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6 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN STRUCTURE WITH FOUR TREATMENTS

6 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN STRUCTURE WITH FOUR TREATMENTS

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16



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 10.14

PROC MIXED Code to Use the Method of Moments Estimates

of the Variance Components by Employing the Hold Option

PROC MIXED CL COVTEST DATA=LONG10;;

CLASS BLOCK TRT;

MODEL Y=TRT X*TRT/NOINT SOLUTION DDFM=KR;

RANDOM BLOCK;

LSMEANS TRT/DIFF AT X=20;

PARMS (5.09) (.99651)/HOLD=1,2;

CONTRAST ‘EQUAL SLOPES’ X*TRT 1 –1 0 0 0, X*TRT 1 0 –1 0 0,

X*TRT 1 0 0 –1 0, x*TRT 1 0 0 0 –1;

CovParm

BLOCK

Residual



Estimate

5.090000

0.996510



Effect

trt

x*trt



NumDF

5

5



DenDF

39

39



FValue

730.39

213.81



ProbF



Label

EQUAL SLOPES



NumDF

4



DenDF

39



FValue

32.52



ProbF

0.0000



trt

1

2

3

4

5

1

2

3

4

5



Estimate

50.8797

52.9537

53.0436

55.0445

52.4730

0.5176

0.6679

0.9284

1.0994

1.4250



StdErr

1.4204

1.2248

1.5662

1.4096

1.5500

0.0610

0.0576

0.0704

0.0608

0.0689



df

1

1

1

1

1

1

1

1

1

1



Effect

trt

trt

trt

trt

trt

x*trt

x*trt

x*trt

x*trt

x*trt



tValue

35.82

43.23

33.87

39.05

33.85

8.49

11.60

13.18

18.08

20.69



TABLE 10.15

Least Squares Means Computed Using the Method

of Moments Estimates of the Variance Components

from Table 10.14

Effect

trt

trt

trt

trt

trt



© 2002 by CRC Press LLC



trt

1

2

3

4

5



x

20.00

20.00

20.00

20.00

20.00



Estimate

61.2307

66.3117

71.6113

77.0333

80.9738



StdErr

0.7122

0.7290

0.7123

0.7122

0.7122



df

1

1

1

1

1



tValue

85.97

90.97

100.53

108.16

113.70



Probt

0.0074

0.0070

0.0063

0.0059

0.0056



Probt

0.0178

0.0147

0.0188

0.0163

0.0188

0.0747

0.0548

0.0482

0.0352

0.0308



More Than Two Treatments in a Blocked Design Structure



17



TABLE 10.16

PROC MIXED Code to Use REML to Estimate the Variance

Components and Complete the Analysis

Proc Mixed CL COVTEST DATA=LONG10;

CLASS BLOCK TRT;

MODEL Y=TRT X*TRT/NOINT SOLUTION DDFM=KR;

CONTRAST ‘=SLOPES’ X*TRT 1 –1 0 0 0, X*TRT 1 0 –1 0 0,

X*TRT 1 0 0 –1 0, X*TRT 1 0 0 0 –1;

CovParm

BLOCK

Residual



Estimate

13.6077

0.9967



StdErr

5.9019

0.2258



ZValue

2.31

4.41



ProbZ

0.0106

0.0000



Effect

trt

x*trt



NumDF

5

5



DenDF

40.89

39.28



FValue

374.54

210.16



ProbF

0.0000

0.0000



Label

=slopes



NumDF

4



DenDF

39.2



FValue

32.28



ProbF

0.0000



trt

1

2

3

4

5

1

2

3

4

5



Estimate

50.9726

52.8946

53.2248

55.2857

52.4010

0.5129

0.6713

0.9192

1.0874

1.4286



StdErr

1.6554

1.4893

1.7832

1.6458

1.7687

0.0613

0.0578

0.0707

0.0611

0.0692



df

40.28

32.64

44.5

39.92

44.11

39.3

39.28

39.31

39.28

39.3



Effect

trt

trt

trt

trt

trt

x*trt

x*trt

x*trt

x*trt

x*trt



Alpha

0.05

0.05



Lower

6.7626

0.6688



tValue

30.79

35.52

29.85

33.59

29.63

8.37

11.61

13.00

17.81

20.65



Probt

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000



Upper

40.1423

1.6435



The estimate statements in Table 10.21 are used to provide predicted values for

treatment 1 evaluated at X = 100 for each of the 16 blocks, even though treatment 1

did not occur in all of the blocks. Only the estimate statements for treatment one

are included in the table, but predicted values for all four treatments were obtained.

The lower part of Table 10.21 lists the predicted values of each of the treatments at

each of the blocks. The last line in Table 10.21 contains the means of the predicted

values in the column. The column means correspond exactly to the least squares

means in Table 10.20. Thus, when an incomplete block design is used, predicted

values for every treatment are obtained for every block, whether or not that treatment

actually occurred in the block, and the least squares means are the means of those

predicted values. So, the least squares means are the means of the predicted responses

as if all treatments had actually occurred in all of the blocks. Because this is in fact

what is being estimated by the computational process, it is very important that the

factors used to form blocks do not interact with the treatments in the treatment

structure (Milliken and Johnson, 1992).



© 2002 by CRC Press LLC



18



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 10.17

Least Squares Means at X=15, 20 and 25 from the REML

Estimates of the Variance Components

LSMEANS TRT/DIFF AT X=15; LSMEANS TRT/DIFF AT X=20;

LSMEANS TRT/DIFF AT X=25;

Effect

trt

trt

trt

trt

trt

trt

trt

trt

trt

trt

trt

trt

trt

trt

trt



trt

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5



x

15.00

15.00

15.00

15.00

15.00

20.00

20.00

20.00

20.00

20.00

25.00

25.00

25.00

25.00

25.00



Estimate

58.6667

62.9643

67.0133

71.5964

73.8306

61.2314

66.3209

71.6095

77.0333

80.9738

63.7961

69.6775

76.2057

82.4702

88.1170



StdErr

1.1474

1.1112

1.1544

1.1447

1.1558

1.1032

1.1142

1.1033

1.1032

1.1032

1.1425

1.1897

1.1627

1.1447

1.1565



df

14

13

15

14

15

12

13

12

12

12

14

16

15

14

15



tValue

51.13

56.66

58.05

62.55

63.88

55.50

59.52

64.91

69.83

73.40

55.84

58.57

65.54

72.05

76.19



Probt

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000



TABLE 10.18

Data Set for Four Treatments in a Balanced Incomplete

Block Design Structure Where Y is the Response and X

is the Possible Covariate

BLOCK

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16



© 2002 by CRC Press LLC



TRT

1

1

1

2

1

1

1

2

1

1

1

2

1

1

1

2



Y

118

101

111

103

123

121

101

122

108

102

108

119

128

113

87

134



X

127

98

90

81

116

96

92

123

110

104

94

109

105

92

83

121



TRT

2

2

3

3

2

2

3

3

2

2

3

3

2

2

3

3



Y

134

109

124

116

99

119

110

114

101

113

128

117

108

117

108

119



X

129

110

128

102

90

121

112

119

91

123

109

116

83

93

95

125



TRT

3

4

4

4

3

4

4

4

3

4

4

4

3

4

4

4



Y

105

121

143

121

108

128

117

124

124

111

142

135

101

118

120

146



X

105

107

123

106

91

108

102

110

127

105

122

120

79

102

104

123



More Than Two Treatments in a Blocked Design Structure



19



TABLE 10.19

PROC MIXED Code to Provide Analysis of BIB Design Structure with

Four Treatments for the Unequal Slopes Model

proc mixed data=bib10 cl covtest; class block trt;

model y=trt x*trt/solution; random block/solution;

CovParm

BLOCK

Residual



Estimate

15.5409

35.9386



StdErr

11.6085

10.2345



ZValue

1.34

3.51



ProbZ

0.0903

0.0002



Effect

TRT

X*TRT



NumDF

3

4



DenDF

25

25



FValue

1.71

16.27



ProbF

0.1904

0.0000



Effect

Intercept

TRT

TRT

TRT

TRT

X*TRT

X*TRT

X*TRT

X*TRT



TRT



Estimate

0.2895

52.8686

51.5408

65.8460

0.0000

0.5676

0.5942

0.4412

1.1433



StdErr

26.5828

30.9324

28.7706

29.1279

0.1627

0.1151

0.1325

0.2389



1

2

3

4

1

2

3

4



Alpha

0.05

0.05



Lower

5.3497

22.0401



df

15

25

25

25



tValue

0.01

1.71

1.79

2.26



Probt

0.9915

0.0998

0.0853

0.0327



25

25

25

25



3.49

5.16

3.33

4.79



0.0018

0.0000

0.0027

0.0001



Upper

153.8921

68.8343



TABLE 10.20

Least Squares Means Computed at X = 100

for the BIB Design Structure

lsmeans trt/at x=100;

TRT

1

2

3

4



X

100.00

100.00

100.00

100.00



Estimate

109.9168

111.2536

110.2540

114.6167



StdErr

2.0331

2.1537

2.3539

3.3181



df

25

25

25

25



tValue

54.06

51.66

46.84

34.54



Probt

0.0000

0.0000

0.0000

0.0000



10.7 EXAMPLE: BALANCED INCOMPLETE BLOCK DESIGN

STRUCTURE WITH FOUR TREATMENTS USING JMP®

The incomplete block data in Table 10.18 is analyzed using JMP® and the data are

displayed in the data screen of Figure 10.1, where Y and X are continuous variables

and TRT and BLOCK are nominal variables. Figure 10.2 is the fit model screen

where the response variable is Y and the model effects are BLOCK, TRT X, and

X*TRT. Specify the BLOCK effects to be RANDOM by selecting BLOCK and use

the Attributes button to select RANDOM. The “no center” option was selected from

© 2002 by CRC Press LLC



20



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 10.21

Estimate Statements Used to Provide Predicted Values for Treatment 1

for Each of the 16 Blocks

estimate ‘11’ intercept 1 trt 1 0 0 0 x*trt 100|block 1;

estimate ‘12’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 1;

estimate ‘13’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 1;

estimate ‘14’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 1;

estimate ‘15’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 1;

estimate ‘16’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 1;

estimate ‘17’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 1;

estimate ‘18’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 1;

estimate ‘19’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 0 1;

estimate ‘110’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 0 0 1;

estimate ‘111’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0 0 0 0 1;

estimate ‘112’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 1;

estimate ‘113’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 1;

estimate ‘114’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 0 1;

estimate ‘115’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 0 0 1;

estimate ‘116’ intercept 1 trt 1 0 0 0 x*trt 100|block 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 1;



Treatment 1

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16



Estimate

108.1873

106.6044

111.8443

111.3143

109.8052

112.3372

108.0669

108.1085

107.9095

103.9985

113.2101

109.8421

114.0777

113.4230

107.5628

112.3774



Means



109.9168



Treatment 2

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16



Estimate

109.5241

107.9412

113.1811

112.6511

111.1420

113.6740

109.4037

109.4453

109.2463

105.3353

114.5469

111.1789

115.4145

114.7598

108.8996

113.7142

111.2536



Treatment 3

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16



Estimate

108.5245

106.9416

112.1815

111.6515

110.1424

112.6743

108.4041

108.4456

108.2466

104.3357

113.5472

110.1793

114.4148

113.7602

107.9000

112.7146

110.2540



Treatment 4

Block

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16



Estimate

112.8872

111.3043

116.5442

116.0142

114.5051

117.0370

112.7668

112.8083

112.6093

108.6984

117.9099

114.5420

118.7775

118.1229

112.2627

117.0773

114.6167



the model specification menu to enable the fitted model to match the models fit by

PROC MIXED. Click on the Run Model button to carry out the analysis. The

parameter estimates are in Figure 10.3 where the ones of interest correspond to

Intercept, TRT[1], TRT[2], TRT[3], X, TRT[1]*X, TRT[2]*X, and TRT[3]*X. The

© 2002 by CRC Press LLC



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