5 EXAMPLE: FIVE TREATMENTS IN RCB DESIGN STRUCTURE
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More Than Two Treatments in a Blocked Design Structure
9
TABLE 10.3
PROC GLM Code to Fit the Within Block Model with
Parameter Estimates
PROC GLM DATA=LONG10; CLASSES TRT BLOCK;
MODEL Y=BLOCK TRT X*TRT/SOLUTION INVERSE;
ESTIMATE “A1-Abar” TRT .8 –.2 –.2 –.2 –.2;
ESTIMATE “A2-Abar” TRT –.2 .8 –.2 –.2 –.2;
ESTIMATE “A3-Abar” TRT –.2 –.2 .8 –.2 –.2;
ESTIMATE “A4-Abar” TRT –.2 –.2 –.2 .8 –.2;
ESTIMATE “A5-Abar” TRT –.2 –.2 –.2 –.2 .8;
Source
Model
Error
Corr Total
df
20
39
59
SS
5724.9054
38.8640
5763.7693
MS
286.2453
0.9965
FValue
287.25
ProbF
0.0000
Source
BLOCK
trt
x*trt
df
11
4
5
SS(III)
658.5902
6.5423
1040.1444
MS
59.8718
1.6356
208.0289
FValue
60.08
1.64
208.76
ProbF
0.0000
0.1834
0.0000
Estimate
–1.3249
0.5019
0.9823
3.0808
0.0000
0.5100
0.6734
0.9135
1.0798
1.4309
Biased
1
1
1
1
1
0
0
0
0
0
StdErr
1.8508
1.7831
1.9378
1.9275
tValue
–0.72
0.28
0.51
1.60
Probt
0.4783
0.7798
0.6151
0.1180
0.0613
0.0579
0.0708
0.0611
0.0693
8.32
11.63
12.90
17.67
20.66
0.0000
0.0000
0.0000
0.0000
0.0000
Parameter
trt 1
trt 2
trt 3
trt 4
trt 5
x*trt 1
x*trt 2
x*trt 3
x*trt 4
x*trt 5
TABLE 10.4
PROC GLM Code for the Within Block Model
Parameter Estimates from Estimate Statements
ESTIMATE
ESTIMATE
ESTIMATE
ESTIMATE
ESTIMATE
Parameter
A1-Abar
A2-Abar
A3-Abar
A4-Abar
A5-Abar
© 2002 by CRC Press LLC
“A1-Abar”
“A2-Abar”
“A3-Abar”
“A4-Abar”
“A5-Abar”
Estimate
–1.9730
–0.1461
0.3343
2.4328
–0.6480
TRT
TRT
TRT
TRT
TRT
.8 –.2 –.2 –.2 –.2;
–.2 .8 –.2 –.2 –.2;
–.2 –.2 .8 –.2 –.2;
–.2 –.2 –.2 .8 –.2;
–.2 –.2 –.2 –.2 .8;
StdErr
1.0980
0.9878
1.2054
1.1397
1.2343
tValue
–1.80
–0.15
0.28
2.13
0.53
Probt
0.0801
0.8832
0.7830
0.0391
–0.6026
10
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 10.5
PROC GLM Code to Fit Model to Provide Expected
Mean Squares and Within Block Test of the Equal
Slopes Hypothesis
PROC GLM DATA=LONG10; CLASSES BLOCK TRT;
MODEL Y=BLOCK TRT X X*TRT;
RANDOM BLOCK;
Source
Model
Error
Corrected Total
df
20
39
59
SS
5724.9054
38.8640
5763.7693
MS
286.2453
0.9965
FValue
287.25
ProbF
0.0000
Source
BLOCK
trt
x
x*trt
df
11
4
1
4
SS(I)
1424.9533
3259.8077
911.6756
128.4688
MS
129.5412
814.9519
911.6756
32.1172
FValue
129.99
817.80
914.87
32.23
ProbF
0.0000
0.0000
0.0000
0.0000
Source
BLOCK
trt
x
x*trt
df
11
4
1
4
SS(III)
658.5902
6.5423
909.9734
128.4688
MS
59.8718
1.6356
909.9734
32.1172
FValue
60.08
1.64
913.16
32.23
ProbF
0.0000
0.1834
0.0000
0.0000
Source
BLOCK
Expected Mean Square (III)
Var(Error) + 4.5455 Var(BLOCK)
TABLE 10.6
Data for the Between Block Analysis
BLOCK
1
2
3
4
5
6
7
8
9
10
11
12
© 2002 by CRC Press LLC
SY
324.8
314.0
344.2
343.3
370.1
324.3
355.8
359.1
389.7
388.4
376.4
372.7
SX1
14.0
15.4
13.5
19.0
12.0
24.1
28.9
18.7
25.6
22.7
25.3
22.6
SX2
11.4
20.6
21.2
9.0
16.9
21.3
23.6
16.0
13.2
28.5
14.8
11.1
SX3
14.4
20.3
15.6
20.8
16.5
14.2
17.8
24.8
27.2
28.5
19.2
18.4
SX4
12.2
11.9
21.6
17.0
31.4
16.2
22.9
19.1
19.4
22.4
26.1
19.8
SX5
21.1
24.3
16.1
17.5
19.0
12.5
26.9
19.8
28.6
16.7
14.9
22.4
More Than Two Treatments in a Blocked Design Structure
11
TABLE 10.7
PROC GLM Code to Fit the Between Block Model, Test
Equality of Slopes and Estimate the Parameters of the Model
PROC GLM data=new; MODEL SY = SX1 SX2
SX5/INVERSE SOLUTION;
CONTRAST ‘SLOPES =’ SX1 1 SX2 –1, SX1 1
1 SX4 –1, SX1 1 SX5 –1;
ESTIMATE “Abar” Intercept 1;
ESTIMATE “B1” SX1 1; ESTIMATE “B2” SX2
“B3” SX3 1;
ESTIMATE “B4” SX4 1; ESTIMATE “B5” SX5
Source
Model
Error
Corrected Total
SX3 SX4
SX3 –1, SX1
1; ESTIMATE
1;
df
5
6
11
SS
6331.0082
793.7584
7124.7667
MS
1266.2016
132.2931
FValue
9.57
ProbF
0.0080
SLOPES =
4
2910.1908
727.5477
5.50
0.0330
Parameter
Abar
B1
B2
B3
B4
B5
Estimate
228.8619
0.9776
–0.9333
2.8539
3.1500
0.1646
StdErr
24.9251
0.6784
0.6271
0.8005
0.6504
0.7776
tValue
9.18
1.44
–1.49
3.57
4.84
0.21
Probt
0.0001
0.1996
0.1872
0.0119
0.0029
0.8394
variance table is a partition of the Type I block sum of squares from the within block
analyses. The block totals are based on five observations; thus the corrected total
sum of squares in Table 10.7 is five times larger than the Type I SS BLOCKS in
Table 10.5. The MSERROR from Table 10.7 provides the estimate of 5(σˆ ε2 + 5σˆ 2b ) =
132.2931 which yields σˆ 2b = 5.0924. Using the Type III expected mean squares in
ˆ 2b = 12.9524. The reason for
Table 10.5, the method of moments estimate of σ b2 is σ
this discrepancy is that the between block covariate variation is contaminating the
between block analysis (which uses Type 1 analysis) but is removed from the within
block analysis (which uses the Type 3 analysis).
Table 10.8 contains the within block estimates of the slopes, the between block
estimates of the slopes, their estimated covariance matrices, and the combined
estimates of the slopes with the corresponding estimated covariance matrix. The
covariance matrices for the within and between block estimates are from the respective
partitions of the inverse of the X′′ X matrix from each model. The statistic to test the
parallelism hypothesis based on the combined information using Equation 10.12 is
*
32.519. Using the approximation in Equation 10.13, the value of F.05,4,df
= 2.623,
c
which provides the approximate number of denominator degrees of freedom as 37.
Again, the test for equal slopes using the combined estimates of the slopes indicates
there is sufficient evidence to conclude the slopes are not all equal.
© 2002 by CRC Press LLC
12
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 10.8
Between and Within Block Estimates of Slopes, Estimated Covariance
Matrices and the Resulting Combined Estimate of the Slopes and
Covariance Matrix
Slope
β1
β2
β3
β4
β5
Within Block Estimate of Slopes and Covariance Matrix
Estimate
Covariance Matrix
0.5100
0.003773
0.000156
0.000376
0.000093
0.6734
0.000156
0.003363
0.000160
0.000175
0.9135
0.000376
0.000160
0.005030
0.000058
1.0798
0.000093
0.000175
0.000058
0.003749
1.4309
0.000227
–0.000138
0.000371
–0.000175
0.000227
–0.000138
0.000371
–0.000175
0.004812
Slope
β1
β2
β3
β4
β5
Between Block Estimate of Slopes and Covariance Matrix
Estimate
Covariance Matrix
0.9776
0.003479
–0.000447
–0.001007
–0.000291
–0.9333
–0.000447
0.002972
–0.000566
–0.000407
2.8539
–0.001007
–0.000566
0.004844
–0.000200
3.1500
–0.000291
–0.000407
–0.000200
0.003198
0.1646
–0.000626
0.000717
–0.001300
0.000653
–0.000626
0.000717
–0.001300
0.000653
0.004571
Slope
β1
β2
β3
β4
β5
Combined Estimates of the Slopes With Standard Errors (STD),
t-Test of Each Slope Equal to Zero (T), and Significance Level (P)
Estimate
STD
T
P
LSD(.05)
0.518
0.060993
8.485
0.0000
0.123574
0.668
0.057599
11.596
0.0000
0.116688
0.928
0.070419
13.184
0.0000
0.142675
1.099
0.060825
18.075
0.0000
0.123220
1.425
0.068891
20.685
0.0000
0.139574
Test of the Equal Slopes Hypothesis Using the Combined Information
UC = 32.519 sign. level < .0001
Next the within block solution vector in Table 10.3 is transformed from the setto-zero restrictions to the sum-to-zero restrictions. The required matrix is in
Table 10.9. The H1 design matrix for the beta-hat model in Equation 10.7 is in
Table 10.9. The set-to-zero estimates (of the intercepts) are in Table 10.10 and the
sum-to-zero estimates (of the intercepts) are in Table 10.11.
–
Table 10.12 contains the between block information where the intercept is 5α.
–
The first step is to transform to α . by multiplying by the transformation matrix in
Table 10.12 with the rescaled between block estimate of the parameters. The resulting combined estimate of θ is in Table 10.13, which was computed by Equation 10.9.
The LSD(.05) values were computed using the method described following
Equation 10.10.
This computation of the combined estimates of the parameters can be accomplished using PROC MIXED. Table 10.14 contains the PROC MIXED code to fit
the desired model where the fixed effects are TRT and X*TRT and the random effect
is BLOCK. The “PARMS (5.09) (.99651)/ HOLD = 1,2” statement was used to
© 2002 by CRC Press LLC
More Than Two Treatments in a Blocked Design Structure
13
TABLE 10.9
Within Block Transformation Matrices
0.0
0.8
–0.2
–0.2
–0.2
–0.2
0.0
0.0
0.0
0.0
0.0
Transformation Matrix to Convert Set-to-Zero Estimates
of Intercepts to Sum-to-Zero Estimates
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
–0.2
–0.2
–0.2
–0.2
0.0
0.0
0.0
0.0
0.0
0.8
–0.2
–0.2
–0.2
0.0
0.0
0.0
0.0
0.0
–0.2
0.8
–0.2
–0.2
0.0
0.0
0.0
0.0
0.0
–0.2
–0.2
0.8
–0.2
0.0
0.0
0.0
0.0
0.0
–0.2
–0.2
–0.2
0.8
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Design Matrix of the Within Block Estimates
Used to Combine the Estimates
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
TABLE 10.10
Within Block Estimates of Regression Model Parameters for Set-to-Zero
Restrictions on the Intercepts
α1 – α5
–1.3249
α2 – α5
0.5019
α3 – α5
0.9823
Parameters and Estimates
α4 – α5
α5 – α5
β1
β2
3.0808
0.0000
0.5100 0.6734
β3
0.9135
β4
1.0798
β5
1.4309
cause PROC MIXED to use those specified values as the estimates of the variance
components. The estimates of the slopes are directly comparable to those in
Table 10.12 and the estimates of the intercepts in Table 10.14 can be computed by
combining the average intercept with each of the estimates of the intercepts deviating
from the average. A contrast statement is included in Table 10.14 to provide a test
of the equal slopes hypothesis. The F statistic has a value of 32.52 with 4 and
© 2002 by CRC Press LLC
14
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 10.11
Within Block Estimates of Regression Model Parameters with Sum-to-Zero
Restrictions on the Intercepts
–
α
0.
–
α1 – α
–1.973
–
α2 – α
–0.146
–
α3 – α
0.334
Parameters and Estimates
–
–
α4 – α
α5 – α
β1
2.432
–0.648
0.510
β2
0.673
β3
0.913
β4
1.07
β5
1.43
TABLE 10.12
Between Block Estimates of Regression Model Parameters
and Estimated Covariance Matrices
Parameters
5α–
β1
228.862
0.9776
4.696085
–0.024078
–0.037388
–0.035901
–0.060125
–0.078422
–0.024078
0.003479
–0.000447
–0.001007
–0.000291
–0.000626
0.2
0
0
0
0
0
0
1
0
0
0
0
β2
β3
Parameter Estimates
–0.9333
2.8539
Covariance Matrix
–0.037388
–0.035901
–0.000447
–0.001007
0.002972
–0.000566
–0.000566
0.004844
–0.000407
–0.000200
0.000717
–0.001300
β4
β5
3.1500
0.1646
–0.060125
–0.000291
–0.000407
–0.000200
0.003198
0.000653
–0.078422
–0.000626
0.000717
–0.001300
0.000653
0.004571
–
Transformation matrix to rescale α
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
Rescaled between block estimate of parameters and covariance matrix.
α–
β1
β2
β3
β4
β5
45.7724
0.9776
–0.9333
2.8539
3.1500
0.1646
39 degrees of freedom. The least squares means from the analysis in Table 10.14
are in Table 10.15. The least squares means are predicted values evaluated at X = 20.
Table 10.16 contains the PROC MIXED code that uses REML estimates of the
variance components to carry out the analysis. The estimates of the block and residual
variance components are 13.6077 and 0.9967, respectively. The estimates of the
slopes and intercepts in Table 10.16 are quite similar to those in Table 10.14, the
difference being that different sets of estimates of variances components were used
in the computations. A contrast statement is included in Table 10.16 to provide a
test of the equal slopes hypothesis. The F statistic has a value of 32.28 with 4 and
39.2 degrees of freedom. The number of denominator degrees of freedom was
© 2002 by CRC Press LLC
More Than Two Treatments in a Blocked Design Structure
15
TABLE 10.13
Combined Estimates of the Parameters with Standard
Errors (STD), t-Test of Each Parameter Equal to Zero (T),
and Significance Level (P) for Example 9.1
Parameter
–
α
–
α1 – α
–
α2 – α
–
α3 – α
–
α4 – α
–
α5 – α
β1
β2
β3
β4
β5
Estimate
52.879
–1.999
0.075
0.165
2.166
–0.406
0.518
0.668
0.928
1.099
1.425
Std Err
0.889635
1.094679
0.983428
1.201541
1.133910
1.228441
0.060993
0.057599
0.070419
0.060825
0.068891
t-value
59.439
–1.826
0.076
0.137
1.910
–0.331
8.485
11.596
13.184
18.075
20.685
Sign Level
0.0000
0.0755
0.9399
0.8916
0.0635
0.7428
0.0000
0.0000
0.0000
0.0000
0.0000
LSD(.05)
2.020255
2.216329
1.991971
2.432836
2.297224
2.488505
0.123574
0.116688
0.142675
0.123220
0.139574
obtained using the KR (Kenward-Roger) option. This test statistic is very similar to
that obtained in Table 10.8. Finally, Table 10.17 contains the least squares means
for the five treatments evaluated at X = 15, 20, and 25, respectively. For this data
set, all pairwise comparisons between pairs of least squares means evaluated at each
of the values of X are significantly different.
10.6 EXAMPLE: BALANCED INCOMPLETE BLOCK
DESIGN STRUCTURE WITH FOUR TREATMENTS
The data in Table 10.18 consist of 4 treatments in blocks of size 3 arranged in
16 blocks to form a BIB design structure where Y is the response variable and X is
the covariate. The results in the previous sections could be used to obtain the within
block and the between block estimates of the parameters of the models and then
those estimates could be combined. That part of the process is left to the reader as
an exercise. The objective of the analysis in this section is to use PROC MIXED to
provide estimates of the model’s parameters and then demonstrate the process of
computing the least squares means for the incomplete block design structure. The
PROC MIXED code in Table 10.19 contains the terms necessary to fit the desired
model. The fixed effects are TRT and X*TRT and the random effect is BLOCK.
The estimates of the block and residual variance components are 15.5409 and
35.9386, respectively. The estimates of the model parameters are included in the
lower part of Table 10.19. The intercept is the estimate of the intercept for treatment 4
and the estimates corresponding to the levels of TRT are the deviations of the
intercepts from that of treatment 4. The least squares means evaluated at X = 100
are in Table 10.20. Since the model has unequal slopes, the least squares means need
to be evaluated at (at least) three values of X (only one was chosen for demonstration
purposes).
© 2002 by CRC Press LLC
16
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 10.14
PROC MIXED Code to Use the Method of Moments Estimates
of the Variance Components by Employing the Hold Option
PROC MIXED CL COVTEST DATA=LONG10;;
CLASS BLOCK TRT;
MODEL Y=TRT X*TRT/NOINT SOLUTION DDFM=KR;
RANDOM BLOCK;
LSMEANS TRT/DIFF AT X=20;
PARMS (5.09) (.99651)/HOLD=1,2;
CONTRAST ‘EQUAL SLOPES’ X*TRT 1 –1 0 0 0, X*TRT 1 0 –1 0 0,
X*TRT 1 0 0 –1 0, x*TRT 1 0 0 0 –1;
CovParm
BLOCK
Residual
Estimate
5.090000
0.996510
Effect
trt
x*trt
NumDF
5
5
DenDF
39
39
FValue
730.39
213.81
ProbF
Label
EQUAL SLOPES
NumDF
4
DenDF
39
FValue
32.52
ProbF
0.0000
trt
1
2
3
4
5
1
2
3
4
5
Estimate
50.8797
52.9537
53.0436
55.0445
52.4730
0.5176
0.6679
0.9284
1.0994
1.4250
StdErr
1.4204
1.2248
1.5662
1.4096
1.5500
0.0610
0.0576
0.0704
0.0608
0.0689
df
1
1
1
1
1
1
1
1
1
1
Effect
trt
trt
trt
trt
trt
x*trt
x*trt
x*trt
x*trt
x*trt
tValue
35.82
43.23
33.87
39.05
33.85
8.49
11.60
13.18
18.08
20.69
TABLE 10.15
Least Squares Means Computed Using the Method
of Moments Estimates of the Variance Components
from Table 10.14
Effect
trt
trt
trt
trt
trt
© 2002 by CRC Press LLC
trt
1
2
3
4
5
x
20.00
20.00
20.00
20.00
20.00
Estimate
61.2307
66.3117
71.6113
77.0333
80.9738
StdErr
0.7122
0.7290
0.7123
0.7122
0.7122
df
1
1
1
1
1
tValue
85.97
90.97
100.53
108.16
113.70
Probt
0.0074
0.0070
0.0063
0.0059
0.0056
Probt
0.0178
0.0147
0.0188
0.0163
0.0188
0.0747
0.0548
0.0482
0.0352
0.0308