Stat 512
Repeated Measures Designs Ch 18
Repeated measures involve the repeated observation of a response on the same
experimental unit. Most often, repeated measures are observed over time.
Occasionally, repeated measures occur on the same experimental unit, but at
different treatment combinations. Most often a completely randomized design is
utilized and repeated observations, over time, occur for each treatment level.
Example:
An experiment involving t = 3 different drugs was conducted to study each drug's
effect on the heart rate of humans. The design consisted of the random selection of
n  r1  r2  r3 subjects that were randomly assigned to the 3 drugs in equal
numbers (e.g., r1  r2  r3  r ). After a drug was administered, the heart rate (Y) was
measured every 5 minutes for a total of p = 4 times.
Neglecting the repeated measures, a partial ANOVA table for this experimental
design is:
Source
Drug
Subject(Drug)
Total
SS
SS Drug
SSSubject  Drug 
SSTotal
df
t-1
(r-1)t
rt-1
MS
MS Drug
MSSubject  Drug 
Note that each subject within a drug is observed at each of the p time periods.
Thus, each subject within a drug is acting as a block with respect to the p time
periods. Thus, the design would appear to be a split plot experimental design, where
the whole plot consisted of a CRD with a one-way treatment structure (drug) and
the sub-plot design was an RCBD with a one-way treatment structure (time).
Recall: RCBDs have all treatments randomly assigned within each blocks.
1
Stat 512
Question: Were the p time periods randomly assigned to each subject within a
drug?
Answer: NO! If they were, the experimental design would be a Split Plot with a
CRD for the whole plot.
The above design is a CRD having a one-way treatment structure with repeated
measures.
Bottom-line:
If the time based observations on a subject have uniform covariance (compound
symmetric covariance) structure, then repeated measures experiments can be
analyzed using the procedures already outlined for the split plot design (or split
block design). The assumption of a compound symmetric covariance can be
assessed using SAS or many other statistical software packages.
2
Stat 512
Example (continued)
Suppose for the experiment already outlined, there were 8 subjects assigned to each
level of the factor DRUG, which has t = 3 levels (AX23, BWW9, and CONTROL).
Further, suppose that p = 4 time measures (RATE1, RATE2, RATE3, and RATE4)
were recorded. The following data for this experiment are taken from Milliken and
Johnson (1984) and appear as follows:
Subject
1
2
3
4
5
6
7
8
DRUG
BWW9
AX23
CONTROL
T1
T2
T3
T4
T1
T2
T3
T4
T1
T2
T3
T4
72
78
71
72
66
74
62
69
86
83
82
83
79
83
73
75
81
88
81
83
77
84
78
76
77
81
75
69
66
77
70
70
85
82
71
83
86
86
79
83
86
86
78
88
85
82
83
84
83
80
70
79
76
83
80
78
80
84
75
81
76
80
81
81
69
66
84
80
72
65
75
71
73
62
90
81
72
62
69
70
72
67
88
77
69
65
69
65
74
73
87
72
70
61
68
65
Linear Model
Yijk    i  eWi  j  k   ik  eSi  jk
 i =1, 2,
, t; j = 1, 2,

, r; k = 1, 2,
,p

grand mean
i ith drug effect
eWi j jth subject effect within the ith drug (Whole Plot Error)
 k kth time effect
 ik drug by time interaction
eSi  jk residual error (Sub-plot Error)
Assumptions: eWi  j iidN  0,  W2  , eSi  jk N  0,  S2  and Corr  Yijk , Yijk    for all i and j
3
Stat 512
Covariance matrix for the ith subject:
1




   S2   .
.

.


 
1 
1

 
 
. . .. .
. . .. .
. . .. .
 





1 
Testing for Compound Symmetric Covariance
Mauchly Sphericity Test is used to assess the assumption of compound symmetry.
This test is produced by SAS Proc GLM.
Huynh and Feldt (1970) showed that a less stringent requirement is required for the
analysis of repeated measures data as a split-plot model. Huynh-Feldt developed a
statistic that implies the compound symmetry condition is satisfied when the
statistic has a value of 1.0. If the statistic differs from 1.0, an adjustment is made to
the degrees of freedom for error to correct for the deviation from compound
symmetry.
If all else fails one can rely on the multivariate analysis of variance provided by
Proc GLM.
4
Stat 512
title 'CRD with Repeated Measures';
title2 'Sample Unit = Subject (1 to 8), Treatment = Drug
CONTROL)';
title3 'Repeated Measure = time (1 to 4)';
data a;
input subject drug $ rate1 rate2 rate3 rate4 @@;
cards;
1 AX23 72 86 81 77 1 BWW9 85 86 83 80 1 CONTROL 69 73 72
2 AX23 78 83 88 81 2 BWW9 82 86 80 84 2 CONTROL 66 62 67
3 AX23 71 82 81 75 3 BWW9 71 78 70 75 3 CONTROL 84 90 88
4 AX23 72 83 83 69 4 BWW9 83 88 79 81 4 CONTROL 80 81 77
5 AX23 66 79 77 66 5 BWW9 86 85 76 76 5 CONTROL 72 72 69
6 AX23 74 83 84 77 6 BWW9 85 82 83 80 6 CONTROL 65 62 65
7 AX23 62 73 78 70 7 BWW9 79 83 80 81 7 CONTROL 75 69 69
8 AX23 69 75 76 70 8 BWW9 83 84 78 81 8 CONTROL 71 70 65
;
proc print data = a;
proc glm data = a;
class subject drug;
model rate1-rate4 = drug / nouni;
repeated time 4 / printe summary;
run;
5
(AX23, BWW9,
74
73
87
72
70
61
68
65
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Obs
subject
drug
rate1
rate2
rate3
rate4
81
83
72
88
80
67
81
70
88
83
79
77
77
76
69
84
83
65
78
80
69
76
78
65
77
80
74
81
84
73
75
75
87
69
81
72
66
76
70
77
80
61
70
81
68
70
81
65
1
1
AX23
72
86
2
1
BWW9
85
86
3
1
CONTROL
69
73
4
2
AX23
78
83
5
2
BWW9
82
86
6
2
CONTROL
66
62
7
3
AX23
71
82
8
3
BWW9
71
78
9
3
CONTROL
84
90
10
4
AX23
72
83
11
4
BWW9
83
88
12
4
CONTROL
80
81
13
5
AX23
66
79
14
5
BWW9
86
85
15
5
CONTROL
72
72
16
6
AX23
74
83
17
6
BWW9
85
82
18
6
CONTROL
65
62
19
7
AX23
62
73
20
7
BWW9
79
83
21
7
CONTROL
75
69
22
8
AX23
69
75
23
8
BWW9
83
84
24
8
CONTROL
71
70
Note: The response is arranged in columns, one for each time of measurement.
CRD with Repeated Measures
2
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
Class Level Information
Class
Levels
Values
subject
8
1 2 3 4 5 6 7 8
drug
3
AX23 BWW9 CONTROL
Number of Observations Read
Number of Observations Used
6
24
24
1
Stat 512
CRD with Repeated Measures
3
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
Repeated Measures Analysis of Variance
Repeated Measures Level Information
Dependent Variable
rate1
rate2
rate3
rate4
Level of time
1
2
3
4
Partial Correlation Coefficients from the Error SSCP Matrix / Prob > |r|
DF = 21
rate1
rate2
rate3
rate4
rate1
1.000000
rate2
0.828050
<.0001
0.825500
<.0001
0.644458
0.0012
0.828050
<.0001
1.000000
0.825500
<.0001
0.837311
<.0001
1.000000
0.644458
0.0012
0.722279
0.0001
0.834635
<.0001
1.000000
rate3
rate4
0.837311
<.0001
0.722279
0.0001
0.834635
<.0001
Note: The partial correlation coefficients provide a subjective measure of the relationship
between the responses at different observation times. The p-values only indicate whether
the correlation is significant, and does not pertain to whether the assumption of compound
symmetry is satisfied. If compound symmetry is satisfied, the correlations should all be
about the same. The correlations printed above range from 0.64 to 0.84, indicating that
they are fairly similar in value.
E = Error SSCP Matrix
time_N represents the contrast between the nth level of time and the last
time_1
time_2
time_3
time_1
time_2
time_3
461.88
309.38
226.38
309.38
418.25
193.38
226.38
193.38
217.25
7
Stat 512
Partial Correlation Coefficients from the Error SSCP Matrix of the
Variables Defined by the Specified Transformation / Prob > |r|
DF = 21
time_1
time_2
time_3
time_1
1.000000
0.703890
0.0003
0.714638
0.0002
time_2
0.703890
0.0003
1.000000
0.641509
0.0013
time_3
0.714638
0.0002
0.641509
0.0013
1.000000
8
Stat 512
CRD with Repeated Measures
4
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
Repeated Measures Analysis of Variance
Sphericity Tests
Variables
DF
Mauchly's
Criterion
Chi-Square
Pr > ChiSq
5
5
0.1952368
0.6693259
32.217081
7.9181595
<.0001
0.1608
Transformed Variates
Orthogonal Components
Note: The Sphericity test based on the orthogonal components is used to assess whether
the assumption of compound symmetry is satisfied. The null hypothesis for the test is that
compound symmetry is satisfied and the alternative hypothesis is that compound
symmetry is not satisfied. For the above analysis, the p-value is 0.1608, so the null
hypothesis is not rejected. The data approximately satisfy the assumption of compound
symmetry.
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time Effect
H = Type III SSCP Matrix for time
E = Error SSCP Matrix
S=1
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0.5
N=8.5
Value
F Value
Num DF
Den DF
Pr > F
0.29886466
0.70113534
2.34599609
2.34599609
14.86
14.86
14.86
14.86
3
3
3
3
19
19
19
19
<.0001
<.0001
<.0001
<.0001
MANOVA Test Criteria and F Approximations for the Hypothesis of no time*drug Effect
H = Type III SSCP Matrix for time*drug
E = Error SSCP Matrix
S=2
Statistic
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
M=0
N=8.5
Value
F Value
Num DF
Den DF
Pr > F
0.10831123
1.09156533
6.38726223
6.08394057
12.91
8.01
19.77
40.56
6
6
6
3
38
40
23.636
20
<.0001
<.0001
<.0001
<.0001
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
NOTE: F Statistic for Wilks' Lambda is exact.
NOTE: F Statistic for Roy's Greatest Root is an upper bound.
Note: The Manova tests assess the components of the repeated measures ANOVA
associated with time (e.g., time and time by drug interaction). If compound symmetry is
not satisfied, one can always rely on the multivariate tests, since this is a more general
approach. However, it is also a less powerful procedure. It is generally better to use the
ANOVA procedures when appropriate.
9
Stat 512
For the above multivariate tests, it is clear that an interaction between the time and drug
factors exists, since each of the tests statistics is significant (p-value < 0.0001). However,
since the assumption of compound symmetry was satisfied for the problem at hand, it is
probably best to use the results from the univariate analysis.
CRD with Repeated Measures
5
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source
DF
Type III SS
Mean Square
F Value
Pr > F
drug
2
1315.083333
657.541667
5.95
0.0090
Error
21
2320.156250
110.483631
Note: Assessing differences between the means effect of the three drugs is shown in the above results. A significant difference among the
three drug means is apparent (p-value = 0.009). However, on should assess the interaction between the drug and time before assessing the
main effects of drug or time.
CRD with Repeated Measures
6
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source
DF
Type III SS
Mean Square
F Value
Pr > F
time
time*drug
Error(time)
3
6
63
282.6145833
531.1666667
458.4687500
94.2048611
88.5277778
7.2772817
12.95
12.16
<.0001
<.0001
Greenhouse-Geisser Epsilon
Huynh-Feldt Epsilon
Adj Pr > F
G - G
H - F
<.0001
<.0001
<.0001
<.0001
0.7986
0.9944
Note: The above results for the effect of time and time by drug interaction indicate that
both are significant (p-value < 0.0001 for both). Of course a significant interaction would
mean the care must be taken when assessing differences between the mean effect of the
three drugs or time, since the effect changes over time. In order to assess the effect of the
different levels of drug correctly one would need to produce the multiple comparisons of
drug at each time point. This is where Proc GLM with the Repeated statement becomes a
liability. Using the Repeated statement in Proc GLM removes the use of the LSMeans
statement for constructing the comparisons among the drug means at each level of time.
Because of this another approach must be utilized.
CRD with Repeated Measures
7
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
Repeated Measures Analysis of Variance
Analysis of Variance of Contrast Variables
10
Stat 512
time_N represents the contrast between the nth level of time and the last
Contrast Variable: time_1
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Mean
drug
Error
1
2
21
2.0416667
103.0833333
461.8750000
2.0416667
51.5416667
21.9940476
0.09
2.34
0.7636
0.1206
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Mean
drug
Error
1
2
21
433.5000000
156.2500000
418.2500000
433.5000000
78.1250000
19.9166667
21.77
3.92
0.0001
0.0357
Source
DF
Type III SS
Mean Square
F Value
Pr > F
Mean
drug
Error
1
2
21
130.6666667
376.0833333
217.2500000
130.6666667
188.0416667
10.3452381
12.63
18.18
0.0019
<.0001
Contrast Variable: time_2
Contrast Variable: time_3
Note: The above are the linear, quadratic and cubic orthogonal polynomials associated
with the time effect.
The graph of the estimated mean heart rate across time, separate for each drug, is presented in
the following graph:
The graph clearly shows the interaction that was indicated by the repeated measure analysis. So,
how do you assess the differences between the mean heart rate for each drug, since lsmeans are
not available for the drug by time components?
11
Stat 512
The following is a reproduction, in part, of the repeated measures analysis produced by SAS
Proc GLM using the Repeated statement. In addition, the SAS code has been modified to a
form suitable for analysis as a Split-Plot Design, so that the LSMeans statement can be used to
assess pairwise differences between the means at the level of the sub-plot.
SAS Code:
options pageno = 1 nodate center;
title 'CRD with Repeated Measures';
title2 'Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9,
CONTROL)';
title3 'Repeated Measure = time (1 to 4)';
data drug;
input person medicine $ rate1 rate2 rate3 rate4 @@;
cards;
1 AX23 72 86 81 77 1 BWW9 85 86 83 80 1 CONTROL 69 73 72 74
2 AX23 78 83 88 81 2 BWW9 82 86 80 84 2 CONTROL 66 62 67 73
3 AX23 71 82 81 75 3 BWW9 71 78 70 75 3 CONTROL 84 90 88 87
4 AX23 72 83 83 69 4 BWW9 83 88 79 81 4 CONTROL 80 81 77 72
5 AX23 66 79 77 66 5 BWW9 86 85 76 76 5 CONTROL 72 72 69 70
6 AX23 74 83 84 77 6 BWW9 85 82 83 80 6 CONTROL 65 62 65 61
7 AX23 62 73 78 70 7 BWW9 79 83 80 81 7 CONTROL 75 69 69 68
8 AX23 69 75 76 70 8 BWW9 83 84 78 81 8 CONTROL 71 70 65 65
;
proc print data = drug;
run;
proc glm data = drug;
class person medicine;
model rate1-rate4 = medicine / nouni;
repeated time 4 / printe summary;
run;
title4 'Analysis Using Proc GLM using a Split Plot Model Structure';
data drug; set drug;
subject = person; drug =medicine; rate = rate1; time = 5; output;
subject = person; drug =medicine; rate = rate2; time = 10; output;
subject = person; drug =medicine; rate = rate3; time = 15; output;
subject = person; drug =medicine; rate = rate4; time = 20; output;
keep subject drug rate time;
run;
proc print data = drug;
run;
proc glm data = drug;
class subject drug time;
model rate = drug subject(drug) time time*drug;
test h = drug e = subject(drug);
lsmeans time*drug / pdiff stderr;
run;
CRD with Repeated Measures
1
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
12
Stat 512
Repeated Measure = time (1 to 4)
Obs
person
medicine
rate1
rate2
rate3
rate4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
6
6
6
7
7
7
8
8
8
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
AX23
BWW9
CONTROL
72
85
69
78
82
66
71
71
84
72
83
80
66
86
72
74
85
65
62
79
75
69
83
71
86
86
73
83
86
62
82
78
90
83
88
81
79
85
72
83
82
62
73
83
69
75
84
70
81
83
72
88
80
67
81
70
88
83
79
77
77
76
69
84
83
65
78
80
69
76
78
65
77
80
74
81
84
73
75
75
87
69
81
72
66
76
70
77
80
61
70
81
68
70
81
65
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
2
The GLM Procedure
Class Level Information
Class
Levels
Values
person
8
1 2 3 4 5 6 7 8
medicine
3
AX23 BWW9 CONTROL
Number of Observations Read
Number of Observations Used
24
24
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
5
The GLM Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source
DF
Type III SS
Mean Square
F Value
Pr > F
medicine
Error
2
21
1315.083333
2320.156250
657.541667
110.483631
5.95
0.0090
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
The GLM Procedure
13
6
Stat 512
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Source
DF
Type III SS
Mean Square
F Value
Pr > F
time
time*medicine
Error(time)
3
6
63
282.6145833
531.1666667
458.4687500
94.2048611
88.5277778
7.2772817
12.95
12.16
<.0001
<.0001
Greenhouse-Geisser Epsilon
Huynh-Feldt Epsilon
Adj Pr > F
G - G
H - F
<.0001
<.0001
<.0001
<.0001
0.7986
0.9944
The above results are only a partial representation of the results computed using
Repeated statement. However, they are sufficient for a comparison to the results when the
data are analyzed using Proc GLM, but based on the model for a split plot design.
14
Stat 512
CRD with Repeated Measures
8
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
Obs
subject
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
1
1
1
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
drug
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
15
rate
time
72
86
81
77
85
86
83
80
69
73
72
74
78
83
88
81
82
86
80
84
66
62
67
73
71
82
81
75
71
78
70
75
84
90
88
87
72
83
83
69
83
88
79
81
80
81
77
72
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
Obs
subject
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
5
5
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
7
7
7
7
8
8
8
8
8
8
8
8
8
8
8
8
drug
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
16
rate
time
66
79
77
66
86
85
76
76
72
72
69
70
74
83
84
77
85
82
83
80
65
62
65
61
62
73
78
70
79
83
80
81
75
69
69
68
69
75
76
70
83
84
78
81
71
70
65
65
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
9
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
10
The GLM Procedure
Class Level Information
Class
Levels
Values
subject
8
1 2 3 4 5 6 7 8
drug
3
AX23 BWW9 CONTROL
time
4
5 10 15 20
Number of Observations Read
Number of Observations Used
96
96
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
11
The GLM Procedure
Dependent Variable: rate
Source
DF
Sum of
Squares
Mean Square
F Value
Pr > F
Model
32
4449.020833
139.031901
19.10
<.0001
Error
63
458.468750
7.277282
Corrected Total
95
4907.489583
R-Square
Coeff Var
Root MSE
rate Mean
0.906578
3.529696
2.697644
76.42708
Source
DF
Type I SS
Mean Square
F Value
Pr > F
drug
subject(drug)
time
drug*time
2
21
3
6
1315.083333
2320.156250
282.614583
531.166667
657.541667
110.483631
94.204861
88.527778
90.36
15.18
12.95
12.16
<.0001
<.0001
<.0001
<.0001
Source
DF
Type III SS
Mean Square
F Value
Pr > F
drug
subject(drug)
time
drug*time
2
21
3
6
1315.083333
2320.156250
282.614583
531.166667
657.541667
110.483631
94.204861
88.527778
90.36
15.18
12.95
12.16
<.0001
<.0001
<.0001
<.0001
Tests of Hypotheses Using the Type III MS for subject(drug) as an Error Term
Source
DF
Type III SS
Mean Square
17
F Value
Pr > F
Stat 512
drug
2
1315.083333
657.541667
5.95
0.0090
The sum of squares for the model, error, drug, subject(drug), time and drug by time interaction are identical to the sum of squares produced
by SAS Proc GLM using the Repeated statement.
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
12
The GLM Procedure
Least Squares Means
drug
time
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
5
10
15
20
5
10
15
20
5
10
15
20
rate LSMEAN
Standard
Error
Pr > |t|
LSMEAN
Number
70.5000000
80.5000000
81.0000000
73.1250000
81.7500000
84.0000000
78.6250000
79.7500000
72.7500000
72.3750000
71.5000000
71.2500000
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
0.9537611
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
1
2
3
4
5
6
7
8
9
10
11
12
Least Squares Means for effect drug*time
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: rate
i/j
1
1
2
3
4
5
6
7
8
9
10
11
12
<.0001
<.0001
0.0561
<.0001
<.0001
<.0001
<.0001
0.1003
0.1694
0.4612
0.5802
2
3
4
5
6
<.0001
<.0001
0.7121
0.0561
<.0001
<.0001
<.0001
0.3576
0.5802
<.0001
<.0001
0.0118
0.0297
<.0001
0.1003
0.7121
<.0001
0.3576
0.0118
0.1694
0.5802
<.0001
<.0001
<.0001
<.0001
<.0001
0.5802
0.0297
0.0831
0.3576
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.0001
<.0001
0.7819
0.5802
0.2328
0.1694
0.1003
0.0238
0.1431
<.0001
<.0001
<.0001
<.0001
0.0002
0.0025
<.0001
<.0001
<.0001
<.0001
Least Squares Means for effect drug*time
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: rate
i/j
1
2
3
7
8
9
10
11
12
<.0001
0.1694
0.0831
<.0001
0.5802
0.3576
0.1003
<.0001
<.0001
0.1694
<.0001
<.0001
0.4612
<.0001
<.0001
0.5802
<.0001
<.0001
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
18
13
Stat 512
The GLM Procedure
Least Squares Means
Least Squares Means for effect drug*time
Pr > |t| for H0: LSMean(i)=LSMean(j)
Dependent Variable: rate
i/j
4
5
6
7
8
9
10
11
12
7
8
9
10
11
12
0.0001
0.0238
0.0002
<.0001
0.1431
0.0025
0.4074
0.7819
<.0001
<.0001
<.0001
<.0001
0.5802
<.0001
<.0001
<.0001
<.0001
0.7819
0.2328
<.0001
<.0001
<.0001
<.0001
0.3576
0.5189
0.1694
<.0001
<.0001
<.0001
<.0001
0.2703
0.4074
0.8536
0.4074
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
0.7819
0.3576
0.2703
0.5189
0.4074
0.8536
NOTE: To ensure overall protection level, only probabilities associated with pre-planned
comparisons should be used.
The LSMeans statement provides one with the ability to assess differences between means at
the sub-plot level. In this case, comparisons can be made between the means for the four time
periods at each level of the Drug (see bold above). You can also compare whole plot means
(Drug) across each level of the subplot (Time), or across different levels of the subplot.
However, the pvalues for the LSD’s produced by the LSMeans statement are not correct. This is
because the standard errors and the degrees of freedom for the pairwise tests are a linear
combination of the whole plot and sub-plot components (see the notes on Split-plot designs in
chapter 14). Therefore, the above LSMeans for the whole plot means (Drugs) should only be
considered a reasonable approximation. Proper computations of the LSDs in this case are
difficult at best, since they are not computed by Proc GLM. The appropriate methods for
computing the estimated standard errors and LSDs when the assumption of compound
symmetry is met are the same as the methods outlined in Chapter 14 (split-plot designs).
Assuming that the number of replicates per whole-plot treatment are the same for all treatments
(r) and there are b and are as follows:
Comparison of Whole Plot Means
19
Stat 512
Each whole plot mean Yi..  is constructed from rb observations and the F test for the global
comparison of whole plot means uses the MSWhole Plot Error , the estimated standard error for the
comparison of two whole plot means is
se Yi..  Yi..  
2  MSWhole Plot Error
r p
The t statistic for the comparison of two whole plot means (e.g., H 0 : i..  i..  0 ) is
t
Yi..  Yi..
2.MSWhole Plot Error
rp
with rejection region: t  t 

 , dfWhole Plot Error  
2

The above LSD formulation is valid for comparison of whole plot marginal means only.
Comparison of Sub-Plot Means
Comparing two sub-plot means Y..k  is based on rt values, along with the sub-plot error. The
estimated standard error for the comparison of two sub-plot treatment means is
se Y..k  Y..k   
2  MSSub- Plot Error
r t
The t statistic for the comparison of two sub-plot means (e.g., Ho:
H 0 : ..k  ..k   0 ) is
t
Y..k  Y..k 
2.MSSub Plot Error
rt
with rejection region: t  t 

 , dfSub- Plot Error  
2

The above LSD formulation is valid for comparison of sub-plot marginal means only.
Comparison of Whole Plot Means within the same or different Sub-Plot Treatments
Comparing two whole plot means at the same level of the sub-plot treatment, or different levels
at different levels of the sub-plot treatment requires a substantial effort individual if the results
of the Proc GLM are used. Cell means Yi.k  within the same whole plot treatment are based on
rt values. In addition, the estimated standard error is a function of both the whole plot error
20
Stat 512
and the sub-plot error. The estimated standard error for the comparison of two whole plot
treatment means at the same or different levels of the sub-plot is
 MSWhole Plot Error   p  1  MS Sub- Plot Error 
se Yi.k  Yi.k    2  

rp


and the appropriate t quantile for the t statistic is
t 
t 
*
 , df 
2



 ,  r 1t 
2

 MSWhole Plot Error  t 

 ,  r 1t  p 1 
2

  p  1  MS Sub- Plot Error
MSWhole Plot Error   p  1  MS Sub- Plot Error
This computation poses a substantial burden if performed by hand. An alternative approach is to
use SAS Proc Mixed.
The LSMeans procedure in SAS Proc Mixed, when used with the Satterwaithe approximation
(or the Kenward and Roger variant), will produce the correct LSD value for multiple
comparisons. For the heart rate drug study the following code is used to implement SAS Proc
Mixed under the compound symmetric covariance structure:
options pageno = 1 nodate center;
title 'CRD with Repeated Measures';
title2 'Sample Unit = Subject (1 to 8), Treatment = Drug (AX23,
BWW9, CONTROL)';
title3 'Repeated Measure = time (1 to 4)';
data drug;
input person medicine $ rate1 rate2 rate3 rate4 @@;
cards;
1 AX23 72 86 81 77 9 BWW9 85 86 83 80 17 CONTROL 69 73 72 74
2 AX23 78 83 88 81 10 BWW9 82 86 80 84 18 CONTROL 66 62 67 73
3 AX23 71 82 81 75 11 BWW9 71 78 70 75 19 CONTROL 84 90 88 87
4 AX23 72 83 83 69 12 BWW9 83 88 79 81 20 CONTROL 80 81 77 72
5 AX23 66 79 77 66 13 BWW9 86 85 76 76 21 CONTROL 72 72 69 70
6 AX23 74 83 84 77 14 BWW9 85 82 83 80 22 CONTROL 65 62 65 61
7 AX23 62 73 78 70 15 BWW9 79 83 80 81 23 CONTROL 75 69 69 68
8 AX23 69 75 76 70 16 BWW9 83 84 78 81 24 CONTROL 71 70 65 65
;
run;
proc print data = drug;
run;
proc glm data = drug;
class person medicine;
model rate1-rate4 = medicine / nouni;
21
Stat 512
repeated time 4 / printe summary;
run;
title4 'Analysis Using Proc GLM using a Split Plot Model
Structure';
data drug; set drug;
subject = person; drug =medicine; rate = rate1; time = 5;
output;
subject = person; drug =medicine; rate = rate2; time = 10;
output;
subject = person; drug =medicine; rate = rate3; time = 15;
output;
subject = person; drug =medicine; rate = rate4; time =
20; output;
keep subject drug rate time;
run;
proc print data = drug;
run;
proc mixed data = drug;
class subject drug time;
model rate = drug time time*drug/ ddfm=satterth;
repeated /type = cs subject=subject r;
lsmeans time*drug / pdiff;
run;
Note that the subject variable (= person) is now numbered from 1 to 24 so that SAS Proc Mixed
will distinguish each replicate (subject). This is necessary for Proc Mixed, but not for Proc
GLM, if the correct analysis is desired.
The output for the above SAS code is as follows:
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
Obs
1
2
subject
1
1
drug
rate
time
AX23
AX23
72
86
5
10
22
1
Stat 512
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
1
1
9
9
9
9
17
17
17
17
2
2
2
2
10
10
10
10
18
18
18
18
3
3
3
3
11
11
11
11
19
19
19
19
4
4
4
4
12
12
12
12
20
20
20
20
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
81
77
85
86
83
80
69
73
72
74
78
83
88
81
82
86
80
84
66
62
67
73
71
82
81
75
71
78
70
75
84
90
88
87
72
83
83
69
83
88
79
81
80
81
77
72
23
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
Obs
subject
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
5
5
5
5
13
13
13
13
21
21
21
21
6
6
6
6
14
14
14
14
22
22
22
22
7
7
7
7
15
15
15
15
23
23
23
23
8
8
8
8
16
16
16
16
24
24
24
24
drug
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
24
rate
time
66
79
77
66
86
85
76
76
72
72
69
70
74
83
84
77
85
82
83
80
65
62
65
61
62
73
78
70
79
83
80
81
75
69
69
68
69
75
76
70
83
84
78
81
71
70
65
65
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
5
10
15
20
2
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
The Mixed Procedure
Model Information
Data Set
Dependent Variable
Covariance Structure
Subject Effect
Estimation Method
Residual Variance Method
Fixed Effects SE Method
Degrees of Freedom Method
WORK.DRUG
rate
Compound Symmetry
subject
REML
Profile
Model-Based
Satterthwaite
Class Level Information
Class
subject
drug
time
Levels
Values
24
1 2 3 4 5 6 7 8 9 10 11 12 13
14 15 16 17 18 19 20 21 22 23
24
AX23 BWW9 CONTROL
5 10 15 20
3
4
Dimensions
Covariance Parameters
Columns in X
Columns in Z
Subjects
Max Obs Per Subject
2
20
0
24
4
Number of Observations
Number of Observations Read
Number of Observations Used
Number of Observations Not Used
96
96
0
Iteration History
Iteration
Evaluations
-2 Res Log Like
Criterion
0
1
1
1
557.24212530
487.17690350
0.00000000
25
3
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
The Mixed Procedure
Convergence criteria met.
4
Estimated R Matrix for subject 1
Row
Col1
Col2
Col3
Col4
1
33.0789
25.8016
25.8016
25.8016
2
25.8016
33.0789
25.8016
25.8016
3
25.8016
25.8016
33.0789
25.8016
4
25.8016
25.8016
25.8016
33.0789
The above covariance matrix is computed under the assumption of compound symmetry.
Covariance Parameter Estimates
Cov Parm
Subject
Estimate
CS
Residual
subject
25.8016
7.2773
Fit Statistics
-2 Res Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
487.2
491.2
491.3
493.5
Null Model Likelihood Ratio Test
DF
Chi-Square
Pr > ChiSq
1
70.07
<.0001
The chi-square test presented above assesses the appropriateness of the selected covariance structure, compound symmetric in this case,
relative to the default model of independence and constant variance.
Type 3 Tests of Fixed Effects
Effect
Num
DF
Den
DF
F Value
Pr > F
drug
2
21
5.95
0.0090
time
3
63
12.95
<.0001
drug*time
6
63
12.16
<.0001
The test statistics computed above are the same as those computed for the split-plot and repeated measures model based on SAS Proc GLM.
This includes the denominator degrees of freedom, which are computed based on the assumption of compound symmetry.
26
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
5
The Mixed Procedure
Least Squares Means
Effect
drug
time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
5
10
15
20
5
10
15
20
5
10
15
20
Estimate
Standard
Error
DF
t Value
Pr > |t|
70.5000
80.5000
81.0000
73.1250
81.7500
84.0000
78.6250
79.7500
72.7500
72.3750
71.5000
71.2500
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
2.0334
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
34.67
39.59
39.83
35.96
40.20
41.31
38.67
39.22
35.78
35.59
35.16
35.04
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
<.0001
Differences of Least Squares Means
Effect
drug
time
_drug
_time
Estimate
Standard
Error
DF
t Value
Pr > |t|
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
5
5
5
5
5
5
5
5
5
5
5
10
10
10
10
10
10
10
10
10
10
15
15
15
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
AX23
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
AX23
BWW9
BWW9
10
15
20
5
10
15
20
5
10
15
20
15
20
5
10
15
20
5
10
15
20
20
5
10
-10.0000
-10.5000
-2.6250
-11.2500
-13.5000
-8.1250
-9.2500
-2.2500
-1.8750
-1.0000
-0.7500
-0.5000
7.3750
-1.2500
-3.5000
1.8750
0.7500
7.7500
8.1250
9.0000
9.2500
7.8750
-0.7500
-3.0000
1.3488
1.3488
1.3488
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
1.3488
1.3488
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
1.3488
2.8757
2.8757
63
63
63
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
63
63
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
63
29.7
29.7
-7.41
-7.78
-1.95
-3.91
-4.69
-2.83
-3.22
-0.78
-0.65
-0.35
-0.26
-0.37
5.47
-0.43
-1.22
0.65
0.26
2.69
2.83
3.13
3.22
5.84
-0.26
-1.04
<.0001
<.0001
0.0561
0.0005
<.0001
0.0084
0.0031
0.4402
0.5194
0.7305
0.7960
0.7121
<.0001
0.6669
0.2331
0.5194
0.7960
0.0115
0.0084
0.0039
0.0031
<.0001
0.7960
0.3053
27
Stat 512
CRD with Repeated Measures
Sample Unit = Subject (1 to 8), Treatment = Drug (AX23, BWW9, CONTROL)
Repeated Measure = time (1 to 4)
Analysis Using Proc GLM using a Split Plot Model Structure
6
The Mixed Procedure
Differences of Least Squares Means
Effect
drug
time
_drug
_time
Estimate
Standard
Error
DF
t Value
Pr > |t|
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
drug*time
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
AX23
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
15
15
15
15
15
15
20
20
20
20
20
20
20
20
5
5
5
5
5
5
5
10
10
10
10
10
10
15
15
15
15
15
20
20
20
20
5
5
5
10
10
15
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
BWW9
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
BWW9
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
BWW9
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
BWW9
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
CONTROL
15
20
5
10
15
20
5
10
15
20
5
10
15
20
10
15
20
5
10
15
20
15
20
5
10
15
20
20
5
10
15
20
5
10
15
20
10
15
20
15
20
20
2.3750
1.2500
8.2500
8.6250
9.5000
9.7500
-8.6250
-10.8750
-5.5000
-6.6250
0.3750
0.7500
1.6250
1.8750
-2.2500
3.1250
2.0000
9.0000
9.3750
10.2500
10.5000
5.3750
4.2500
11.2500
11.6250
12.5000
12.7500
-1.1250
5.8750
6.2500
7.1250
7.3750
7.0000
7.3750
8.2500
8.5000
0.3750
1.2500
1.5000
0.8750
1.1250
0.2500
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
1.3488
1.3488
1.3488
2.8757
2.8757
2.8757
2.8757
1.3488
1.3488
2.8757
2.8757
2.8757
2.8757
1.3488
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
2.8757
1.3488
1.3488
1.3488
1.3488
1.3488
1.3488
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
63
63
63
29.7
29.7
29.7
29.7
63
63
29.7
29.7
29.7
29.7
63
29.7
29.7
29.7
29.7
29.7
29.7
29.7
29.7
63
63
63
63
63
63
0.83
0.43
2.87
3.00
3.30
3.39
-3.00
-3.78
-1.91
-2.30
0.13
0.26
0.57
0.65
-1.67
2.32
1.48
3.13
3.26
3.56
3.65
3.98
3.15
3.91
4.04
4.35
4.43
-0.83
2.04
2.17
2.48
2.56
2.43
2.56
2.87
2.96
0.28
0.93
1.11
0.65
0.83
0.19
0.4154
0.6669
0.0075
0.0054
0.0025
0.0020
0.0054
0.0007
0.0655
0.0284
0.8971
0.7960
0.5763
0.5194
0.1003
0.0238
0.1431
0.0039
0.0028
0.0013
0.0010
0.0002
0.0025
0.0005
0.0003
0.0001
0.0001
0.4074
0.0500
0.0379
0.0191
0.0156
0.0211
0.0156
0.0075
0.0061
0.7819
0.3576
0.2703
0.5189
0.4074
0.8536
Notice that the degrees of freedom for the pairwise comparisons shown above are either
29.7 or 63. Those comparisons with 63 degrees of freedom are comparisons of sub-plot
treatments (time) means at the same level of the whole plot. Those comparisons with 29.7
degrees of freedom are comparisons of whole plot treatments (drug) means at the same or
different levels of the sub-plot. The 29.7 degrees of freedom represent the Satterwaithe
approximation to the degrees of freedom outlined on page 25 of these notes. In addition,
28
Stat 512
the estimated standard errors listed along side these comparisons are the Satterwaithe
approximations to the standard error for the comparison of two means (also listed on page
25).
29