ANOVA Designs Involving
Repeated Measures
46-511: One-Way Repeated Measures and
Groups by Trials
1
Learning Objectives
 Be able to identify the advantages and disadvantages
of repeated measures designs
 Understand how variance is partitioned in RM
designs
 Comprehend variations possible with such designs
 Be familiar with post-hoc & planned comparisons
using RM Designs

Including Trend Analysis
2
The Design
 Repeated Measures Designs, a.k.a.

Dependent Measures Designs

Within-Subjects Designs
 Mixed Randomized-Repeated Designs, a.k.a.

Groups by Trials

Split Plot Factorial
 What are they
 Relative Advantages
 Relative Disadvantages
3
Some Examples
 Completely Within Designs


1-Way
N-Way
 Mixed Randomized-Repeated Designs


1-Between by 1-Within
N-Between by N-Within
 Characteristics of repeated measures
designs


Nature of the repeated measures
Duration between measures
4
1-Way Repeated Measures ANOVA:
Sources of Variation
 Between Subjects
 Between Treatments
 Within Subjects
 Within Treatments
5
One-Way Example
Person
Drug 1
Drug 2
Drug 3
Drug 4
Pi
Mean
1
2
3
4
5
30
14
24
38
26
28
18
20
34
28
16
10
18
20
14
34
22
30
44
30
108
64
92
136
98
27.0
16.0
23.0
34.0
24.5
Tj
132
128
78
160
G = 498
Mean
26.4
25.6
15.6
32.0
24.9
GM=24.9
Five subjects, all are tested for reaction time after taking each of the
four drugs, over a period of four days.
6
One-Way Example
Person
Drug 1
Drug 2
Drug 3
Drug 4
Pi
Mean
1
2
3
4
5
30
14
24
38
26
28
18
20
34
28
16
10
18
20
14
34
22
30
44
30
108
64
92
136
98
27.0
16.0
23.0
34.0
24.5
132
128
78
160
G = 498
Tj
.
.
.
Between Treatments
Within Subjects
Between Subjects Effects
Where does Within Treatment variation come from?
7
Two Structural Models
The rosy additive model:
Xij = μ + πi + τj + εij
The model that assumes people x
treatment interaction:
Xij = μ + πi + τj + πτij + εij
8
Partitioning Sums of Squares:
or, here we go again
Between People
SSB.PEOPLE  k(Pi  G)
2
Between Treatments
SSTREAT  n(Tj  G)
2
9
Sums of Squares Within
Within People
SSW .PEOPLE   ( X ij  Pi )
i
2
j
Within Treatments
SSW .TREATMENT  ( X ij  Tj )
2
j i
10
The error term
 Two ways to get it:


SSRES = SSW.PEOPLE – SSTREAT
SSRES = SSW.TREATMENT – SSB.PEOPLE
 How the error term differs from Between
Subjects Design
 What the error term represents/contains
11
Source Table
Source
SS
df
MS
SSB.PEOPLE
680.80
4
170.20
SSW.PEOPLE
811.00
15
54.07
SSTREAT
698.20
3
232.73
SSRES
112.80
12
9.40
1,491.80
19
78.52
SSTOT
F
p
24.759
0.000020
12
Missing Data in Within Subjects
Designs
 Due to such things as
 Equipment failure
 Experimenter or subject error
 Loss of questionnaires
 Usual missing data solutions ignore design
Y 
*
ij
sSi'  aA'j  T '
(a  1)(s  1)
•Y*ij = predicted (missing) score
•s = number of subjects
•S’i = sum of known values for the case
•a = number of levels of A
•A’j = sum of known values of A
•T’ = sum of all known values
13
Example
 Say subject #3 didn’t return to take drug 4
#3’s sum is now 62
 Sum for A4 is 130 (160-30)
 Sum of known scores for entire table = 49830=468
5(62)  4(130)  468
*
Y3,4 
 30.167
(4  1)(5  1)

Error term must be reduced by number of imputed values (imputed
values are not independent)
14
Assumptions
 Observations within each treatment cell are independent.
 Population treatment within each treatment must be normally
distributed.
 Variances for the population treatments should be equivalent.
 Sphericity – that the variance of the difference scores for each
pair of conditions is the same in the population.
 Alternatives if assumptions do not hold.
15
Mean Comparison Procedures
 Tukey

Same as in 1-way between, substitute MS error
(residual) for MS within & df error for df within
 Scheffe’

Same as in 1-way between, substitute MS error for MS
within & df error for df within
 Bonferroni procedure
 Šidák procedure
16
Contrasts
 Unfortunately, contrasts are not quite a logical
extension of contrasts from between subjects
designs

Affected by mild violations of sphericity
 Must determine variability specific to each
contrast.
 Two methods, t and F.
 Let’s test the following contrast:

C1: .5*Drug1+.5*Drug2 – Drug3=0.
17
Method One
Ci
t
sM C
Single Sample t-test:
2
where
and
sMc
sC

n
2
2

C

(

C
)
/n
2
i
i
sC 
n 1
18
Method 1 (Cont’d)
(single sample t-test)
Effect of Different Drugs on Reaction Time
Person
Drug 1
Drug 2
Drug 3
C1 (Using t-test)
1
30
28
16
13
2
14
18
10
6
3
24
20
18
4
4
38
34
20
16
5
26
28
14
13
Mean
26.4
25.6
15.6
ΣCi
52.00
Mean Ci
10.40
ΣCi2
646.00
19
Calculations for t:
2
646.00

(52)
/5
2
sC 
 26.3
5 1
sM c
26.3

 2.293
5
10.40
t
 4.535
2.293
df = n-1; t(.05,4) = 2.78
20
Method 2: Using the F statistic
2
Sum of squares for contrast:
Sum of squares for error term:
5(10.40)2
SSC 
1.5
SSCerror
nC
SSC 
C 2j
SSCerror 
C  nC
C 2j
2
i
2
646.00  5(10.40) 2

 105.20
1.5
21
Method 2 (Cont’d)
df = 1, (n – 1)
MSCerror
105.20

 26.3
4
540.8
F
 20.563
26.3
Recalling that t2 = F; 4.5352 = 20.566
22
Effect sizes
Partial ω2:
Partial η2 / R2:
(a  1)( FA  1)
 
(a  1)( FA  1)  an
2
p
SS A
 
SS A  SS RES
2
p
Cohen’s d: similar, have to decide on proper standard deviation
23
Power
Use partial effect size
2
f 
1   p2
^
p
Use G*Power or power charts. Assume ρ=.50 unless you
know different.
24
Trend Analysis Example
Mean
Subject #
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
1
43.00
31.00
10.00
9.00
4.00
4.00
2
40.00
30.00
9.00
6.00
4.00
5.00
3
49.00
33.00
12.00
5.00
6.00
3.00
4
39.00
26.00
11.00
8.00
5.00
5.00
5
41.00
28.00
8.00
6.00
5.00
5.00
6
44.00
34.00
9.00
7.00
7.00
6.00
42.67
30.33
9.83
6.83
5.17
4.67
Experiment on forgetting: six participants master a list of 50 words, then are
asked to recall them the next day (time 1), one week later (time 2), and so on. 25
SPSS Output: Test of Sphericity &
Summary Table
Epsilon(a)
Within Subjects
Effect
Mauchly's
W
factor1
.004
Error(factor1)
16.948
df
Greenhous
e-Geisser
Sig.
14
Type III Sum of
Squares
Source
factor1
Approx.
Chi-Square
.394
df
.373
Mean Square
HuynhFeldt
.587
F
Lowerbound
.200
Sig.
Sphericity Assumed
7694.250
5
1538.850
364.369
.000
Greenhouse-Geisser
7694.250
1.867
4121.540
364.369
.000
Huynh-Feldt
7694.250
2.937
2620.068
364.369
.000
Lower-bound
7694.250
1.000
7694.250
364.369
.000
Sphericity Assumed
105.583
25
4.223
Greenhouse-Geisser
105.583
9.334
11.311
Huynh-Feldt
105.583
14.683
7.191
Lower-bound
105.583
5.000
21.117
26
SPSS Output Polynomial Contrasts
Source
factor1
Type III Sum of
Squares
factor1
Linear
6179.336
1
6179.336
532.505
.000
Quadratic
1292.161
1
1292.161
502.740
.000
.112
1
.112
.125
.738
Order 4
143.006
1
143.006
39.040
.002
Order 5
79.636
1
79.636
33.393
.002
Linear
58.021
5
11.604
Quadratic
12.851
5
2.570
4.471
5
.894
Order 4
18.315
5
3.663
Order 5
11.924
5
2.385
Cubic
Error(factor1)
Cubic
df
Mean Square
F
Sig.
27
Graph of linear & quadratic trend
Repeated Measures Trend Analysis
Repeated Measures Trend Analysis
50.00
45.00
45.00
40.00
40.00
35.00
35.00
Recall N Correct
Recall N Correct
30.00
25.00
20.00
y = -7.6714x + 43.433
R2 = 0.8031
15.00
30.00
Series1
Poly. (Series1)
25.00
Series1
Linear (Series1)
20.00
y = 2.4018x2 - 24.484x + 65.85
R2 = 0.971
15.00
10.00
10.00
5.00
5.00
0.00
Time 1
Time 2
Time 3
Time 4
-5.00
Time 5
Time 6
0.00
Time 1
Time 1
Time 2
Time 3
Time 4
Time 5
Time 6
Time 1
28
Groups by Trials Design:
What it is?
 Combines between and within designs
 Yields effects for…



Trials
Groups
Trials by Groups interaction
 Can be used to answer questions such as…



Do test pattern scores (e.g., pre-post) differ by experimental
vs. control group…
Do women and men differ significantly in their ability to
detect smell in varying conditions.
Do patients receiving drug A have a different course of
improvement than those receiving drug B?
29
Assumptions
 The usual assumptions for between subjects
designs
 The usual assumptions for repeated measures
designs
 Homogeneity of variance/covariance matrices by
group
30
Partitioning Variance!
 Between Groups Variance –
 Subjects within Groups Variance –
 Between Trials Variance –
 Group by Trials Variance –
 Subjects within Groups within Trials Variance (residual) –
31
Between Group Variance
Definitional Formula
SS A  nk ( Aj  G)
2
Computational Formula
A
2
j
2
G
SS A 

nk njk
32
Subjects within Groups
Definitional Formula
SSsubj _ w. groups  k ( Pi  Aj )
2
Computational Formula
Pi A


k
nk
2
SSsubj _ w. groups
2
j
33
Between Trials Variance
Definitional Formula
SSB  nj(Bk  G)
2
Computational Formula
B
G
SS B 

nj njk
2
k
2
34
Group by Trials Variance
Definitional Formula
SSAB  n( AB jk  A j  Bk  G)
2
Computational Formula
SS AB 
AB2jk
n
A2j
Bk2 G 2



nk
nj njk
35
Subjects within Groups within
Trials (residual)
Definitional Formula
SS BSubjects _W / In _ Groups   [( ABijk  Pi )  ( B k  A j )]
2
i k
Computational Formula
SS BSubjects _W / In _ Groups  X ijk 
2
AB 2jk
n
A2j
Pi


k
nk
2
36
Numerical Example
Subject #
B1: Month 1
B2: Month 2
B3: Month 3
1
2
3
4
5
1
1
3
5
2
3
4
3
5
4
6
8
6
7
5
A2: Mystery
6
7
8
9
10
3
4
5
4
4
1
4
3
2
5
0
2
2
0
3
A3: Romance
11
12
13
14
15
4
2
3
6
3
2
6
3
2
3
0
1
3
1
2
A1: Scifi.
37
Between Subjects Effects
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
Source
Intercept
novtype
Error
Type III Sum
of Squares
473.689
20.578
26.400
df
1
2
12
Mean Square
473.689
10.289
2.200
F
215.313
4.677
Sig .
.000
.031
Estimates
Measure: MEASURE_1
Genre of Novel
Science Fiction
Mystery
Romance
Mean
4.200
2.800
2.733
Std. Error
.383
.383
.383
95% Confidence Interval
Lower Bound
Upper Bound
3.366
5.034
1.966
3.634
1.899
3.568
38
Within Subjects & Interaction
Tests of W ithin-Subjects Effects
Measure: MEASURE_1
Source
factor1
factor1 * novtype
Error(factor1)
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
.711
.711
.711
.711
71.422
71.422
71.422
71.422
37.200
37.200
37.200
37.200
df
2
1.651
2.000
1.000
4
3.303
4.000
2.000
24
19.818
24.000
12.000
Mean Square
.356
.431
.356
.711
17.856
21.624
17.856
35.711
1.550
1.877
1.550
3.100
F
.229
.229
.229
.229
11.520
11.520
11.520
11.520
Sig .
.797
.755
.797
.641
.000
.000
.000
.002
39
Interaction Effect
Estimated Marginal Means of MEASURE_1
3. Genre of Novel * factor1
Genre of Novel
7.00
Science Fiction
Measure: MEASURE_1
Mystery
Romance
Estimated Marginal Means
6.00
Genre of Novel
Science Fiction
5.00
4.00
Mystery
3.00
Romance
2.00
1.00
1
2
factor1
1
2
3
1
2
3
1
2
3
Mean
Std. Error
2.400
.611
3.800
.627
6.400
.542
4.000
.611
3.000
.627
1.400
.542
3.600
.611
3.200
.627
1.400
.542
95% Confidence Interval
Lower Bound Upper Bound
1.069
3.731
2.434
5.166
5.220
7.580
2.669
5.331
1.634
4.366
.220
2.580
2.269
4.931
1.834
4.566
.220
2.580
3
factor1
40
Two-Within Subject Factors
 Brief Example
 Effects



Main effects for Factor A
Main effects for Factor B
Interaction effect for A x B
41
Numerical Example
Number of Books Read each Month by Genre
B1: Science Fiction
B2: Mystery
A1: Month
1
A2: Month
2
A3: Month
3
A1: Month
1
A2: Month
2
A3: Month
3
s1
1
3
6
5
4
1
s2
1
4
8
8
8
4
s3
3
3
6
4
5
3
s4
5
5
7
3
2
0
s5
2
4
5
5
6
3
A 3(month) x 2(genre) way within subjects ANOVA, where n=5
42
Summary Table
Tests of W ithin-Subj ects Effects
Measure: MEASURE_1
Source
genre
Error(genre)
month
Error(month)
genre * month
Error(genre*month)
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Type III Sum
of Squares
.133
.133
.133
.133
33.533
33.533
33.533
33.533
2.867
2.867
2.867
2.867
4.133
4.133
4.133
4.133
64.467
64.467
64.467
64.467
9.867
9.867
9.867
9.867
df
1
1.000
1.000
1.000
4
4.000
4.000
4.000
2
1.661
2.000
1.000
8
6.645
8.000
4.000
2
1.144
1.303
1.000
8
4.577
5.212
4.000
Mean Square
.133
.133
.133
.133
8.383
8.383
8.383
8.383
1.433
1.726
1.433
2.867
.517
.622
.517
1.033
32.233
56.344
49.479
64.467
1.233
2.156
1.893
2.467
F
.016
.016
.016
.016
Sig.
.906
.906
.906
.906
2.774
2.774
2.774
2.774
.122
.136
.122
.171
26.135
26.135
26.135
26.135
.000
.004
.003
.007
43
And, the Interaction Plot
44
Two-way Repeated Measures
ANOVA: Main effects
Main effect for A
B1: Science Fiction
B2: Mystery
A1:
Month 1
A2:
Month 2
A3:
Month 3
A1:
Month 1
A2:
Month 2
A3:
Month 3
s1
1
3
6
5
4
1
s2
1
4
8
8
8
4
s3
3
3
6
4
5
3
s4
5
5
7
3
2
0
s5
2
4
5
5
6
3
A1: Month 1
A2: Month 2
A3: Month 3
s1
1+5=6
3+4=7
6+1=7
s2
1+8=9
4+8=12
8+4=12
s3
3+4=7
3+5=8
6+3=9
s4
5+3=8
5+2=7
7+0=7
s5
2+5=7
4+6=10
5+3=8
45
Why different error terms?
 Recall SSRES=SSW.PEOPLE-SSTREAT
A1: Month 1
A2: Month 2
A3: Month 3
SSW.PEOPLE
s1
6
7
7
0.667
s2
9
12
12
6.000
s3
7
8
9
2.000
s4
8
7
7
0.667
s5
7
10
8
4.667
=14.000/2=7
SSRES=7-2.867=4.133
46
For B Main effect…
B1: SciFi
B2: Mystery
SSW.PEOPLE
s1
10
10
0.0
s2
13
20
24.5
s3
12
12
0.0
s4
17
5
72.0
s5
11
14
4.5
=101/3=33.667
SSRES=33.667-0.133=33.534
47
Finally, for AxB…
B1: Science Fiction
B2: Mystery
A1:
Month 1
A2:
Month 2
A3:
Month 3
A1:
Month 1
A2:
Month 2
A3:
Month 3
SSW.PEOPLE
s1
1
3
6
5
4
1
21.33
s2
1
4
8
8
8
4
43.50
s3
3
3
6
4
5
3
8.00
s4
5
5
7
3
2
0
31.33
s5
2
4
5
5
6
3
10.83
115.00
SSRES=SSW.PEOPLE – SSA – SSB – SSAB – SSAS – SSBS
SSRES=115.0 – 2.867 – 0.133 – 64.467 – 4.133 – 33.533 = 9.867
48