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Chapter 6
Repeated Measures Analysis of Variance
6
...
The different occasions are the levels of a factor, which is referred to as a
repeated measures factor or within-subjects factor
...
These types of data are usually analysed with a paired t-test
...
Disadvantages: Problems with
i) The carry-over effect: The carry-over effect occurs when a new treatment is
administered before the effect of the previous treatment has worn off
ii) The latent effect: The latent effect occurs when one treatment may activate the
dormant effect of the previous treatment or interact with the previous treatment
...
The data: Typically the data in a repeated measure design have the following form :
Subjects
1
2
Treatment 1
Treatment 2
Treatment 3
N
This looks like a two-way analysis of variance data, but the problem here is that each row
represents the same subject so the observations in each row are not independent
...
While the two errors are independent the Y’s are not longer independent
...
6
...
The response variable was the rating a picture received from a panel of judges
...
Subject
1
2
3
4
5
6
7
8
9
10
Crayon
10
18
20
12
19
25
18
22
17
23
Implement
Paint
12
10
15
10
20
22
16
18
14
20
Felt tip
14
16
16
12
21
20
17
18
12
18
Here the repeated-measures factor or within-subjects factor is type of implement used
...
To conduct a repeated measures ANOVA use the following procedure:
1
...
2
...
3
...
4
...
79
5
...
6
...
7
...
8
...
9
...
10
...
11
...
Options for producing profile plots and for saving residuals are also available within the
GLM – Repeated Measures procedure
...
Mean
Deviation
18
...
65
15
...
27
16
...
06
N
10
10
10
b
Multivariate Tests
Effect
IMPLEMEN
a
...
503
...
012
1
...
047
2
...
000
a
4
...
000
8
...
047
2
...
000
a
4
...
000
8
...
...
061
...
061
Exact statistic
b
...
W
Chi-Square
...
760
a
df
2
Epsilon
Sig
...
684
...
000
...
a
...
Corrected tests are displayed in the layers
(by default) of the Tests of Within Subjects Effects table
...
Design: Intercept
Within Subjects Design: IMPLEMEN
82
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
IMPLEMEN
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Error(IMPLEMEN)
Type III
Sum of
Squares
39
...
267
39
...
267
72
...
733
72
...
733
df
2
1
...
000
1
...
505
18
...
000
Mean
Square
19
...
412
19
...
267
4
...
407
4
...
081
F
Estimated Marginal Means
IMPLEMEN
Measure: MEASURE_1
IMPLEMEN
1
2
3
95% Confidence Interval
Lower
Upper
Mean
Std
...
400
1
...
075
21
...
700
1
...
645
18
...
400
...
209
18
...
0
18
...
0
17
...
0
16
...
0
15
...
859
4
...
859
4
...
...
024
...
055
6
...
3
...
When both approaches lead to similar results choosing between them is not of much
importance
...
The multivariate approach considers the measurements on a subject to be a sample from a
multivariate Normal distribution and it does not make special assumptions about the variance
covariance matrix of the response variable
...
These assumptions are often called the symmetry
assumptions
...
6
...
2 Hypothesis Tests
The hypothesis of interest is whether the rating of a picture depends on the type of implement
used
...
These tests are provided in SPSS by the Multivariate Tests and Tests of Within-Subjects
Effects respectively
...
3
...
The first column in the Multivariate Tests table shows the value for the four tests
...
For a one-way within-subjects ANOVA, all multivariate tests yield the same results
...
061
...
3
...
Specifically, these tests require certain patterns
of variance-covariance matrices
...
In the
repeated measures model situation, measures made on the same subject are usually
correlated
...
The independence assumption is replaced by the
symmetry assumption based on sphericity test
...
A related concept is that of compound symmetry
...
The sphericity test is more restrictive than the compound symmetry
...
For our example, the value of the test statistic is 0
...
This test has an equivalent
approximate Chi-Square test value of 0
...
The Chi-square value is not significant (P=0
...
The sphericity test should be viewed with the following reservations: It has low power for
small sample sizes
...
The sphericity test can be very sensitive
to outliers
...
3
...
859 with 2 and 18 degrees of freedom, P = 0
...
Thus the type of implement
used does affect the ratings that a picture receive
...
4 Experiments with Within-Subjects and Between-Subjects
Factors
It is possible to have between-subjects factors in repeated measures design
...
Each subject has only one
value for a between-subject factor
...
4 of Howell (1997)
The experiment investigates motor activity in rats following injection of the drug midazolam
...
Three groups were used
...
On the day of test one group (2) was
injected with midazolam in the same environment in which it had earlier been injected, the
other group (3) was injected in a different environment, and finally the control group was
injected for the first time
...
The dependent
variable is a measure of ambulatory behaviour, in arbitrary units
...
362
104
144 114
115
127 3
338
132
91
77
108
169
3
263
94 141 142
120
195
3
138
38
16
95
39
55
3
329
62
62
6
93
67
3
292
139
104 184
193
122 3
In this example, all 6 intervals occur within each rat, hence Interval is a within-subjects
factor
...
SPSS Procedure
Using the General linear model/GLM Repeated measures we obtain a dialogue box where
the within-subject factor has to be declared
...
Next we
tick define and we define the between-subject factor group
...
General Linear Model
Within-Subjects Factors
Measure: MEASURE_1
INTERVAL
1
2
3
4
5
6
86
Dependent
Variable
I1
I2
I3
I4
I5
I6
Between-Subjects Factors
GROUP
1
2
3
Value
Label
control
same
environme
nt
different
environme
nt
N
8
8
8
a
Box's Test of Equality of Covariance Matrices
Box's M
F
df1
df2
Sig
...
236
...
477
Tests the null hypothesis that the observed covariance matrices
of the dependent variables are equal across groups
...
Design: Intercept+GROUP
Within Subjects Design: INTERVAL
c
Multivariate Tests
Effect
INTERVAL
Value
...
151
5
...
621
...
367
1
...
109
F
Hypothesis
df
Error df
a
19
...
000
17
...
110
5
...
000
a
19
...
000
17
...
110
5
...
000
2
...
000
36
...
216
10
...
000
2
...
000
32
...
992
5
...
000
Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
INTERVAL * GROUP Pillai's Trace
Wilks' Lambda
Hotelling's Trace
Roy's Largest Root
a
...
The statistic is an upper bound on F that yields a lower bound on the significance level
...
Design: Intercept+GROUP
Within Subjects Design: INTERVAL
87
Sig
...
000
...
000
...
043
...
041
...
W
Chi-Square
...
698
a
df
14
Epsilon
Sig
...
009
...
867
...
a
...
Corrected tests are displayed in the layers
(by default) of the Tests of Within Subjects Effects table
...
Design: Intercept+GROUP
Within Subjects Design: INTERVAL
Tests of Within-Subjects Effects
Measure: MEASURE_1
Source
INTERVAL
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
INTERVAL * GROUP Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
Error(INTERVAL)
Sphericity Assumed
Greenhouse-Geisser
Huynh-Feldt
Lower-bound
88
Type III
Sum of
Mean
Squares
df
Square
F
399736
...
313
29
...
563
3
...
641
29
...
563
4
...
251
29
...
563
1
...
563
29
...
958
10
8081
...
018
80819
...
569
12302
...
018
80819
...
674
9317
...
018
80819
...
000
40409
...
018
281199
...
089
281199
...
979
4076
...
313
91
...
401
281199
...
000
13390
...
...
000
...
000
...
009
...
070
Tests of Within-Subjects Contrasts
Measure: MEASURE_1
Type III
Sum of
Squares
df
170066
...
446
53116
...
095
2986
...
873
11883
...
256
12449
...
538
83359
...
759
66040
...
741
25802
...
315
42
...
446
60
...
181
16
...
095
2
...
524
2
...
936
5
...
814
2
...
128
2
...
832
2
...
269
1
...
511
2742
...
797
2304
...
683
Tests of Between-Subjects Effects
Measure: MEASURE_1
Transformed Variable: Average
Source
Intercept
GROUP
Error
89
Type III
Sum of
Squares
df
4113798
...
042
384722
...
4113798
...
551
...
521
7
...
003
18320
...
...
000
...
103
...
016
...
152
...
266
Estimated Marginal Means
GROUP * INTERVAL
Measure: MEASURE_1
GROUP
control
same environment
different environment
INTERVAL
1
2
3
4
5
6
1
2
3
4
5
6
1
2
3
4
5
6
95% Confidence Interval
Lower
Upper
Mean
Std
...
875
28
...
109
272
...
250
31
...
050
158
...
500
22
...
511
142
...
625
23
...
750
178
...
875
21
...
046
167
...
125
25
...
964
183
...
625
28
...
859
413
...
250
31
...
050
331
...
000
22
...
011
266
...
000
23
...
125
219
...
625
21
...
796
243
...
625
25
...
464
231
...
125
28
...
359
348
...
250
31
...
050
163
...
500
22
...
511
154
...
000
23
...
125
158
...
500
21
...
671
168
...
625
25
...
464
191
...
Residual Plots
Dependent Variable: interval 1
Observed
Dependent Variable: interval 2
Observed
Predicted
Dependent Variable: interval 3
Observed
Predicted
Std
...
Residual
Std
...
Residual
Model: Intercept + GROUP
90
Predicted
Std
...
Residual
Model: Intercept + GROUP
Profile Plots
Estimated Marginal Means of MEASURE_1
400
Estimated Marginal Means
300
200
GROUP
control
100
same environment
different environmen
0
t
1
2
3
4
5
6
INTERVAL
6
...
5
...
For tests that involve only between-subjects effects both the multivariate
and the univariate approaches give rise to the same tests
...
(F statistic 7
...
003)
...
6
...
2 Within Subjects Multivariate
For within subjects effects and for within subject by between subject interaction effects, the
univariate and multivariate approaches yield different tests
...
The interaction
between INTERVAL and group is marginally significant
...
5
...
When between-subjects factors are included in a repeated measures design
an additional assumption is necessary
...
The hypothesis that the variance-covariance matrices are equal across all levels of the
between-subjects factor can be examined using the multivariate generalization of the Box’s
M test which is based on all the original variables for the within-subjects effect
...
236 with p-value
= 0
...
The Mauchly’s spericity test is highly significant (P = 0
...
6
...
4 Within subjects Univariate
When there is reason to doubt the symmetry assumption tests of within-subjects effects can
be made by reducing the degrees of freedom contributed by the within factors
...
They are based on a
degrees of freedom adjustment factor known as (epsilon)
...
When this happens the epsilon is
set to one
...
The lowest value possible for is also shown
...
The adjustment to the F-test is shown below
...
563
5
79947
...
852
399736
...
285
121695
...
852
399736
...
337
92166
...
852
399736
...
000
399736
...
852
80819
...
996
3
...
958
6
...
398
3
...
958
8
...
227
3
...
958
2
...
979
3
...
313
105
2678
...
313
68
...
581
281199
...
080
3087
...
313
21
...
443
Sig
...
000
...
000
...
002
...
004
...
92
6
...
5 Within-Subjects Contrasts
If the factor which defines the repeated measures (i
...
INTERVAL) is found to be significant
further investigation is required to found out which of the levels contributed to the significant
result
...
The following output is produced using the ‘simple’ option
...
Level 6
Level 2 vs
...
Level 6
Level 4 vs
...
Level 6
Level 1 vs
...
Level 6
Level 3 vs
...
Level 6
Level 5 vs
...
Level 6
Level 2 vs
...
Level 6
Level 4 vs
...
Level 6
Type III Sum
of Squares
df
451004
...
042
1
1218
...
500 1
15
...
333 2
85057
...
000 2
3420
...
583 2
163121
...
625 21
53434
...
750 21
146294
...
451004
...
062
...
042
...
829
1218
...
479
...
500 1
...
295
15
...
002
...
167 2
...
120
42528
...
056
...
000 5
...
010
1710
...
467
...
792
...
682
7767
...
554
2544
...
798
6966
...
93
Using the option ‘difference’ we get the following table
...
Level 1
Level 3 vs
...
Previous
Level 5 vs
...
Previous
Level 2 vs
...
Previous
Level 4 vs
...
Previous
Level 6 vs
...
Level 1
Level 3 vs
...
Previous
Level 5 vs
...
Previous
Type III Sum
of Squares
df
428535
...
760 1
79964
...
667 1
13680
...
000 2
5017
...
787 2
7897
...
210 2
102046
...
406 21
109133
...
766 21
68420
...
428535
...
188
...
760 42
...
000
79964
...
387
...
667 8
...
007
13680
...
199
...
500 4
...
022
2508
...
737
...
894 4
...
029
3948
...
519
...
605 2
...
076
4859
...
829
5196
...
179
3258
...
Polynomial Contrasts
If the levels of a within-subject factor represent quantitative values that are equally spaced,
then it may be appropriate to conduct polynomial contrasts
...
94
Practical 6: Repeated measures
1
...
4
...
The data are in Shared (K):\Sctms\som\ma2013\data\prac7
...
b) To explore the data, use ANALYZE/DESCRIPTIVES/EXPLORE and also tick the
box PLOTS
...
c) Use correlate to produce the correlation coefficient for the intervals
...
the
intervals
can
be
explored
using
e) Now use the Generalised Linear model/Repeated Measures procedure to produce
the repeated measures analysis
...
2) This is the same problem as the two sample pair test example of practical 1
...
sav
...
95
Questions for practical 6
General question
i) What distinguishes the repeated measures analysis from the simple analysis of variance?
ii) What is the basic model used in the analysis of repeated measures
...
iv)What is the basic difference between the Multivariate and the univariate approach in the
analysis of repeated measures?
v) In which ANOVA table are the two approaches different and in which are they the same
...
Howell example:
i) What is the factor INTERVAL and how is it defined?
ii) Is the factor F important in the analysis? (use the between-subjects effects table)
iii)Is the sphericity assumption valid?
iv)Given the result from iii above what F values are appropriate for testing the significance of
INTERVAL and its interaction with F
...
vii)
Is there any evidence that the residuals are not normal?
The two sample paired test
i) What is the question of interest here and how it can be tested?
ii) Compare the results with the ones you obtain from the Howell example and in particular
try to identify
a) the within subjects factor
b) the between subjects factor
...
96