MX2 =4
Anthony Greene
6-month
follow-up
1
4
G=48
9
N=12
6
k=3
T3=20
n3=4
MX3 =5
29
RepeatedMeasures
ANOVA
Compute the
treatment SSs
Before
After
Therapy
Therapy
5 (5-3)2=4 3
2 (2-3)2=1 5
4 (4-3)2=1 6
1 (1-3)2=4 2
T1=12 sum T2=16
n1=4
n2=4
MX1=3
MX2 =4
SS=10
SS=10
Anthony Greene
6-month
follow-up
1
4
G=48
9
N=12
6
k=3
T3=20
n3=4
MX3 =5
SS=34
30
Before
Repeated- Therapy
5
Measures
2
ANOVA
4
Compute all X2s
1
After
Therapy
3
5
6
2
6-month
follow-up
1
4
9
6
and sum them
T1=12
n1=4
MX1=3
SS=10
T2=16
n2=4
MX2 =4
SS=10
Anthony Greene
T3=20
n3=4
MX3 =5
SS=34
G=48
N=12
k=3
X2= 254
31
Before
After
6-month
Therapy Therapy followRepeatedup
Measures
5
3
1
2
5
4
ANOVA
4
6
9
1
2
6
Add in Person totals
T1=12
T2=16
T3=20
column
n1=4
n2=4
n3=4
P totals
9
11
19
9
G=48
N=12
MX1=3
MX2 =4
MX3 =5
k=3
SS=10
SS=10
SS=34
X2=
254
Anthony Greene
32
RepeatedMeasures
ANOVA
Compute SSTotal
Before
Therapy
5
2
4
1
After
Therapy
3
5
6
2
T1=12
n1=4
MX1=3
SS=10
T2=16
n2=4
MX2 =4
SS=10
SSTotal= X2 – G2/N
SSTotal=62
Anthony Greene
6-month
follow-up
1
4
9
6
T3=20
n3=4
MX3 =5
SS=34
P totals
9
11
19
9
G=48
N=12
k=3
X2= 254
33
RepeatedMeasures
ANOVA
Compute SSbetween
SSbetween=
T2/n – G2/N
SSbetween=8
Before
After
6-month
Therapy Therapy follow-up
5
3
1
2
5
4
4
6
9
1
2
6
T1=12
T2=16
T3=20
n1=4
n2=4
n3=4
MX1=3
MX2 =4
MX3 =5
SS1=10
SS2=10
SS3=34
T2/n=36 T2/n=64 T2/n=100
Anthony Greene
P totals
9
11
19
9
G=48
N=12
k=3
X2= 254
34
RepeatedMeasures
ANOVA
Compute SSWithin
SSWithin= SSi
SSWithin = 54
Before
Therapy
5
2
4
1
T1=12
n1=4
MX1=3
SS1=10
After
Therapy
3
5
6
2
T2=16
n2=4
MX2 =4
SS2=10
Anthony Greene
6-month
follow-up
1
4
9
6
T3=20
n3=4
MX3 =5
SS3=34
P totals
9
11
19
9
G=48
N=12
k=3
X2= 254
35
RepeatedMeasures
ANOVA
Compute SSBtw sub
SSBtw Sub=
P2/k – G2/N
SSBtw Sub =22.66
Before
Therapy
5
2
4
1
T1=12
n1=4
MX1=3
SS1=10
After
Therapy
3
5
6
2
T2=16
n2=4
MX2 =4
SS2=10
6-month
follow-up
1
4
9
6
T3=20
n3=4
MX3 =5
SS3=34
P
9
11
19
9
G=48
N=12
k=3
X2= 254
SSerror = SSwithin-SSbtw sub = 54-22.66 = 31.34
Anthony Greene
36
Before
Repeated-Therapy
Measures 5
2
ANOVA 4
Determine d.f.s
1
T1=12
n1=4
d.f. Within=N-k
MX1=3
SS1=10
d.f. Between=k-1
After
Therapy
3
5
6
2
T2=16
n2=4
MX2 =4
SS2=10
d.f. Total=N-1
d.f. error = (N-k) – (n-1)
Anthony Greene
6-month
follow-up
1
4
9
6
T3=20
n3=4
MX3 =5
SS3=34
G=48
N=12
k=3
X2= 254
d.f. Within=9
d.f. Between=2
d.f.bet sub= 3
d.f.error=6
d.f. Total=11
37
Before
Repeated-Therapy
Measures 5
2
ANOVA 4
Ready to move it
1
to a source table
T1=12
n1=4
MX1=3
SS1=10
After
Therapy
3
5
6
2
T2=16
n2=4
MX2 =4
SS2=10
Anthony Greene
6-month
follow-up
1
4
9
6
T3=20
n3=4
MX3 =5
SS3=34
G=48
N=12
k=3
X2= 254
SSWithin=54
d.f. Within=9
d.f. Between=2
d.f.bet sub= 3
d.f.error=6
d.f. Total=11
38
Repeated Measures ANOVA
Source
Between
Within
Subject
Error
SS
8
54
d.f.
2
9
22.66 3
31.34 6
MS
4
F
0.766
5.22
Total
62
11
Compare to critical value of F(2,6) = 5.14
Fail to reject (no significant difference)
Anthony Greene
39