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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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