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6 EXAMPLE: SYSTOLIC BLOOD PRESSURE STUDY WITH COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT

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



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