6 EXAMPLE: SYSTOLIC BLOOD PRESSURE STUDY WITH COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT
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10
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 16.1
Data for the Blood Pressure Study Where IBP is the Initial Systolic Blood
Pressure and Week 1 to Week 6 are the Weekly Systolic Blood Pressure
Measurements (in mmHg)
Drug
No
No
No
No
No
No
No
No
Exercise
No
No
No
No
No
No
No
No
Person
1
2
3
4
5
6
7
8
IBP
133
137
148
136
140
139
154
152
Week 1
146
136
148
139
141
140
170
146
Week 2
144
135
148
137
143
136
166
147
Week 3
142
134
148
138
145
135
168
146
Week 4
141
134
143
139
147
135
167
145
Week 5
142
134
143
139
147
132
165
146
Week 6
143
137
144
142
147
135
166
143
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
No
No
No
No
No
No
No
1
2
3
4
5
6
7
8
150
147
142
144
140
137
143
137
145
147
133
139
132
133
144
122
134
139
121
128
121
122
133
111
136
139
122
128
121
122
131
111
134
132
120
122
115
118
124
104
132
134
121
119
117
119
123
101
134
135
116
116
116
121
124
101
No
No
No
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
1
2
3
4
5
6
7
8
150
151
151
142
149
132
134
151
143
144
155
149
142
133
136
150
142
137
149
144
140
127
129
145
139
134
147
140
136
123
125
141
137
131
140
140
132
122
122
137
134
128
136
137
126
117
123
131
128
121
131
129
126
116
115
130
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
1
2
3
4
5
6
7
8
131
148
151
150
144
151
152
135
119
138
146
137
129
140
149
123
109
130
134
126
123
138
141
115
102
124
129
119
118
134
138
109
100
120
130
119
117
131
132
107
100
122
134
120
118
126
132
111
103
123
134
121
117
129
131
109
would be between 28 and 168, where those numbers will be reduced depending on
the form of the covariance matrix and the form of the covariate part of the model.
It is not reasonable to just assume that the split-plot assumptions are adequate without
first looking at the fit of several of the possible covariance structures. In order to
pursue the investigation into an adequate covariance structure, the covariate part of
the model must be addressed. Plots of the measured blood pressure (BP) values
© 2002 by CRC Press LLC
Analysis of Covariance for Repeated Measures Designs
11
TABLE 16.2
Analysis of Variance Table without Covariate Information for the Blood
Pressure Study
Source
df
EMS
Drug
1
σ + 6 σ + φ (Drug)
Exercise
1
σ ε2 + 6 σ p2 + φ 2 (Exercise)
Drug*Exercise
1
σ ε2 + 6 σ p2 + φ 2 (Drug ∗ Exercise)
Error(person) = Person( Drug*Exercise)
28
2
ε
2
p
2
σ ε2 + 6 σ p2
Time
5
σ ε2 + φ 2 (Time)
Time*Drug
5
σ ε2 + φ 2 (Time ∗ Drug)
Time*Exercise
5
σ ε2 + φ 2 (Time ∗ Exercise)
Time*Drug*Exercise
5
σ ε2 + φ 2 (Time ∗ Drug ∗ Exercise)
Error(week interval) = Time*Person(Drug*Exercise)
140
σ ε2
against the initial blood pressure (IBP) values for each combination of exercise by
drug by time indicates that a linear relationship is adequate to describe the relationship between BP and IBP (plots not shown). The form of the model used to investigate the selection of the covariance structure for the repeated measurements uses
different intercepts and different slopes for each of the combinations of exercise by
drug by time as:
BPijkm = α ijm + βijm IBPijk + p ijk + ε ijkm
(16.4)
The random person effects are specified by the random statement in PROC MIXED
as “Random person(exercise drug);” and repeated statement is used to specify the
covariance structure of the repeated measurements as “Repeated Time/type=xxx
subject=person(exercise drug);” where xxx is one of the covariance structures that
can be selected when using PROC MIXED. As discussed in Section 16.3, the
inclusion of the random statement depends on which covariance structure has been
selected to fit the data. Tables 16.3 through 16.10 contain the PROC MIXED code
and results for fitting eight different covariance structures to the repeated measurements. The eight covariance structures are split-plot (Table 16.3), compound symmetry (Table 16.4), heterogeneous compound symmetry (Table 16.5), first-order
auto-regressive (Table 16.6), heterogeneous first-order auto-regressive (Table 16.7),
ante-dependence (Table 16.8), Toeplitz (Table 16.9), and unstructured (Table 16.10).
The random statement was not needed for compound symmetry, heterogeneous
compound symmetry, ante-dependence, Toeplitz, and unstructured, as discussed in
Section 16.3. Each of the tables contains the PROC MIXED code, estimates of the
covariance structure parameters, tests that the intercepts are all equal, tests that the
© 2002 by CRC Press LLC
12
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 16.3
Analysis of Variance Table for the Blood Pressure Data Using
the Split-Plot Assumptions for the Error Terms
Proc Mixed cl ic covtest DATA=E165;
Class Drug Exercise person Time;
Model BP= Drug*Exercise*Time IBP*Drug*Exercise*Time/ddfm=kr;
Random person(Drug*Exercise);
Neg2LogLike
896.94
Parameters
2
AIC
900.94
AICC
901.03
HQIC
901.91
CovParm
person(drug*exercis)
Residual
Estimate
28.4266
4.2917
StdErr
8.4131
0.5541
ZValue
3.38
7.75
ProbZ
0.0004
0.0000
Effect
drug*exercise*TIME
IBP*drug*exercise*TIME
NumDF
23
24
DenDF
123.2
121.7
FValue
1.09
3.23
ProbF
0.3626
0.0000
BIC
903.87
CAIC
905.87
TABLE 16.4
Analysis of Variance Table with Covariance Parameter (CovParm) Estimates Using
Compound Symmetry Covariance Structure
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
repeated Time/type=cs subject=person(Exercise*Drug);
Neg2LogLike
896.94
CovParm
CS
Residual
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
Parameters
2
AIC
900.94
AICC
901.03
HQIC
901.91
BIC
903.87
Subject
person(exercise*drug)
Estimate
28.4266
4.2917
StdErr
8.4131
0.5541
ZValue
3.38
7.75
ProbZ
0.0007
0.0000
NumDF
23
24
DenDF
123.2
121.7
FValue
1.09
3.23
ProbF
0.3626
0.0000
CAIC
905.87
slopes are all equal to zero, and a list of information criteria. For the information
criteria, let Q= –2 Log(likelihood), then AIC = Q + 2d (Akaike, 1974), AICC =
Q + 2dn/(n – d – 1) (Burnham and Anderson, 1998), HQIC = Q + 2d log(log(n))
(Hannan and Quinn, 1979), BIC = Q + d log(n) (Schwarz, 1978), and CAIC = q +
d (log(n) + 1) (Bozdogan, 1987) where d is the effective number of parameters in
the covariance structure, n is the number of observations for maximum likelihood
estimation, and n–p where p is the rank of the fixed effects part of the model for
REML estimation (SAS Institute Inc., 1999). Using this form of the information
© 2002 by CRC Press LLC
Analysis of Covariance for Repeated Measures Designs
13
TABLE 16.5
Analysis of Variance Table with Covariance Parameter (CovParm) Estimates
Using Heterogeneous Variance Compound Symmetry Covariance Structure
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
repeated Time/type=csh subject=person(Exercise*Drug);
Neg2LogLike
893.07
CovParm
Var(1)
Var(2)
Var(3)
Var(4)
Var(5)
Var(6)
CSH
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
Parameters
7
AIC
907.07
AICC
907.89
HQIC
910.47
BIC
917.33
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
33.8673
27.8799
30.6665
29.6744
37.7676
36.9608
0.8725
StdErr
9.8103
8.0000
8.7844
8.5013
10.9035
10.6445
0.0360
ZValue
3.45
3.49
3.49
3.49
3.46
3.47
24.21
ProbZ
0.0003
0.0002
0.0002
0.0002
0.0003
0.0003
0.0000
NumDF
23
24
DenDF
85.7
81.2
FValue
0.95
3.68
ProbF
0.5378
0.0000
CAIC
924.33
TABLE 16.6
Analysis of Variance Table with Covariance Parameter (CovParm) Estimates
Using First-Order Auto-Regressive Covariance Structure
PROC MIXED cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
Random person(Exercise*Drug);
repeated Time/type=ar(1) subject=person(Exercise*Drug);
Neg2LogLike
864.01
CovParm
person(exercise*drug)
AR(1)
Residual
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
© 2002 by CRC Press LLC
Parameters
3
AIC
870.01
AICC
870.19
HQIC
871.47
BIC
874.41
Subject
Estimate
25.0939
0.7104
8.3096
StdErr
8.9805
0.1295
3.6323
ZValue
2.79
5.49
2.29
ProbZ
0.0026
0.0000
0.0111
DenDF
111.7
111.2
FValue
0.87
2.91
ProbF
0.6341
0.0001
person(exercise*drug)
NumDF
23
24
CAIC
877.41
14
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 16.7
Analysis of Variance Table with Covariance Parameter (CovParm) Estimates
Using Heterogeneous First-Order Auto-Regressive Covariance Structure
PROC MIXED cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
Random person(Exercise*Drug);
repeated Time/type=arh(1) subject=person(Exercise*Drug);
Neg2LogLike
847.41
CovParm
person(exercise*drug)
Var(1)
Var(2)
Var(3)
Var(4)
Var(5)
Var(6)
ARH(1)
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
Parameters
8
AIC
863.41
AICC
864.48
HQIC
867.30
BIC
875.13
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
28.6767
10.4832
1.7199
3.1345
8.7116
14.4857
11.9145
0.7486
StdErr
8.7041
3.7663
1.5233
1.4346
3.2948
5.3342
4.2686
0.0972
ZValue
3.29
2.78
1.13
2.18
2.64
2.72
2.79
7.70
ProbZ
0.0005
0.0027
0.1294
0.0144
0.0041
0.0033
0.0026
0.0000
NumDF
23
24
DenDF
74.7
75.9
FValue
1.07
3.08
ProbF
0.3983
0.0001
CAIC
883.13
criteria, the better model is the one that has the smaller value within a criteria. Each
of the criteria penalize the value of the likelihood function in different ways by a
function of the number of parameters in the covariance matrix and the sample size.
The AIC has been used by many modelers, so the AIC value is used here as the
criteria to select the most adequate covariance matrix, i.e, covariance matrix 1 is
more adequate than covariance matrix 2 if the AIC from the fit of 1 is less than the
AIC from the fit of 2. The split-plot analysis in Table 16.3 and the analysis in
Table 16.4 with the compound symmetry covariance structure provide identical
analyses, but the compound symmetry model is a little more general since the CS
parameter can be estimated with a negative value where as the person(drug*exercise)
variance component cannot take on negative values since it is a variance. The
heterogeneous compound symmetry covariance structure (see Table 16.5) has seven
parameters and the corresponding AIC is larger than the one for the compound
symmetry structure, i.e., 900.94 for CS and 907.07 for CSH. The AR(1) covariance
structure is fit to the data by using the PROC MIXED code in Table 16.6. The
repeated measures covariance structure has two parameters along with the person(drug*exercise) variance component. The estimate of the autocorrelation coefficient is 0.7104. The AIC for the AR(1) structure is 870.01, which is considerably
smaller than the AIC for the CS and CSH structures. The estimate of the autocorrelation coefficient from fitting the ARH(1) model is 0.7486, as shown in
Table 16.7. This model has eight parameters in the covariance structure, including
© 2002 by CRC Press LLC
Analysis of Covariance for Repeated Measures Designs
15
TABLE 16.8
Analysis of Variance Table with Covariance Parameter (CovParm)
Estimates Using First-Order Ante-Dependence Covariance Structure
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
repeated Time/type=ANTE(1) subject=person(Exercise*Drug);
Neg2LogLike
848.70
CovParm
Var(1)
Var(2)
Var(3)
Var(4)
Var(5)
Var(6)
Rho(1)
Rho(2)
Rho(3)
Rho(4)
Rho(5)
Parameters
11
AIC
870.70
AICC
872.70
HQIC
876.04
BIC
886.82
CAIC
897.82
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
StdErr
32.0944
28.4515
31.7284
30.6694
36.7300
36.6363
0.9062
0.9782
0.9253
0.9245
0.9073
ZValue
9.2648
8.2133
9.1592
8.8535
10.6030
10.5760
0.0365
0.0088
0.0293
0.0297
0.0361
ProbZ
3.46
3.46
3.46
3.46
3.46
3.46
24.83
111.02
31.54
31.15
25.13
0.0003
0.0003
0.0003
0.0003
0.0003
0.0003
0.0000
0.0000
0.0000
0.0000
0.0000
FValue
1.04
2.83
ProbF
0.4281
0.0005
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
NumDF
23
24
DenDF
65.1
62.6
the person(drug*exercise) variance component, six residual variances, one for each
time period, and the auto-correlation coefficient. The AIC value for ARH(1) is
863.41, which is the smallest value for the covariance structures discussed so far.
The analyses in Tables 16.8, 16.9, and 16.10 fit the ANTE(1), TOEP, and UN
covariance structures, respectively. The AIC values are all larger than that for the
ARH(1) structure. Table 16.11 contains the values of the AIC from each of the
covariance structures evaluated where the column of number of parameters contains
of the number of covariance parameters estimated that have nonzero values (values
not on the boundary). The covariance structure ARH(1) has the smallest AIC value;
thus, based on AIC, the ARH(1) was selected to be the most adequate covariance
structure among the set of covariance structures evaluated. The next step in the
process is to express the slopes as factorial effects and attempt to simplify the
covariate part of the model while using ARH(1) as the repeated measures covariance
structure. The model with factorial effects for both the intercepts and slopes is
(
)
+ (λ
BPijkm = µ + E i + D j + ( ED)ij + θ + ζ i + δ j + φij IBPijk + p ijk
+ Tm + (TE )im + (TD) jm + (TED)ijm
© 2002 by CRC Press LLC
m
(16.5)
)
+ κ im + ηjm + ω ijm IBPijk + ε ijkm
16
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 16.9
Analysis of Variance Table with Covariance Parameter (CovParm)
Estimates Using Toplitz Covariance Structure
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
repeated Time/type=TOEP subject=person(Exercise*Drug);
Neg2LogLike
863.21
CovParm
TOEP(2)
TOEP(3)
TOEP(4)
TOEP(5)
TOEP(6)
Residual
Parameters
6
AIC
875.21
AICC
875.82
HQIC
878.12
BIC
884.00
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
30.9604
29.1006
27.8108
27.3703
26.8067
33.3667
StdErr
8.5780
8.5684
8.5641
8.5581
8.5861
8.5835
ZValue
3.61
3.40
3.25
3.20
3.12
3.89
ProbZ
0.0003
0.0007
0.0012
0.0014
0.0018
0.0001
DenDF
87.5
88.4
FValue
0.91
2.82
ProbF
0.5908
0.0002
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
NumDF
23
24
CAIC
890.00
where Ei, Dj, (ED)ij, Tm, (TE)im, (TD)jm and TED)ijk denote the fixed effects for
exercise, drug, time, and their interactions, and θ + ζi + δj + φij and λm + κim + ηjm +
ωijm are the factorial effects for the slopes for the person experimental unit and the
within person experimental unit, respectively. Table 16.12 contains the PROC MIXED
code to fit Model 16.5 using the ARH(1) structure for the covariance part of the
model. Several of the significance levels corresponding to the terms involving IBP
have significance levels that are quite large, which indicates the model can be
simplified. Since the IBP*exercise*drug*time term has a significance level of 0.9162,
it was the first term to be deleted. The terms deleted and the order they were deleted
are IBP*Exercise*Drug*Time, IBP*Time*Exercise, IBP*Time*Drug,
IBP*Drug*Exercise, IBP*Time, and IBP*Exercise, leaving the covariate part of the
model with just IBP and IBP*drug. The final model after reducing the covariate part
is
BPijkm = µ + E i + D j + ( ED)ij + θ IBPijk + δ j IBPijk + p ijk
+ Tm + (TE )im + (TD) jm + (TED)ijm + ε ijkm
(16.6)
Table 16.13 contains the PROC MIXED code to fit Model 16.6 using the ARH(1)
covariance structure. The significance level corresponding to IBP*drug in Table 16.13
is 0.1045, so there is some indication that the slopes may not be equal. If a common
slope model is fit to the data set, the additional variability causes PROC MIXED to
fail to converge. Thus the model with unequal slopes for the two levels of drug was
selected for further analyses. A full rank expression of Model 16.6 with intercepts
© 2002 by CRC Press LLC
Analysis of Covariance for Repeated Measures Designs
17
TABLE 16.10
Analysis of Variance Table with Covariance Parameter (CovParm) Estimates
Using Unstructured Covariance Matrix
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise*Drug*Time IBP*Exercise*Drug*Time/ddfm=kr;
repeated Time/type=UN subject=person(Exercise*Drug);
Neg2LogLike
836.05
CovParm
UN(1,1)
UN(2,1)
UN(2,2)
UN(3,1)
UN(3,2)
UN(3,3)
UN(4,1)
UN(4,2)
UN(4,3)
UN(4,4)
UN(5,1)
UN(5,2)
UN(5,3)
UN(5,4)
UN(5,5)
UN(6,1)
UN(6,2)
UN(6,3)
UN(6,4)
UN(6,5)
UN(6,6)
Effect
exercise*drug*TIME
IBP*exercise*drug*TIME
Parameters
21
AIC
878.05
AICC
885.62
HQIC
888.25
BIC
908.83
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
32.0944
27.3839
28.4515
27.9428
29.3897
31.7284
25.6162
26.2251
28.8656
30.6694
26.9072
26.2718
28.6413
31.0279
36.7300
27.4034
27.2981
29.2933
30.8522
33.2808
36.6363
StdErr
9.2648
8.3242
8.2133
8.6581
8.5792
9.1592
8.2677
8.0631
8.6754
8.8535
8.9042
8.5030
9.0960
9.3301
10.6030
8.9600
8.6302
9.1754
9.2994
10.1104
10.5760
ZValue
3.46
3.29
3.46
3.23
3.43
3.46
3.10
3.25
3.33
3.46
3.02
3.09
3.15
3.33
3.46
3.06
3.16
3.19
3.32
3.29
3.46
ProbZ
0.0003
0.0010
0.0003
0.0012
0.0006
0.0003
0.0019
0.0011
0.0009
0.0003
0.0025
0.0020
0.0016
0.0009
0.0003
0.0022
0.0016
0.0014
0.0009
0.0010
0.0003
NumDF
23
24
DenDF
36.8
36.3
FValue
0.89
2.47
ProbF
0.6045
0.0067
CAIC
929.83
for each combination of exercise by drug by time and with slopes for each level of
drug is
BPijkm = µ ijm + δ j IBPijk + p ijk + ε ijkm
(16.7)
Table 16.14 contains the PROC MIXED code needed to fit model 16.7 with the
ARH(1) covariance structure to the data. Table 16.15 contains the estimates of the
models intercepts and slopes, where the slope for those using no drug is estimated
to be 0.9103 and for those using drug is 1.4138. Table 16.16 contains the adjusted
means for each combination of exercise by drug by time evaluated at three values
© 2002 by CRC Press LLC
18
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 16.11
Summary of the Akiake
Information Criterion (AIC)
for Eight Covariance Structures
Covariance
Structure
Split-plot
CS
CSH
AR(1)
ARH(1)
ANTE(1)
TOEP
UN
Number of
Parameters
2
2
7
3
8
11
6
21
AIC
900.94
900.94
907.07
870.01
863.41
870.07
875.21
878.05
of IBP: 140, 150, and 160 mmHg. Since the slopes of the regression lines are not
dependent on the levels of exercise, the levels of exercise can be compared within
each level of drug and exposure time at a single value of IBP. The pairwise comparisons of the levels of exercise, where the simulate adjustment for multiple comparisons was used, are in Table 16.17. There are significantly lower mean systolic
blood pressure values for those persons using an exercise program than those not
using an exercise program for times of 3 weeks and beyond when drug was not in
their regime. There are no significant differences between exercise and no exercise
at any time when the persons included the drug in their regime. Pairwise comparisons
between the drug and no-drug regimes at each time for exercise and no exercise at
the three values of IBP are displayed in Table 16.18. All denominator degrees of
freedom for each test statistic were between 27.9 and 31.2, and thus were not
included in the table. Those including the drug in their regime had significantly
lower mean systolic blood pressure values for (1) both no exercise and exercise
when IBP = 140 mmHg at times 1 through 5, (2) for no exercise when IBP =
150 mmHg for times 2 through 6, (3) for exercise when IBP = 150 mmHg at time 3,
and (4) for no exercise when IBP = 150 at times 6 for IBP = 140 mmHg. Figures 16.1
through 16.3 are graphs of the adjusted systolic blood pressure means for the exercise
by drug combinations against the values of IBP at times 1, 3, and 6. On each graph,
two lines with the same level of drug are parallel and lines with different levels of
drug are not parallel.
The analysis of covariance strategy for this repeated measures design is identical
to that for the split-plot or any other design that involves a factorial treatment
structure, except the starting point is to model the covariance structure of the repeated
measures part of the model. Eight different structures were considered here, but
there are others that can be very meaningful covariance structures. In particular, the
parameters for a given covariance structure could be unequal for different levels of
one or more of the factors in the treatment structure. For example, the AR(1) structure
might be different for the two levels of drug. To specify such a structure in PROC
© 2002 by CRC Press LLC
Analysis of Covariance for Repeated Measures Designs
19
TABLE 16.12
Analysis of Variance Table with the Factorial Effects for the Intercepts
and the Slopes with Covariance Parameter (CovParm) Estimates Using
ARH(1) Covariance Structure for the Repeated Measurement Errors
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise|Drug|Time IBP IBP*Drug IBP*Exercise
IBP*Drug*Exercise IBP*Time IBP*Time*Drug IBP*Time*Exercise
IBP*Exercise*Drug*Time/ddfm=kr;
Random person(Exercise*Drug);
repeated Time/type=arh(1) subject=person(Exercise*Drug);
Neg2LogLike
847.41
CovParm
person(exercise*drug)
Var(1)
Var(2)
Var(3)
Var(4)
Var(5)
Var(6)
ARH(1)
Effect
exercise
drug
exercise*drug
Time
exercise*Time
drug*Time
exercise*drug*Time
IBP
IBP*drug
IBP*exercise
IBP*exercise*drug
IBP*Time
IBP*drug*Time
IBP*exercise*Time
IBP*exercise*drug*Time
Parameters
8
AIC
863.41
AICC
864.48
HQIC
867.30
BIC
875.13
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
28.6767
10.4832
1.7199
3.1345
8.7116
14.4857
11.9145
0.7486
StdErr
8.7041
3.7663
1.5233
1.4346
3.2948
5.3342
4.2686
0.0972
ZValue
3.29
2.78
1.13
2.18
2.64
2.72
2.79
7.70
ProbZ
0.0005
0.0027
0.1294
0.0144
0.0041
0.0033
0.0026
0.0000
NumDF
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
DenDF
26.2652
26.2652
26.2652
41.9903
41.9903
41.9903
41.9903
26.2652
26.2652
26.2652
26.2652
41.9903
41.9903
41.9903
41.9903
FValue
0.75
6.14
0.17
1.97
0.20
0.73
0.45
46.45
4.38
1.02
0.11
1.45
0.69
0.14
0.29
ProbF
0.3953
0.0200
0.6824
0.1036
0.9600
0.6037
0.8092
0.0000
0.0461
0.3210
0.7459
0.2266
0.6364
0.9812
0.9162
CAIC
883.13
MIXED, use the repeated statement “Repeated time/type=AR(1) subject=person(exercise*drug) group=drug;”. The group = drug option is the specification that
the covariance structures for the two levels of drug could have different values.
However, one thing to keep in mind is that with more parameters in the covariance
structure, it is more difficult computationally to obtain the maximum likelihood or
REML estimators of the parameters.
© 2002 by CRC Press LLC
20
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 16.13
Analysis of Variance Table for the Reduced Model with Factorial Effects
for the Slopes and Intercepts Using the ARH(1) Covariance Structure
for the Repeated Measurement Errors
Proc Mixed cl ic covtest DATA=E165;
Class Exercise Drug person Time;
Model BP= Exercise|Drug|Time IBP IBP*Drug/ddfm=kr;
Random person(Exercise*Drug);
repeated Time/type=arh(1) subject=person(Exercise*Drug);
Neg2LogLike
814.34
CovParm
person(exercise*drug)
Var(1)
Var(2)
Var(3)
Var(4)
Var(5)
Var(6)
ARH(1)
Effect
exercise
drug
exercise*drug
Time
exercise*Time
drug*Time
exercise*drug*Time
IBP
IBP*drug
Parameters
8
AIC
830.34
AICC
831.26
HQIC
834.23
BIC
842.07
Subject
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
person(exercise*drug)
Estimate
26.9435
10.5648
2.3860
3.7448
8.8289
14.3004
12.4418
0.7642
StdErr
8.0844
4.1654
2.0324
1.7661
3.7106
5.8853
4.9743
0.1052
ZValue
3.33
2.54
1.17
2.12
2.38
2.43
2.50
7.27
ProbZ
0.0004
0.0056
0.1202
0.0170
0.0087
0.0076
0.0062
0.0000
NumDF
1
1
1
5
5
5
5
1
1
DenDF
26.2
24.3
26.2
47.4
47.4
47.4
47.4
24.3
24.3
FValue
19.23
4.39
3.57
68.82
22.21
14.90
12.59
60.61
2.84
ProbF
0.0002
0.0467
0.0701
0.0000
0.0000
0.0000
0.0000
0.0000
0.1045
CAIC
850.07
16.7 EXAMPLE: OXIDE LAYER DEVELOPMENT EXPERIMENT
WITH THREE SIZES OF EXPERIMENTAL UNITS WHERE
THE REPEATED MEASURE IS AT THE MIDDLE SIZE OF
EXPERIMENTAL UNIT AND THE COVARIATE IS
MEASURED ON THE SMALL SIZE EXPERIMENTAL UNIT
One of the many steps in the fabrication of semiconductors involves putting a wafer
of silicon into a furnace set to a specific temperature to enable a layer of oxide to
accumulate. A chemical engineer ran an experiment to study the effect of temperature
of the furnace, position within the furnace, and wafer type on the thickness of the
resulting layer of oxide for each wafer. The process was to measure the thickness
of each wafer before putting the wafer into the furnace and then measuring the
thickness of the wafer after the furnace run. The difference, or delta, between the
© 2002 by CRC Press LLC