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9 EXAMPLE: FOUR TREATMENTS IN BIB

9 EXAMPLE: FOUR TREATMENTS IN BIB

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20



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 11.17

Least Squares Means and Comparisons at wt = 0.596, 0.510, and 0.0680

lsmeans ration/diff at means;

lsmeans ration/diff at wt=.510;

lsmeans ration/diff at wt=.680;

Effect

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration



Ration

1

2

3

4

1

2

3

4

1

2

3

4



wt

0.596

0.596

0.596

0.596

0.510

0.510

0.510

0.510

0.680

0.680

0.680

0.680



Estimate

2.0417

2.1475

2.8692

2.4575

2.0838

2.3639

2.8847

2.6533

2.0007

1.9369

2.8541

2.2670



StdErr

0.0742

0.0742

0.0742

0.0742

0.1317

0.1317

0.1317

0.1317

0.1293

0.1293

0.1293

0.1293



df

13.1

13.1

13.1

13.1

13.1

13.1

13.1

13.1

13.1

13.1

13.1

13.1



tValue

27.52

28.95

38.68

33.13

15.82

17.95

21.90

20.15

15.48

14.98

22.08

17.54



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



Effect

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration

Ration



Ration

1

1

1

2

2

3

1

1

1

2

2

3

1

1

1

2

2

3



_Ration

2

3

4

3

4

4

2

3

4

3

4

4

2

3

4

3

4

4



wt

0.596

0.596

0.596

0.596

0.596

0.596

0.510

0.510

0.510

0.510

0.510

0.510

0.680

0.680

0.680

0.680

0.680

0.680



Estimate

–0.1058

–0.8275

–0.4158

–0.7217

–0.3100

0.4117

–0.2802

–0.8009

–0.5695

–0.5207

–0.2893

0.2314

0.0638

–0.8534

–0.2663

–0.9172

–0.3301

0.5870



StdErr

0.0436

0.0436

0.0436

0.0436

0.0436

0.0436

0.0775

0.0775

0.0775

0.0775

0.0775

0.0775

0.0761

0.0761

0.0761

0.0761

0.0761

0.0761



df

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30

30



tValue

–2.42

–18.96

–9.53

–16.54

–7.10

9.43

–3.62

–10.34

–7.35

–6.72

–3.73

2.99

0.84

–11.22

–3.50

–12.06

–4.34

7.72



Probt

0.0215

0.0000

0.0000

0.0000

0.0000

0.0000

0.0011

0.0000

0.0000

0.0000

0.0008

0.0056

0.4083

0.0000

0.0015

0.0000

0.0001

0.0000



cutting tools. Three pieces could be turned or cut from each rod. The rods varied in

hardness; thus the rods were considered as blocks and the measure of hardness

(Hard) of a given rod was used as a covariate. Since there were four tool types and

blocks of size three, a balanced incomplete block (BIB) design structure was used

as shown in Table 11.19.

The model using a linear relationship between surface finish and hardness is

Finish ij = α i + βi Hard j + b j + ε ij .

© 2002 by CRC Press LLC



(11.17)



Covariate Measured on the Block in RCB



21



TABLE 11.18

PROC MIXED Code and Analysis of Variance with Comparisons

of the Four Rations without the Covariate Information

proc mixed cl covtest data=III118;

class blk ration;

model adg=ration/solution ddfm=kr;

lsmeans ration/diff;

random blk;

CovParm

blk

Residual



Estimate

0.0553

0.0158



StdErr

0.0253

0.0039



ZValue

2.19

4.06



ProbZ

0.0144

0.0000



Alpha

0.05

0.05



Lower

0.0266

0.0103



Effect

Ration



NumDF

3



DenDF

33



FValue

104.78



ProbF

0.0000



Effect

Ration

Ration

Ration

Ration



Ration

1

2

3

4



Estimate

2.0417

2.1475

2.8692

2.4575



StdErr

0.0770

0.0770

0.0770

0.0770



df

15.6

15.6

15.6

15.6



tValue

26.53

27.90

37.28

31.93



Probt

0.0000

0.0000

0.0000

0.0000



Effect

Ration

Ration

Ration

Ration

Ration

Ration



Ration

1

1

1

2

2

3



_Ration

2

3

4

3

4

4



Estimate

–0.1058

–0.8275

–0.4158

–0.7217

–0.3100

0.4117



StdErr

0.0513

0.0513

0.0513

0.0513

0.0513

0.0513



df

33

33

33

33

33

33



tValue

–2.06

–16.13

–8.10

–14.06

–6.04

8.02



Upper

0.1759

0.0274



Probt

0.0471

0.0000

0.0000

0.0000

0.0000

0.0000



FIGURE 11.1 Fit model screen for the BIB where the center model specification is not used

and blk is declared as a random effect.

© 2002 by CRC Press LLC



22



Analysis of Messy Data, Volume III: Analysis of Covariance



FIGURE 11.2 Estimates of the variance components and tests for the effects in the model

using JMP®.



TABLE 11.19

Data for Balanced Incomplete Block Design

Structure with Four Types of Tools

BLOCK

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16



HARD

23

44

40

27

26

33

39

50

20

44

28

26

27

38

44

48



TOOL1

63

78

66

71

85

87

66

97

82

87

87

84



TOOL2

73

72

73

77

87

88

67

91

72

85

90

104



TOOL3

64

69

75

88

72

87

76

84

76

86

72

103



TOOL4

85

86

97

103

101

94

97

90

91

107

96

128



Table 11.20 contains the PROC GLM code to extract the within block analysis. The

same problems as observed in the previous two examples occur in this analysis. The

sum of squares corresponding to Hard is zero since the average value of the covariate

is completely confounded with the blocks. Only least squares means at the average

hardness value are estimable by PROC GLM and they are displayed in fourth part

of Table 11.20 and the p-values for pairwise differences are in the lower part of

Table 11.20. The best approach is to use mixed models software to carry out the

analysis because it will provide the combined within and between block analysis.

© 2002 by CRC Press LLC



Covariate Measured on the Block in RCB



23



TABLE 11.20

PROC GLM Code and Within Block Analysis for the BIB Example

(The LSMEANS for HARD = 25 and 45 are nonestimable.)

PROC GLM data=bibhard; CLASS

MODEL Finish=BLOCK Tool Hard

LSMEANS Tool/STDERR PDIFF AT

LSMEANS Tool/STDERR PDIFF AT

LSMEANS Tool/STDERR PDIFF AT

RANDOM BLOCK/TEST;



BLOCK Tool;

Hard*Tool/SOLUTION;

MEAN;

HARD=25;

HARD=45;



Source

Model

Error

Corrected Total



df

21

26

47



SS

7517.9156

639.8969

8157.8125



MS

357.9960

24.6114



FValue

14.55



ProbF

0.0000



Source

BLOCK

Tool

HARD

HARD*Tool



df

15

3

0

3



SS (I)

5389.1458

1893.4375

0.0000

235.3322



MS

359.2764

631.1458



FValue

14.60

25.64



ProbF

0.0000

0.0000



78.4441



3.19



0.0403



Source

BLOCK

Tool

HARD

HARD*Tool



df

14

3

0

3



SS (III)

3350.9528

354.8133

0.0000

235.3322



MS

239.3538

118.2711



FValue

9.73

4.81



ProbF

0.0000

0.0086



78.4441



3.19



0.0403



LSMean

81.6313

80.3581

79.9083

97.0543



StdErr

1.5133

1.5100

1.5285

1.7047



Probt

0.0000

0.0000

0.0000

0.0000



LSMean#

1

2

3

4



_1



_2

0.5636



_3

0.4403

0.8388



_4

0.0000

0.0000

0.0000



Tool

1

2

3

4

RowName

1

2

3

4



0.5636

0.4403

0.0000



0.8388

0.0000



0.0000



Table 11.21 contains the PROC MIXED code to provide the combined analysis. The

NOINT option was not used in this model so the estimates of the intercepts satisfy

the set-to-zero restriction. The test of the slopes equal to zero hypothesis is provided

by the F value corresponding to HARD*Tool, which has a significance level of

0.0253. The contrast “b1 = b2 = b3 = b4” provides the F value to test the equal

slopes hypothesis which has a significance level of 0.0565. Two of the slopes are

not significantly different from zero, HARD*Tool 3 and 4. A simpler model possibly

could be obtained, but it was not attempted here. Least squares means were computed

at Hard = 34.8125 (mean), 25, and 45 and those results are in Table 11.22. The least

squares means are graphed in Figure 11.3. The means for Tool 4 are significantly

© 2002 by CRC Press LLC



24



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 11.21

PROC MIXED Code and Combined Within and Between Block Analysis

with Parameter Estimates for BIB

PROC MIXED CL COVTEST data=bibhard; CLASS BLOCK Tool;

MODEL Finish=Tool hard Hard*Tool/SOLUTION DDFM=KR;

CONTRAST ‘b1=b2=b3=b4’ HARD*TOOL 1 –1, HARD*TOOL 1 0 –1, HARD*TOOL

1 0 0 –1;

RANDOM BLOCK/solution;

CovParm

BLOCK

Residual



Estimate

75.5758

24.6198



StdErr

31.9598

6.8304



ZValue

2.36

3.60



ProbZ

0.0090

0.0002



Alpha

0.05

0.05



Lower

38.1043

15.2668



Effect

Tool

HARD*Tool



NumDF

3

4



DenDF

26.8

29.5



FValue

4.52

3.25



ProbF

0.0109

0.0253



Effect

Intercept

Tool

Tool

Tool

Tool

HARD*Tool

HARD*Tool

HARD*Tool

HARD*Tool



Tool

1

2

3

4

1

2

3

4



Estimate

84.3294

–36.1543

–24.4205

–14.0808

0.0000

0.9557

0.5939

0.2779

0.3630



StdErr

11.5223

10.6914

9.5039

9.4157



df

31.1

27.8

27.1

27.0



tValue

7.32

–3.38

–2.57

–1.50



Probt

0.0000

0.0022

0.0160

0.1464



0.2966

0.2744

0.2757

0.3059



26.3

21.1

21.4

28.3



3.22

2.16

1.01

1.19



0.0034

0.0421

0.3248

0.2452



Label

b1=b2=b3=b4



NumDF

3



DenDF

26.8



FValue

2.84



ProbF

0.0565



Upper

215.4836

46.2483



larger than the means of the other three tools for all hardness values. For Hard =

34.8125 and 25, the adjusted means for Tools 1, 2, and 3 are not significantly

different. For Hard = 45, the mean of Tool 1 is significantly larger than the mean

for Tool 3. The mean of Tool 1 is not significantly different from the mean of Tool

2 and the means of Tools 2 and 3 are not significantly different. From the graph in

Figure 11.3, it can be concluded that Tool 4 provides the largest mean finish value

for any of the hardness values in the range of the data.



11.10 SUMMARY

When blocking is used in a design, information about the parameters can be extracted

from more than one source. The usual analysis extracts only the within block

information and ignores the between block information. The process described in

this chapter extracts both the within block and between block information and then

combines it when possible. The discussion here considered equal block size designs

and could be extended to unequal block size design structures. For unequal block

size designs, the process would be to group blocks of the same size together and

© 2002 by CRC Press LLC



Covariate Measured on the Block in RCB



25



TABLE 11.22

Least Squares Means for the Four Levels of Tool Evaluated

at the Mean Hardness (34.81) and at 25 and 45

LSMEANS Tool/DIFF at MEAN;

LSMEANS Tool/DIFF at HARD=25;

LSMEANS Tool/DIFF at HARD=45;

Effect

Tool

Tool

Tool

Tool



Tool

1

2

3

4



HARD

34.81

34.81

34.81

34.81



Estimate

81.4451

80.5840

79.9222

96.9651



StdErr

2.6459

2.6440

2.6543

2.7565



df

23.1

23.1

23.3

25.9



tValue

30.78

30.48

30.11

35.18



Probt

0.0000

0.0000

0.0000

0.0000



Tool

Tool

Tool

Tool



1

2

3

4



25.00

25.00

25.00

25.00



72.0674

74.7564

77.1955

93.4035



3.8324

3.7019

3.6655

4.4639



23.1

20.7

20.1

32.8



18.80

20.19

21.06

20.92



0.0000

0.0000

0.0000

0.0000



Tool

Tool

Tool

Tool



1

2

3

4



45.00

45.00

45.00

45.00



91.1812

86.6344

82.7531

100.6628



4.1165

3.9199

3.9873

3.7233



26.6

23.3

24.4

19.8



22.15

22.10

20.75

27.04



0.0000

0.0000

0.0000

0.0000



Tool

1

1

1

2

2

3



_Tool

2

3

4

3

4

4



HARD

34.81

34.81

34.81

34.81

34.81

34.81



Estimate

0.8610

1.5229

–15.5200

0.6618

–16.3811

–17.0429



StdErr

2.1684

2.1891

2.2904

2.1806

2.3043

2.3036



df

26.7

26.8

26.9

26.7

27.0

26.9



tValue

0.40

0.70

–6.78

0.30

–7.11

–7.40



Probt

0.6945

0.4926

0.0000

0.7639

0.0000

0.0000



1

1

1

2

2

3



2

3

4

3

4

4



25.00

25.00

25.00

25.00

25.00

25.00



–2.6890

–5.1282

–21.3361

–2.4391

–18.6471

–16.2080



2.9267

2.8751

4.0252

2.7050

3.7731

3.7058



26.4

26.4

27.7

26.2

27.3

27.2



–0.92

–1.78

–5.30

–0.90

–4.94

–4.37



0.3665

0.0860

0.0000

0.3754

0.0000

0.0002



1

1

1

2

2

3



2

3

4

3

4

4



45.00

45.00

45.00

45.00

45.00

45.00



4.5468

8.4281

–9.4816

3.8813

–14.0284

–17.9097



3.5010

3.6082

3.2099

3.3224

2.9209

3.0118



27.1

27.2

26.7

26.8

26.3

26.4



1.30

2.34

–2.95

1.17

–4.80

–5.95



0.2050

0.0271

0.0065

0.2530

0.0001

0.0000



obtain between block information from each group. Then the estimators from each

block size analysis and the within block analysis would be combined. This process

is tedious when there are several different block sizes, but the mixed models approach

can provide the appropriate analysis. It is important to provide a combined analysis

© 2002 by CRC Press LLC



26



Analysis of Messy Data, Volume III: Analysis of Covariance



Tools Effect on Finish



Finish Score



110.00



100.00

90.0000



x



20.00



+

*



+

*



80.0000



70.0000



x



x



+*

25.00



30.00



35.00



40.00



45.00



50.00



Hardness Measure



+++ 1



***



2



3



x x x 4



FIGURE 11.3 Predicted finish scores for the four tools over a range of hardness measures.

The vertical line is the mean hardness measure, 34.8125.



since experiments involving split-plot and repeated measures designs are composed

of many small blocks and much of the covariate information is contained in the

between block comparisons.

Contrary to common thought and method of analysis, even when the covariate

is used to form blocks, including the covariate in the analysis can improve the

comparisons of the treatments when there is a relationship between the mean of the

treatments and the values of the covariate.



REFERENCES

Kenward, M. G. and Roger, J. H. (1997). Small sample inference for fixed effects from

restricted maximum likelihood, Biometrics 54:983.

Milliken, G. A. and Johnson, D. E. (1992). Analysis of Messy Data, Volume I: Design

Experiments, Chapman & Hall, London.



EXERCISES

EXERCISE 11.1: Determine the proper form for the analysis of covariance model

to describe the following data. Construct the appropriate analysis of variance table.

Provide estimates of the intercepts and slopes for each treatment. Plot the regression

lines. Compare the regression lines at x = .40, x = .60, and x = .80. Determine the

LSMEANS, estimate their standard errors, and construct 95% confidence intervals

about each.

© 2002 by CRC Press LLC



Covariate Measured on the Block in RCB



27



BLK

1

1

1

1

1



TRT

1

2

3

4

5



Y

84.6

86.8

90.8

90.3

90.2



X

0.834

0.834

0.834

0.834

0.834



BLK

6

6

6

6

6



TRT

1

2

3

4

5



Y

90.4

91.6

91.4

95.5

97.3



X

0.579

0.579

0.579

0.579

0.579



2

2

2

2

2



1

2

3

4

5



86.8

86.7

91.2

85.1

92.1



0.163

0.163

0.163

0.163

0.163



7

7

7

7

7



1

2

3

4

5



82.2

93.3

88.0

94.8

94.1



0.603

0.603

0.603

0.603

0.603



3

3

3

3

3



1

2

3

4

5



84.6

87.8

93.3

89.0

97.6



0.356

0.356

0.356

0.356

0.356



8

8

8

8

8



1

2

3

4

5



94.0

89.0

95.4

92.3

96.7



0.649

0.649

0.649

0.649

0.649



4

4

4

4

4



1

2

3

4

5



89.0

86.9

89.6

85.8

87.7



0.593

0.593

0.593

0.593

0.593



9

9

9

9

9



1

2

3

4

5



71.9

81.3

74.5

84.7

86.8



0.174

0.174

0.174

0.174

0.174



5

5

5

5

5



1

2

3

4

5



80.8

76.6

81.2

81.0

83.2



0.395

0.395

0.395

0.395

0.395



10

10

10

10

10



1

2

3

4

5



84.4

86.9

87.3

89.9

99.2



0.824

0.824

0.824

0.824

0.824



EXERCISE 11.2: Determine the proper form for the analysis of covariance model

to describe the following data. Construct the appropriate analysis of variance table.

Provide estimates of the intercepts and slopes for each treatment. Plot the regression

lines. Compare the regression lines at x = 6, x = 12, and x = 18. Determine the

LSMEANS, estimate their standard errors, and construct 95% confidence intervals

about each.

BLK

1

1

2

2

3

3

4

4



TRT

1

2

1

3

2

3

1

2



Y

15.4

20.7

13.4

13.5

17.3

20.9

15.4

19.9



© 2002 by CRC Press LLC



X

17.1

17.1

6.1

6.1

9.3

9.3

13.1

13.1



BLK

5

5

6

6

7

7

8

8



TRT

1

3

2

3

1

2

1

3



Y

9.2

15.3

19.3

23.3

16.7

17.7

20.5

28.5



X

9.9

9.9

12.9

12.9

13.3

13.3

14.1

14.1



BLK

9

9

10

10

11

11

12

12



TRT

2

3

1

2

1

3

2

3



Y

7.7

8.7

15.1

19.8

10.3

15.2

10.0

10.2



X

6.3

6.3

16.9

16.9

13.6

13.6

7.4

7.4



12



Random Effects Models

with Covariates



12.1 INTRODUCTION

A treatment is called a random effect when the levels of the treatment are a random

sample of levels from a population of possible levels (Milliken and Johnson, 1992).

Assume that there is one covariate and that a simple linear model describes the

relationship between the mean of the response and the covariate for each treatment.

Since the levels of the treatments are randomly selected from a population of possible

levels, the mean of the response given the value of the covariate or the model for a

selected treatment is also a random variable. More specifically, the coefficients of

the model corresponding to the randomly selected treatment are random variables.

These models are referred to as random coefficient models. Three examples are used

to demonstrate the use of these models.



12.2 THE MODEL

To describe the model, assume the relationship between the response variable, y,

and the value of a single covariate, x, is the simple linear regression model. The

model for the jth observation from the ith randomly selected level of the population

of treatments (called a random effect) is

y ij = a i + b i x ij + ε ij

i = 1, 2, …, t, j = 1, 2, …, n i



(12.1)



where the intercept and slope, ai and bi, are random variables with joint distribution

 α



 ai 

  ~ iid N   , Σ m 

 β 



 bi 

where

 σ a2

Σm = 

σ ab



© 2002 by CRC Press LLC



σ ab 

.

σ b2 



2



Analysis of Messy Data, Volume III: Analysis of Covariance



To carry

out an experiment, involving the random effect, a random sample of

t

size N = iΣ=1 ni experimental units is selected from a population of experimental units.

Measure the values of the covariates on each experimental unit. The values of the

covariate are the same no matter which level of the treatment is to be assigned to

an experimental unit. In fact, the values of the covariate do not depend on whether

the treatment structure is a set of fixed effects, random effects, or mixed effects.

Thus the analysis for a random effects treatment structure is similar to the analysis

of a fixed effects treatment structure in that the models are compared for the given

values of the covariate. The analysis of a random effects treatment structure also

involves estimating the variances and covariances of Σm and the experimental unit

variance, σε2.

The model in Equation 12.1 can also be expressed as

y ij = α + βx ij + e ij

(12.2)



i = 1, 2, …, t, j = 1, 2, …, n i

where e ~ N {0, X (Σm ⊗ It ) X′ + σ ε2 IN ],

 X1

 0

X=

 M

 0





0

X2

M

0



L

L

O

L



0

0

,

M

X t 



1

1

Xi = 

M



1



x i1 

Σ m

 0

x i 2 

, and Σ m ⊗ I t = 

M 

 M





x in i 

 0





0

Σm

M

0



L

L

O

L



0 

0 

.

M 



Σ m 



(See Graybill (1969) for a discussion of direct products like A ⊗ B.) The model of

12.1 can be expressed as



(



) (



)



y ij = α + a *i + β + b*i x ij + ε ij

 a *i 

 0



  ~ iid N   , Σ m 

 0



 b*i 

where

 σ a2

Σm = 

σ ab



σ ab 

.

σ b2 



Models 12.1 and 12.2 can be extended to multiple covariates as

y ij = a i + b1i x1ij + b 2 x 2 ij + … + b ki x kij + ε ij

i = 1, 2, …, t, j = 1, 2, …, n i

© 2002 by CRC Press LLC



(12.3)



Random Effects Models with Covariates



3



where εij ~ iid N(0, σε2 ) and



 α 



 

 ai 



 β1 

 



 

 b1i 

  ~ iid N  β2  , Σ m 



 

 M





M

 

 



 

 b ki 



 β 



 k

where

 σ a2



 σ ab

 1

Σ m = σ ab2



 M



σ

 abk



σ ab1



σ ab2



L



σ b21



σ b1b2



L



σ b1b2



σ b22



L



M



M



σ b1b k



σ b2 b k



σ abk 



σ b1b k 



σ b2 b k 



M 



σ b2 

k 





L



or

y ij = α + β1x1ij + β2 x 2 ij + … + β k x kij + e ij



(12.4)



where e ~ N {0, X (Σm ⊗ It ) X′ + σ ε2 IN ]

 X1



 0

X=

 M



 0



0



L



X2



L



M



O



0



L



1

0





1

0



,

and

X

=



i

M

M





1

X t 





x1i1



L



x1i 2



L



x ki1 



x ki 2 



M 



x ik n 

i 



M

x1i n



L



i



The model for a two-way random effects treatment structure with one covariate

in a linear form can be expressed similar to the model in Equation 12.1 as



(

) (

)

= (α + βx ) + (a + b x ) + (c + d x ) + (f



y ijk = α + a i + c j + fij + β + b i + d j + g ij x ijk + ε ijk

ijk



i



i ijk



j



j ijk



i = 1, 2, …, r, j = 1, 2, …, s, k = 1, 2, …, n ij

© 2002 by CRC Press LLC



ij



)



+ g ijx ijk + ε ijk



(12.5)



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