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10 EXAMPLE: TREATMENTS IN MULTI-LOCATION TRIAL

10 EXAMPLE: TREATMENTS IN MULTI-LOCATION TRIAL

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30



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.19

Simulate Adjusted Significance Levels for Comparing the Levels

of Feed within Each Level of rpm for the Three Levels of Hardness

Effect

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm

Feed*rpm



rpm

100

100

100

100

100

100

200

200

200

200

200

200

400

400

400

400

400

400

800

800

800

800

800

800



Feed

0.01

0.01

0.01

0.02

0.02

0.04

0.01

0.01

0.01

0.02

0.02

0.04

0.01

0.01

0.01

0.02

0.02

0.04

0.01

0.01

0.01

0.02

0.02

0.04



_Feed

0.02

0.04

0.08

0.04

0.08

0.08

0.02

0.04

0.08

0.04

0.08

0.08

0.02

0.04

0.08

0.04

0.08

0.08

0.02

0.04

0.08

0.04

0.08

0.08



Adjp(40)

0.0028

0.0000

0.0000

0.0000

0.0000

0.0000

0.0005

0.0000

0.0000

0.0000

0.0000

0.0000

0.0170

0.0000

0.0000

0.0021

0.0000

0.0000

0.0413

0.0000

0.0000

0.0190

0.0000

0.0000



Adjp(mn)

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0051

0.0000

0.0000

0.0000

0.0000

0.0000

0.0344

0.0000

0.0000

0.0017

0.0000

0.0000



Adjp(70)

0.0079

0.0000

0.0000

0.0000

0.0000

0.0000

0.0285

0.0000

0.0000

0.0000

0.0000

0.0000

0.0107

0.0000

0.0000

0.0000

0.0000

0.0000

0.0586

0.0000

0.0000

0.0002

0.0000

0.0000



ADG ijk = δ i + β i WTijk + L j + b j WTijk + LTij + c ij WTijk + ε ijk



(14.8)



i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,

0   σ L2

L j 

 b  ~ iid N   , 

0  σ Lb

 j



 0   σ 2

σ Lb  LTij 

~ iid N   ,  LT

,

2   c 

σ b   ij 

0  σ LTc



σ LTc 

2

 , ε ~ N 0, σ

σ c2  ijk



(



)



The following random statements are needed with PROC MIXED to specify the

above covariance structure:

random int wt/type=un subject=loc;

random int wt/type=un subject=loc*dose;

However, PROC MIXED could not fit the model to the data set as it was exceeding

the likelihood evaluation limits and was not converging. A simpler model could be

fit to the data set that involves setting σLb = 0, σb2 = 0, σLTc = 0, and σc2 = 0. The

© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



31



TABLE 14.20

Sets of PROC MIXED Code to Fit Models Using Adjusted Values

of the Covariate Using the Unequal Variance Model with

Different Variances for Each Level of rpm

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard40*feed/ddfm=satterth;

repeated/group=rpm;

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard70*feed/ddfm=satterth;

repeated/group=rpm;

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hardmn*feed/ddfm=satterth;

repeated/group=rpm;

CovParm

Residual

Residual

Residual

Residual



Group

rpm 100

rpm 200

rpm 400

rpm 800



Estimate

10.92

14.27

93.88

232.83



StdErr

3.06

3.95

25.15

62.46



ZValue

3.57

3.62

3.73

3.73



Effect

Feed

rpm

Feed*rpm

hard40*Feed



NumDF

3

3

9

4



DenDF

100.3

38.5

38.7

54.5



FValue

361.62

767.09

8.18

48.20



ProbF

0.0000

0.0000

0.0000

0.0000



Effect

Feed

rpm

Feed*rpm

hardmn*Feed



NumDF

3

3

9

4



DenDF

54.7

38.5

38.7

54.5



FValue

808.45

767.09

8.18

48.20



ProbF

0.0000

0.0000

0.0000

0.0000



Effect

Feed

rpm

Feed*rpm

hard70*Feed



NumDF

3

3

9

4



DenDF

98.63

38.52

38.68

54.51



FValue

518.29

767.09

8.18

48.20



ProbF

0.0000

0.0000

0.0000

0.0000



Lower

6.75

8.86

59.06

146.41



Upper

20.67

26.74

172.01

427.00



PROC MIXED code in Table 14.22 fits Model 14.8 with the four variances and

covariances set to zero to the data set. Table 14.22 contains the estimates of the

variance components and the information criteria as well as tests for the fixed effects.

At this point the within location residual variances have been assumed to be equal.

Table 14.23 contains the PROC GLM code to fit a one-way treatment structure in a

CRD design structure with one covariate with unequal slopes to the data from each

location. The table contains the mean square error from each location’s model which

are each based on 8 degrees of freedom. Since the data from each location is

© 2002 by CRC Press LLC



32



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.21

Data from a Multilocation Experiment with Four Levels

of Dose Where Average Daily Gain is the Response Variable

and Initial Weight is the Possible Covariate

Dose = 0 mg

loc

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

4

5

5

5

5

6

6

6

6



adg1

1.21

1.47

1.19

1.11

1.18

1.48

1.49

1.29

1.40

1.51

1.51

1.56

1.82

1.53

1.07

1.35

1.20

1.46

1.61

1.72

1.39

1.84

1.68

1.75



Dose = 1.2 mg



wt1

415

444

413

410

412

442

446

409

425

439

442

432

436

445

406

426

429

449

449

449

411

428

420

442



adg2

1.27

1.43

1.31

1.27

1.40

1.23

1.25

1.20

1.55

1.31

1.43

1.76

1.71

1.74

1.41

1.33

1.92

1.67

1.74

1.63

1.51

1.78

1.86

1.73



wt2

422

437

422

420

425

412

417

407

432

423

421

433

438

448

405

409

450

433

448

435

427

422

448

430



Dose = 2.4 mg

adg3

1.20

1.42

1.37

1.14

1.33

1.38

1.68

1.23

1.74

1.49

1.56

1.45

1.71

1.46

1.56

1.62

2.16

1.70

1.63

1.39

1.81

1.88

1.78

1.74



wt3

407

429

425

404

413

417

437

407

433

429

415

402

428

419

411

402

446

440

418

410

411

444

428

411



Dose = 3.6 mg

adg4

1.36

1.22

1.15

1.20

1.41

1.56

1.45

1.65

1.51

1.70

1.59

1.74

1.80

1.63

1.85

2.33

1.56

1.81

1.47

1.70

2.53

2.21

1.96

2.15



wt4

420

402

402

407

411

421

425

435

408

417

416

443

426

401

447

440

422

447

413

421

448

428

416

447



independent of the data from any other location, Hartley’s Max-F test can be used

to test the equality of the within location variances. The value of the test statistic is

Max-F = 0.03754/0.00079 = 47.52. The 0.01 percentage point for the Max-F distribution comparing six variances each based on eight degrees of freedom is 30;

thus there is sufficient evidence to conclude the variances are not equal. The unequal

slope model with unequal within location variances is

ADG ijk = δ i + βi WTijk + L j + LTij + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, …, 6, k = 1, 2, 3, 4,



(



)



(



)



(14.9)



(



)



2

L i ~ iid N 0, σ L2 , LTij ~ iid N 0, σ LT

, ε ijk ~ N 0, σ j2 .



© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



33



TABLE 14.22

PROC MIXED Code to Fit the Model with Four Variance and Covariance

Parameters Set to Zero

proc mixed cl covtest ic data=ex148;

class loc dose;

model adg=dose wt wt*dose/ddfm=kr;

random loc dose*loc;

CovParm

loc

loc*dose

Residual

Neg2LogLike

–53.2



Estimate

0.02380

0.00281

0.01424



StdErr

0.01612

0.00267

0.00248



ZValue

1.48

1.05

5.73



ProbZ

0.0699

0.1467

0.0000



Alpha

0.05

0.05

0.05



Lower

0.00882

0.00079

0.01040



Parameters

3



AIC

–47.2



AICC

–46.9



HQIC

–49.7



BIC

–47.8



CAIC

–44.8



NumDF

3

1

3



DenDF

82.3

78.1

82.2



FValue

0.08

106.34

0.12



ProbF

0.9701

0.0000

0.9466



Effect

dose

wt

wt*dose



Upper

0.17254

0.08253

0.02070



TABLE 14.23

PROC GLM Code to Compute the within Location

Variances Based on a One-Way Analysis of Covariance

Model with Unequal Slopes at Each Location

proc sort data=ex148; by loc;

proc glm data=ex148; by loc;

class dose;

model adg=dose wt*dose;

Location

1

0.00079



2

0.00288



3

0.00979



4

0.03754



5

0.01734



6

0.02234



and the PROC MIXED code in Table 14.24 fits this model to the ADG data. The

results consist of the estimates of the variance components and the information

criteria where the value of AIC is –75.4. The AIC value for the unequal variance

model is considerably less than the value of AIC for the equal variance model in

Table 14.22, indicating that the unequal variance model is more adequate than the

model with equal variances. The statistic corresponding to wt*dose tests the equal

slopes hypothesis, H0: β1 = β2 = β3 = β4 vs. Ha: (not Ho:) with significance level

0.4332, indicating there is not sufficient evidence to conclude that the slopes are not

equal, so a common slope model is adequate to describe the data. The common

slope model with unequal within location variances is



© 2002 by CRC Press LLC



34



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.24

PROC MIXED Code to Fit the Unequal Slopes Model with

Unequal Variances within Each of the Locations

proc mixed cl covtest ic data=ex148;

class loc dose;

model adg=dose wt wt*dose/ddfm=kr;

random loc dose*loc;

repeated/group=loc;

CovParm

loc

loc*dose

Residual

Residual

Residual

Residual

Residual

Residual



Group



Estimate

0.0243

0.0017

0.0008

0.0029

0.0107

0.0310

0.0182

0.0258



StdErr

0.0163

0.0020

0.0004

0.0012

0.0041

0.0121

0.0076

0.0118



ZValue

1.49

0.84

2.04

2.33

2.61

2.55

2.40

2.18



Lower

0.0091

0.0004

0.0004

0.0014

0.0057

0.0163

0.0093

0.0124



Upper

0.1702

0.2450

0.0027

0.0083

0.0271

0.0802

0.0508

0.0824



Parameters

8



AIC

–75.4



AICC

–73.6



HQIC

–82.1



BIC

–77.1



CAIC

–69.1



NumDF

3

1

3



DenDF

26.4

26.1

26.3



FValue

0.63

227.95

0.94



ProbF

0.6006

0.0000

0.4332



loc

loc

loc

loc

loc

loc



Neg2LogLike

–91.4

Effect

dose

wt

wt*dose



1

2

3

4

5

6



ADG ijk = δ i + β WTijk + L j + LTij + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, …, 6, k = 1, 2, 3, 4,



(



)



(



)



(14.10)



(



)



2

L i ~ iid N 0, σ L2 , LTij ~ iid N 0, σ LT

, ε ijk ~ N 0, σ j2 .



The PROC MIXED code to fit Model 14.10 to the ADG data is in Table 14.25 where

2

the random statement specifies the variance components σL2 and σLT

and the repeated

2

2

2

statement specifies the residual variances, σ1 , σ 2 , …, σ 6 . The estimates of the variance components, estimates of the intercepts and common slope and the tests for

the fixed effects are in Table 14.25. The significance level for wt indicates there is

sufficient evidence to conclude the common slope is not zero and should be included

in the model. The significance level for dose indicates the intercepts are significantly

different, which also means the models are different at any fixed value of the

covariate, wt, since the models are a set of parallel lines. The estimate of the common

slope is 0.0100 lb/day per lb of initial weight. Table 14.26 contains the adjusted

means for the levels of dose evaluated at the mean of the initial pen weights across

all of the locations, which was 425.33 lb. The estimated standard errors of the



© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



35



TABLE 14.25

PROC MIXED Code to Fit the Common Slope Model with

Unequal within Location Variances

proc mixed cl covtest ic data=ex148;

class loc dose;

model adg=dose wt/solution ddfm=kr;

random loc dose*loc;

repeated/group=loc;

CovParm

loc

loc*dose

Residual

Residual

Residual

Residual

Residual

Residual



Group



Effect

Intercept

dose

dose

dose

dose

wt



Dose



Effect

dose

wt



loc

loc

loc

loc

loc

loc



1

2

3

4

5

6



1

2

3

4



NumDF

3

1



Estimate

0.0248

0.0028

0.0006

0.0036

0.0103

0.0283

0.0184

0.0222



StdErr

0.0167

0.0022

0.0002

0.0014

0.0039

0.0109

0.0076

0.0099



ZValue

1.49

1.28

2.38

2.53

2.62

2.58

2.41

2.24



Lower

0.0092

0.0009

0.0003

0.0019

0.0055

0.0150

0.0094

0.0108



Upper

0.1757

0.0326

0.0017

0.0094

0.0258

0.0722

0.0511

0.0682



Estimate

–2.5795

–0.2501

–0.1780

–0.0762

0.0000

0.0100



StdErr

0.2400

0.0448

0.0447

0.0446



df

26.1

8.5

8.4

8.3



tValue

–10.75

–5.58

–3.98

–1.71



Probt

0.0000

0.0004

0.0037

0.1247



0.0005



22.1



18.47



0.0000



DenDF

8.4

22.1



FValue

12.04

340.98



ProbF

0.0021

0.0000



adjusted means are all equal to 0.072 lb, where all of the variance components are

used to provide the estimated standard errors (something a non-mixed models

approach cannot do). The lower part of Table 14.26 contains the pairwise comparisons of the dose means using the simulate adjustment for multiple comparisons.

The results indicate that dose levels 2.4 and 3.6 mg have means significantly greater

than the mean of dose 0 mg. The mean of dose 3.6 mg is significantly larger than

the mean of dose 1.2 mg. The estimate statements in Table 14.27 provide information

about the trend the ADG means as a function of the levels of dose. The orthogonal

polynomial coefficients are selected for four equally spaced levels. The results of

the estimate statements indicate there is a significant linear trend (p = 0.0002), but

there is no evidence of curvature since the quadratic and cubic effects are not

significantly different from zero. All of the tests and comparisons of the treatments

are carried out using KR approximate numbers of degrees of freedom in the denominator which are less than the loc*dose interaction degrees of freedom. If the type III

sums of squares and an equal variance model were used to describe the data, then



© 2002 by CRC Press LLC



36



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.26

LSMEAN Statement to Provide Adjusted Means Evaluated

at the Mean Initial Pen Weight Using the Simulate Multiple

Comparison Method

lsmeans dose/diff at means adjust=simulate;

Effect

dose

dose

dose

dose



Dose

1

2

3

4



wt

425.33

425.33

425.33

425.33



Estimate

1.428

1.500

1.601

1.678



StdErr

0.072

0.072

0.072

0.072



df

6.5

6.5

6.5

6.5



tValue

19.90

20.92

22.33

23.40



Probt

0.0000

0.0000

0.0000

0.0000



Effect

dose

dose

dose

dose

dose

dose



Dose

1

1

1

2

2

3



_dose

2

3

4

3

4

4



Estimate

–0.072

–0.174

–0.250

–0.102

–0.178

–0.076



StdErr

0.045

0.045

0.045

0.045

0.045

0.045



df

8.3

8.4

8.5

8.3

8.4

8.3



tValue

–1.62

–3.87

–5.58

–2.27

–3.98

–1.71



Adjp

0.4310

0.0171

0.0025

0.1802

0.0148

0.3854



TABLE 14.27

Estimate Statements to Evaluate the Linear,

Quadratic, and Cubic Trends across

the Four Equally Spaced Dose Levels

estimate ‘linear’ dose –3 –1 1 3;

estimate ‘quad’ dose 1 –1 –1 1;

estimate ‘cubic’ dose 1 –3 3 –1;

Label

linear

quad

cubic



Estimate

0.8520

0.0041

0.0551



StdErr

0.1420

0.0631

0.1413



DF

8.5

8.3

8.3



tValue

6.00

0.07

0.39



Probt

0.0002

0.9495

0.7064



the LOC*DOSE interaction would be the error term selected to make comparisons

among the adjusted means.

This example demonstrates the process of selecting a model to fit a mixed effects

treatment structure. First the form of the residual variances was selected, followed

by selecting the form of the variances and covariances associated with the covariate

part of the model. Finally the form of the fixed effects of the covariate part of the

model was simplified and then comparisons among the levels of the fixed effects in

the treatment structure were made using this final model.



© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



37



14.11 SUMMARY

The problem of unequal variances in the analysis of covariance model is discussed

where several strategies are developed. First, the simplest form of the variance

structure should be determined to improve the power of the resulting tests involving

the fixed effects. Second, once the form of the variances is determined, then the

usual strategies of analysis of covariance described in previous chapters are applied

to the fixed effects of the model. The examples demonstrate various aspects of the

developed methods. Larholt and Sampson (1995) discuss the estimation of the point

of intersection of two regression lines when the variances are not equal. They provide

a confidence interval about the point of intersection and a simultaneous confidence

region about the difference between the two regression lines. Using the process

described in this and the previous chapter, the results of Larholt and Sampson (1995)

can be extended to models involving random effects in either the design structure

or the treatment structure.



REFERENCES

Larholt, K. M. and Sampson, A. R. (1995) Effects of Heteroscedasticity Upon Certain

Analyses When the Regression Lines Are Not Parallel, Biometrics 51, 731-737.

Littell, R., Milliken, G. A., Stroup, W., and Wolfinger, R. (1996), SAS System for Mixed

Models. SAS Institute Inc., Cary, NC.

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

Experiments, Chapman & Hall, London.

SAS Institute Inc. (1999) SAS/STAT ® Users Guide, Version 8, SAS Institute Inc., Cary, NC.

Westfall, P. H., Tobias, R. D., Rom D., Wolfinger, R. D., and Hochberg, Y. (1999) Multiple

Comparisons and Multiple Tests Using the SAS ® System, SAS Institute Inc., Cary, NC.



EXERCISES

EXERCISE 14.1: Use Levene’s procedure to test for equality of variances for the

data in Problem 3.1. Use the unequal variance model to compare the treatments

(even though the variances may not be unequal).

EXERCISE 14.2: Use Levene’s procedure to test for equality of variances for the

data in Problem 5.1. Use the unequal variance model to compare the treatments

(even though the variances may not be unequal).

EXERCISE 14.3: Use the following data set to carry out the analysis of covariance

strategy of finding the simplest covariance part of the model and then the simplest

form of the covariate part of the model before comparing the treatments.



© 2002 by CRC Press LLC



38



Analysis of Messy Data, Volume III: Analysis of Covariance



Data for Exercise 14.3

Treatment 1

Y

59

64

60

53

53

61

59

62



X

23

31

31

15

17

11

20

15



Treatment 2

Y

60

57

63

60

61

59

43



X

26

11

20

21

30

17

9



Treatment 3

X

72

62

70

68

66

61

70

60



Y

37

30

25

38

20

16

36

27



Treatment 4

X

63

79

72

78

64

65



Y

11

29

17

32

13

18



Treatment 5

X

70

73

72

72

73

72

70



Y

22

27

29

34

32

32

11



EXERCISE 14.4: Build a response surface model for the roughness data in

Section 14.7. Use an unequal variance model with the variance structure selected in

Section 14.7.



© 2002 by CRC Press LLC



15



Analysis of Covariance

for Split-Plot and StripPlot Design Structures



15.1 INTRODUCTION

In the previous chapters, discussions were centered around specifying the model, specifying the variances and covariances for the covariate part of the model, and modeling

the variances of the residual part of the model. This chapter involves the split-plot and

strip-plot type of design structures and the issues are similar to those in the previous

chapters, except the treatment structures considered are factorial arrangements with at

least two factors. In Chapter 16, the repeated measures design structures and the additional concerns about modeling the correlation structure of the repeated measures are

discussed. Repeated measures and split-plot design structures are similar in structure

except for the repeated measures design where the levels of one or more of the factors

cannot be randomly assigned to the experimental units, thus the name repeated measures

(Milliken and Johnson, 1992). This chapter starts with a description of the split-plot

design structure and a discussion of the different types of models that provide estimates

of the fixed effects parameters.

Repeated measures or split-plot designs involve more than one size of experimental unit (Milliken and Johnson, 1992, Chapter 5) and the covariate can be

measured on any of the different sizes of experimental units. The experimental unit

on which the covariate is measured must be considered when constructing the

covariance model in order to understand the sources of the information used in an

appropriate analysis. This chapter presents an introduction to the analysis of covariance by describing three different cases for split-plot designs with two sizes of

experimental units where whole-plot is used to describe the large size experimental

unit and sub-plot is used to describe the small size experimental unit. Specifically,

the situations discussed are a model with the covariate measured on the whole plot

or larger sized experimental unit, a model with the covariate measured on the subplot

or smaller sized experimental unit, and a model with two covariates where one is

measured on the whole plot and the other is measured on the subplot. There are two

levels of analysis: the whole plot analysis and the sub-plot analysis. The discussion

in the next section explores the two types of analyses on the covariate part of the

model detailing the cases for different assumptions on the form of the slopes. The

chapter ends with detailed analyses of four examples.

A word of caution is that you must be sure the values of the covariates measured

on smaller sized experimental units are not affected by the applications of the

treatments on the larger sized experimental units. This problem exists when a



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2



Analysis of Messy Data, Volume III: Analysis of Covariance



repeated measures design is used and the smaller sized experimental unit is a time

interval. It is very easy to incorrectly select variables as possible covariates that have

been affected by the application of treatments to the larger sized experimental units.



15.2 SOME CONCEPTS

The analysis of covariance models for split-plot design structures take on the relevant

characteristics pertaining to the analysis covariance models which were discussed in

the previous chapters and the problems encountered when analyzing split-plot designs

that are discussed in Chapters 5 and 24 of Milliken and Johnson (1992). In particular,

the concept of size of experimental unit must be used to specify an appropriate model

and is used in understanding the analysis. For example, if an experimental design

involves three sizes of experimental units, it is possible for a covariate to be measured

on any of the three sizes of experimental units. Thus, in the terminology of a split-split

plot experiment, the covariate could be measured on a whole plot (largest experimental

unit), or a covariate could be measured on a subplot (middle size experimental unit),

or a covariate could be measured on a sub-sub-plot (smallest size experimental unit).

As described in Milliken and Johnson (1992), there is an analysis for each size

of experimental unit since the model can be partitioned into groups of terms corresponding to each size of experimental unit. The situations discussed in this chapter

involve two different sizes of experimental units, but the discussion can be extended

to designs involving more than two sizes of experimental units. Kempthorne (1952)

discusses the split-plot model where two covariates with corresponding slopes were

constructed with one covariate and slope for the whole plot part of the model and

one covariate and slope for the sub-plot part of the model. The models described

here will have just one slope for each covariate since there is no distinction between

the whole plot information and the sub-plot information about the slope parameter.

The general thought is the covariate will have an affect on the analysis of the

size of experimental unit on which the covariate is measured as well as on larger

sizes of experimental units. However, because of the possibility of heterogeneous

slopes, the covariate can have an affect on the analysis of all sizes of experimental

units. Even when the slopes are homogeneous, the covariate will have an affect on

all sizes of experimental units larger than the one on which the covariate is measured.

When constructing an appropriate covariance model, the covariate term is

included with the segment of the model corresponding to the size of the experimental

unit on which the covariate is measured. Expressing the model in this manner enables

the analyst to determine the number of degrees of freedom removed by estimating

the covariate parameters from each of the error terms. The following three situations

illustrate these concepts.



15.3 COVARIATE MEASURED ON THE WHOLE PLOT

OR LARGE SIZE OF EXPERIMENTAL UNIT

A split-plot design involves a factorial arrangement treatment structure and an

incomplete block design structure. What distinguishes the split-plot and strip-plot

© 2002 by CRC Press LLC



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