8 EXAMPLE: FOUR TREATMENTS IN RCB
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12
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 11.8
Least Squares Means and Comparisons at x = 26.98 (Mean), 15, 20, 25, 30,
35, and 40
lsmeans
lsmeans
lsmeans
lsmeans
lsmeans
lsmeans
lsmeans
treat/diff
treat/diff
treat/diff
treat/diff
treat/diff
treat/diff
treat/diff
TREAT
1
2
1
2
1
2
1
2
1
2
1
2
1
2
X
26.98
26.98
15
15
20
20
25
25
30
30
35
35
40
40
TREAT
1
1
1
1
1
1
1
_TREAT
2
2
2
2
2
2
2
at
at
at
at
at
at
at
means;
x=15;
x=20;
x=25;
x=30;
x=35;
x=40;
Estimate
65.5875
80.0375
50.8673
59.6062
57.0135
68.1370
63.1597
76.6678
69.3060
85.1986
75.4522
93.7294
81.5985
102.2602
X
26.98
15
20
25
30
35
40
StdErr
1.8205
1.8205
4.3747
4.3747
2.9466
2.9466
1.9351
1.9351
2.0794
2.0794
3.2281
3.2281
4.6941
4.6941
df
6
6
6
6
6
6
6
6
6
6
6
6
6
6
tValue
36.03
43.96
11.63
13.63
19.35
23.12
32.64
39.62
33.33
40.97
23.37
29.04
17.38
21.78
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
Estimate
–14.4500
–8.7390
–11.1235
–13.5081
–15.8927
–18.2772
–20.6618
StdErr
0.7262
1.7450
1.1754
0.7719
0.8294
1.2876
1.8724
df
6
6
6
6
6
6
6
tValue
–19.90
–5.01
–9.46
–17.50
–19.16
–14.19
–11.04
Probt
0.0000
0.0024
0.0001
0.0000
0.0000
0.0000
0.0000
The model assuming there is a linear relationship between ADG and WT is
ADG ij = α i + βi WTj + b j + ε ij .
(11.10)
The within block analysis provides estimates of the contrasts of the αi’s and contrasts
of the βi’s. Table 11.11 contains the PROC GLM code to extract the within block
information. The results are in Tables 11.11 and 11.12 which includes that partition
of (X.X)– and parameter estimates corresponding to the α’s and β’s. The solutions
ˆ i’s and βˆ i’s satisfy the set-to-zero restrictions. The sum of squares correfor the α
sponding to wt*Ration tests the equal slopes hypothesis (df = 3) instead of testing
© 2002 by CRC Press LLC
Covariate Measured on the Block in RCB
13
TABLE 11.9
PROC MIXED Code and Analysis of Variance without the Covariate
proc mixed cl covtest data=III117;
class blk treat;
model yield=treat/solution;
lsmeans treat/diff;
random blk;
CovParm
BLK
Residual
Estimate
92.9077
5.7114
StdErr
51.2104
3.0529
ZValue
1.81
1.87
ProbZ
0.0348
0.0307
Alpha
0.05
0.05
Lower
39.8015
2.4968
Effect
TREAT
NumDF
1
DenDF
7
FValue
146.23
ProbF
0.0000
Effect
TREAT
TREAT
TREAT
1
2
Estimate
65.5875
80.0375
StdErr
3.5110
3.5110
df
7
7
tValue
18.68
22.80
Probt
0.0000
0.0000
Effect
TREAT
TREAT
1
_TREAT
2
Estimate
–14.5500
StdErr
1.1949
df
7
tValue
–12.09
Upper
408.7899
23.6586
Probt
0.0000
TABLE 11.10
Data for Example in Section 11.8
blk
1
2
3
4
5
6
7
8
9
10
11
12
wt
0.51
0.52
0.54
0.55
0.57
0.59
0.61
0.62
0.64
0.65
0.67
0.69
ration1
2.19
2.11
2.06
1.70
2.00
2.59
2.06
1.91
1.98
1.73
1.80
2.37
ration2
2.44
2.47
2.28
1.91
1.99
2.59
2.34
1.99
2.06
1.72
1.86
2.12
ration3
3.02
3.01
2.82
2.60
2.42
3.37
3.16
2.62
2.96
2.65
2.81
2.99
ration4
2.66
2.96
2.36
2.35
2.22
2.90
2.59
2.29
2.39
2.26
2.10
2.41
Note: Rationi corresponds to the average daily gain for the
ith ration and wt is initial weight in pounds divided by 1000.
the slopes equal to zero hypothesis (df = 4). The significance level corresponding
to the equal slopes hypothesis is 0.0052, indicating there is sufficient evidence to
conclude the slopes are not equal. The least squares means were requested at the
© 2002 by CRC Press LLC
14
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 11.11
PROC GLM Code and Within Block Analysis
for Data of Section 11.8
proc glm data=III118;
class blk ration;
model adg=blk ration wt*ration/solution i;
random blk/test;
Source
Model
Error
Corrected Total
df
17
30
47
SS
7.7508
0.3429
8.0936
MS
0.4559
0.0114
FValue
39.89
ProbF
0.0000
Source
blk
Ration
wt*Ration
df
11
3
3
SS (I)
2.6066
4.9657
0.1785
MS
0.2370
1.6552
0.0595
FValue
20.73
144.84
5.21
ProbF
0.0000
0.0000
0.0052
Source
blk
Ration
wt*Ration
df
10
3
3
SS (III)
2.2988
0.1585
0.1785
MS
0.2299
0.0528
0.0595
FValue
20.12
4.62
5.21
ProbF
0.0000
0.0090
0.0052
mean of the wt values and at wt = 0.510 and 0.680. Only the adjusted means at the
mean of the wt values (0.596) are estimable and are provided in Table 11.13.
The block sum or total model is
ADG. j = α. + β.WTj + e j
(11.11)
where ej = 4bj + ε.j ~ N(0, 4(σ ε2 + 4σ 2b )), which is a simple linear regression model.
The block total data are in Table 11.14 (note that model depends on WT not 4*WT)
and the PROC REG code to fit the blocked total model is in Table 11.15. This
ˆ = 12.7662, βˆ . = –5.4521
analysis provides estimates of α., β. and 4(σ ε2 + 4σ 2b ) as α
2
2
–1
and 4(σ ε + 4σ b ) = 0.9195. The X′′ X matrix is also included in Table 11.15. Using
the between block and the within block information, combined estimators of the
slopes and intercepts can be computed using the process outlined in Chapter 10.
The next step is to combine the within block information and between block information. The vector of parameters to be estimate is
(
)′
= α., β., α 1 − α., α 2 − α., α 3 − α., α 4 − α., β1 − β., β 2 − β., β 3 − β., β 4 − β.
(11.12)
1* = (α 1 − α 4 , α 2 − α 4 , α 3 − α 4 , α 4 − α 4 , β1 − β 4 , β 2 − β 4 , β 3 − β 4 , β 4 − β 4 )′
(11.13)
Let
© 2002 by CRC Press LLC
Parameter
Ration 1
Ration 2
Ration 3
Ration 4
wt*Ration 1
wt*Ration 2
wt*Ration 3
wt*Ration 4
Ration_1
17.332451
8.666225
8.666225
0.000000
–28.793599
–14.396799
–14.396799
0.000000
Ration_2
8.666225
17.332451
8.666225
0.000000
–14.396799
–28.793599
–14.396799
0.000000
Ration_3
8.666225
8.666225
17.332451
0.000000
–14.396799
–14.396799
–28.793599
0.000000
Ration_
4
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Parameter
Ration 1
Ration 2
Ration 3
Ration 4
wt*Ration 1
wt*Ration 2
wt*Ration 3
wt*Ration 4
Estimate
–1.4789
–0.1669
–0.8355
0.0000
1.7832
–0.2400
2.0919
0.0000
StdErr
0.4451
0.4451
0.4451
tValue
–3.32
–0.37
–1.88
Probt
0.0024
0.7103
0.0702
0.7429
0.7429
0.7429
2.40
–0.32
2.82
0.0228
0.7489
0.0085
wt_Ration_1
–28.793599
–14.396799
–14.396799
0.000000
48.297901
24.148951
24.148951
0.000000
wt_Ration_2
–14.396799
–28.793599
–14.396799
0.000000
24.148951
48.297901
24.148951
0.000000
wt_Ration_3
–14.396799
–14.396799
–28.793599
0.000000
24.148951
24.148951
48.297901
0.000000
wt_Ration_4
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
0.000000
Covariate Measured on the Block in RCB
TABLE 11.12
Partition of Inverse of X′ X Corresponding to the Intercepts and Slopes and Within Block Estimates
of the Slopes and Intercepts for the Data of Section 11.8
15
© 2002 by CRC Press LLC
16
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 11.13
Least Squares Means and Comparisons at the
Mean Value of wt (0.596) as Others are Not
Estimable from the Within Block Information
lsmeans ration/pdiff at means stderr;
lsmeans ration/pdiff at wt=.510 stderr;
lsmeans ration/pdiff at wt=.680 stderr;
Ration
1
2
3
4
RowName
1
2
3
4
LSMean
2.0417
2.1475
2.8692
2.4575
StdErr
0.0309
0.0309
0.0309
0.0309
Probt
0.0000
0.0000
0.0000
0.0000
LSMean#
1
2
3
4
_1
_2
0.0215
_3
0.0000
0.0000
_4
0.0000
0.0000
0.0000
0.0215
0.0000
0.0000
0.0000
0.0000
0.0000
TABLE 11.14
Sums of the ADG Values within
Each of the Blocks, Information
for the Between Block Analysis
BLK
1
2
3
4
5
6
7
8
9
10
11
12
sADG
10.31
10.55
9.52
8.56
8.63
11.45
10.15
8.81
9.39
8.36
8.57
9.89
mWT
0.505
0.520
0.539
0.551
0.569
0.585
0.606
0.619
0.640
0.654
0.674
0.692
denote the parameter estimated by the set to zero restriction solution. Convert *1 to
a sum-to-zero restriction 1 by 1 = T1 *1 where
I − 1 J
4 4 4
T1 =
0
© 2002 by CRC Press LLC
1
I4 − J4
4
0
Covariate Measured on the Block in RCB
17
TABLE 11.15
PROC REG Code and Between Block Analysis with the Inverse
of X′ X and the Parameter Estimates
proc reg data=blktotal;
model sadg=mwt/xpx i;
Variable
Intercept
mWT
Intercept
8.666225278
–14.3967995
mWT
–14.3967995
24.14895073
Source
Model
Error
Corrected Total
df
1
10
11
SS
1.2309
9.1954
10.4263
MS
1.2309
0.9195
FValue
1.34
ProbF
0.2742
Variable
Intercept
mWT
df
1
1
Estimate
12.7662
–5.4521
StdErr
2.8229
4.7123
tValue
4.52
–1.16
Probt
0.0011
0.2742
(
)
′
and 1 = α1 − α., α 2 − α., α 3 − α., α 4 − α., β1 − β., β2 − β., β3 − β., β 4 − β. ,
The beta-hat model relating ˆ to is
1
ˆ 1 = H1 + e1
(11.14)
where
0
0
0
0
H1 =
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
(11.15)
with Var(e1) = Σˆ 2.
The between
block information estimates θ 2* = (α., β.) ′ which can be transformed
–
–
to 2 = (α., β.)′ by 2 = T2 where T2 = (1/4)I2.
The beta-hat model relating ˆ 2 to is ˆ 2 = H2 + e2 where
1
H2 =
0
and Var(e2) = Σˆ b.
© 2002 by CRC Press LLC
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
18
Analysis of Messy Data, Volume III: Analysis of Covariance
The two beta-hat models are combined as
ˆ 1
ˆ 2
=
H1
H2
+
e 1
e 2
and the results in
Section 11.3 are applied to provide the combined estimate of and its corresponding
estimated covariance matrix. In this chapter the combining process is accomplished
by using the mixed models approach. The intercepts and slopes are computed as
αˆ
βˆ
= H0 ˆ where
1
1
1
1
H=
0
0
0
0
0
0
0
0
1
1
1
1
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
(11.16)
The adjusted mean for, say, ration 1 at X = X0 is µˆ 1|X0 = a′X0ˆ where a′X0 = (1 X0
1 0 0 0 X0 0 0 0) and differences between, say rations 1 and 2 at X = Xo is µˆ 1|X0 –
µˆ 2|X0 = b′X0ˆ where b′X0 = (0 0 1 –1 0 0 X0 –X0 0 0).
In this chapter the combining process is accomplished by using the mixed models
approach. The PROC MIXED code in Table 11.16 fits the unequal slopes model to
the data set with contrast statements to provide tests of the equal slopes and of the
equal intercepts hypotheses. Since the results in Table 11.16 are based on combined
estimators, the Kenward-Roger adjustment to the denominator of freedom was used
(Kenward and Roger, 1997). The lower partition of Table 11.16 contains the combined estimates of the intercepts and slopes of the four regression models. The
F-statistic corresponding to wt*Ration tests the slopes equal to zero hypothesis and
the contrast with label “b1 = b2 = b3 = b4” tests the equal slopes hypotheses. There
is sufficient information to reject both hypotheses as the significance levels are
0.0095 and 0.0052, respectively. The least squares means evaluated at wt = 0.596,
0.510, and 0.680 are included in Table 11.17 where the lower part of the table
contains the pairwise comparisons of the ration means within each level of wt. All
adjusted means within a value of wt are significantly different (p ≤ 0.05) except for
rations 1 and 2 at wt = 0.680 which has a significance level of 0.4083. Finally,
Table 11.18 contains the PROC MIXED code to carry out an analysis of variance
on the data set. The estimate of the blk variance component is 0.0553 for the analysis
of variance and is 0.0546 when the covariate is taken into account. The least squares
means in the fourth partition of Table 11.18 are identical to the least squares means
in Table 11.17 evaluated at wt = 0.596. The analysis of covariance has smaller
estimated standard errors for the least squares means and for the comparison of the
least squares means. The analysis of covariance is providing additional information
© 2002 by CRC Press LLC
Covariate Measured on the Block in RCB
19
TABLE 11.16
PROC MIXED Code and Combined Within-Between Block Analysis with
Parameter Estimates for the Unequal Slopes Model
proc mixed cl covtest data=III118;
class blk ration;
model adg=ration wt*ration/solution noint ddfm=kr;
contrast ‘a1=a2=a3=a4’ ration 1 –1, ration 1 0 –1, ration 1 0 0 –1;
contrast ‘b1=b2=b3=b4’ wt*ration 1 –1, wt*ration 1 0 –1,
wt*ration 1 0 0 –1;
random blk;
CovParm
blk
Residual
Estimate
0.0546
0.0114
StdErr
0.0257
0.0030
ZValue
2.12
3.87
ProbZ
0.0168
0.0001
Effect
Ration
wt*Ration
NumDF
4
4
DenDF
28.6
28.6
FValue
8.30
4.10
ProbF
0.0001
0.0095
Label
a1=a2=a3=a4
b1=b2=b3=b4
NumDF
3
3
DenDF
30
30
FValue
4.62
5.21
ProbF
0.0090
0.0052
Ration
1
2
3
4
1
2
3
4
Estimate
2.3330
3.6450
2.9764
3.8119
–0.4886
–2.5119
–0.1799
–2.2718
StdErr
0.7565
0.7565
0.7565
0.7565
1.2629
1.2629
1.2629
1.2629
df
13.1
13.1
13.1
13.1
13.1
13.1
13.1
13.1
Effect
Ration
Ration
Ration
Ration
wt*Ration
wt*Ration
wt*Ration
wt*Ration
Alpha
0.05
0.05
Lower
0.0259
0.0073
tValue
3.08
4.82
3.93
5.04
–0.39
–1.99
–0.14
–1.80
Probt
0.0086
0.0003
0.0017
0.0002
0.7050
0.0680
0.8889
0.0951
Upper
0.1817
0.0204
about the relationship among the four ration means than is obtained by ignoring the
covariate because it was used as a blocking factor.
Figure 11.1 contains the model specification screen for analyzing the data using
JMP®. ADG was selected as the response variable. The model effects are blk
(specified as being random), ration, wt and wt*ration. Do not use the center model
option from the model specification menu (deactivate it), then click on run model.
Figure 11.2 contains the estimates of the variance components and the test of the
effects in the model. These are identical to those of PROC MIXED in Table 11.16.
The custom test menu can be used to provide least squares means at various values
of wt as was done in Chapter 10, but those results are not included here.
11.9 EXAMPLE: FOUR TREATMENTS IN BIB
The data in Table 11.19 are surface finish measurements (Finish) of rods turned or
cut on a lathe. Sixteen rods were available for the study involving four types of
© 2002 by CRC Press LLC
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)