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)