7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH MISSING CELLS
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Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.19
PROC GLM Code to Fit the Means Model
for the Intercepts and Slopes
PROC GLM; CLASSES TRT;
MODEL Rough=TRT BHN*TRT/E NOINT SOLUTION;
Source
Model
Error
Uncorrected Total
df
14
31
45
SS
1175.109
4.226
1179.336
MS
83.936
0.136
FValue
615.68
ProbF
0.0000
Source
trt
BHN*trt
df
7
7
SS
50.341
39.844
MS
7.192
5.692
FValue
52.75
41.75
ProbF
0.0000
0.0000
as there are 7 intercepts and 7 slopes. There are 31 degrees of freedom for estimating
the error. The estimate of the error is obtained by pooling the error sums of squares
across the seven observed treatment combinations. There are two basic sets of
estimable functions as discussed by Milliken and Johnson (1992). First, the estimable
functions of Chapter 13 based on the means are the mean of speed 2 averaged over
the level of depth, the two 2 × 2 table differences measuring interaction and a
comparison of the levels of speed using two contrasts, one averaging over depths 1
and 3 and the other averaging over depth 1 and 2. The linear combinations are
–
(a) “MU 2 DOT EX 13.1” = µ2. = (µ21 + µ22 + µ23)/3
(b) “Inter Ex 13.2” is measuring interaction by testing
µ11 – µ13 – µ21 + µ23 = 0 and µ21 – µ22 – µ31 + µ32 = 0
and
(c) “TEST IN EX 13.3” is measuring the main effect of speed by testing
µ11 + µ13 µ 21 + µ 23
=
2
2
and
µ 21 + µ 22 µ 31 + µ 32
=
.
2
2
The linear combinations of the above form can be constructed using the intercepts,
the slopes, or the models evaluated at some value of BHN.
Second, the Type IV estimable hypotheses for speed (from Figure 14.2 of
Milliken and Johnson, 1992) and for depth (from Figure 14.3 of Milliken and
Johnson, 1992) are investigated. Again, the above contrasts are evaluated to compare
intercepts, slopes, and/or models evaluated at some value of BHN. Tables 5.20 and
5.21 contain the estimate statements to evaluate the two types of estimable functions
for the intercepts and the slopes. The contrasts in Table 5.20 are constructed from
the intercepts as they are functions of TRT. The contrasts in Table 5.21 are constructed from the slopes as they are functions of BHN*TRT. The only difference
between Tables 5.20 and 5.21 is that the contrasts in Table 5.20 use TRT and the
contrasts in Table 5.21 use BHN*TRT. The results are in Table 5.22. Within each
set, there is an indication of differences between the intercepts and slopes for speed
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Two-Way Treatment Structure and Analysis of Covariance
149
TABLE 5.20
Contrast Statements Used to Evaluate the Two Sets of Estimable Functions
for the Intercepts of the Model
* COMPARISONS OF INTERCEPTS AS IN CHAPTER 13 OF AMD I;
ESTIMATE ‘MU BAR 2 DOT EX 13.1’ TRT 0 0 .3333333 .3333333 .3333333 0 0;
CONTRAST ‘INTER EX 13.2’ TRT 1 –1 –1 0 1 0 0, TRT 0 0 1 –1 0 –1 1;
CONTRAST ‘TEST IN EX 13.3’ TRT 1 1 –1 0 –1 0 0, TRT 0 0 1 1 0 –1 –1;
* INTERCEPT COMPARISONS FOR CHAPTER 14 — Depth;
CONTRAST ‘(M11+M21–M13–M23)/2’ TRT .5 –.5 .5 0 –.5 0 0;
CONTRAST ‘(M21+M31–M22–M32)/2’ TRT 0 0 .5 –.5 0 .5 –.5;
CONTRAST ‘M11–M13’ TRT 1 –1 0 0 0 0 0;
CONTRAST ‘M21–M22’ TRT 0 0 1 –1 0 0 0;
CONTRAST ‘M21–M23’ TRT 0 0 1 0 –1 0 0;
CONTRAST ‘M22–M23’ TRT 0 0 0 1 –1 0 0;
CONTRAST ‘M31–M32’ TRT 0 0 0 0 0 1 –1;
CONTRAST ‘SS Depth Type IV’ TRT 0 0 0 1 –1 0 0, TRT .5 –.5 .5 0 –.5 0 0;
* COMPARISONS FOR CHAPTER 14 — Speed;
CONTRAST ‘(M11+M13–M21-M23)/2’ TRT .5 .5 -.5 0 -.5 0 0;
CONTRAST ‘(M21+M22-M31-M32)/2’ TRT 0 0 .5 .5 0 -.5 -.5;
CONTRAST ‘M11-M21’ TRT 1 0 -1 0 0 0 0;
CONTRAST ‘M11-M31’ TRT 1 0 0 0 0 -1 0;
CONTRAST ‘M21-M31’ TRT 0 0 1 0 0 -1 0;
CONTRAST ‘M22-M32’ TRT 0 0 0 1 0 0 -1;
CONTRAST ‘M13-M23’ TRT 0 1 0 0 -1 0 0;
CONTRAST ‘SS Speed Type IV’ TRT 1 0 0 0 0 -1 0, TRT 0 0 .5 .5 0 -.5 -.5;
TABLE 5.21
Selected Contrast Statements Used to Evaluate the Two Sets of Estimable
Functions for the Slopes of the Model
* COMPARISONS OF SLOPES AS IN CHAPTER 13 OF AMD I;
ESTIMATE ‘Slope BAR 2 DOT like EX 13.1’ BHN*TRT 0 0 .3333333
.3333333 .3333333 0 0;
CONTRAST ‘BHN*SPEED*DEPTH-EX 13.2’ BHN*TRT 1 -1 -1 0 1 0 0, BHN*TRT
0 0 1 -1 0 -1 1;
CONTRAST ‘Slope TEST like IN EX 13.3’ BHN*TRT 1 1 -1 0 -1 0 0,
BHN*TRT 0 0 1 1 0 -1 -1;
* Slope COMPARISONS AS IN CHAPTER 14 for Depth;
CONTRAST ‘(Slope 11+21-13-23)/2’ BHN*TRT .5 -.5 .5 0 -.5 0 0;
CONTRAST ‘(Slope 21+31-22-32)/2’ BHN*TRT 0 0 .5 -.5 0 .5 -.5;
CONTRAST ‘Slope 11-13’ BHN*TRT 1 -1 0 0 0 0 0;
CONTRAST ‘Slope 21-22’ BHN*TRT 0 0 1 -1 0 0 0;
CONTRAST ‘Slope 21-23’ BHN*TRT 0 0 1 0 -1 0 0;
CONTRAST ‘Slope 22-23’ BHN*TRT 0 0 0 1 -1 0 0;
CONTRAST ‘Slope 31-32’ BHN*TRT 0 0 0 0 0 1 -1;
CONTRAST ‘SS BHN*Depth Type IV’ BHN*TRT 0 0 0 1 -1 0 0, BHN*TRT
.5 -.5 .5 0 -.5 0 0;
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Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.22
Results of the Contrast Statements for Both Types of Estimable Functions
for the Intercepts and the Slopes
Intercepts
Slopes
Source
INTER EX 13.2
TEST IN EX 13.3
(M11+M21-M13-M23)/2
(M21+M31-M22-M32)/2
M11-M13
M21-M22
M21-M23
M22-M23
M31-M32
SS Depth Type IV
df
2
2
1
1
1
1
1
1
1
2
SS
0.547
15.813
4.274
0.281
0.847
0.341
4.265
1.065
0.017
4.484
MS
0.274
7.907
4.274
0.281
0.847
0.341
4.265
1.065
0.017
2.242
FValue
2.01
58.00
31.35
2.06
6.21
2.50
31.29
7.81
0.13
16.45
SS
0.536
12.351
6.503
0.431
1.597
0.425
5.692
1.415
0.061
6.701
MS
0.268
6.175
6.503
0.431
1.597
0.425
5.692
1.415
0.061
3.351
FValue
1.97
45.30
47.70
3.16
11.72
3.12
41.75
10.38
0.45
24.58
(M11+M13-M21-M23)/2
(M21+M22-M31-M32)/2
M11-M21
M11-M31
M21-M31
M22-M32
M13-M23
SS Speed Type IV
1
1
1
1
1
1
1
2
4.541
8.487
4.538
15.583
4.206
4.373
0.916
18.346
4.541
8.487
4.538
15.583
4.206
4.373
0.916
9.173
33.31
62.26
33.28
114.30
30.85
32.08
6.72
67.28
3.545
6.740
3.837
12.355
3.315
3.496
0.620
14.433
3.545
6.740
3.837
12.355
3.315
3.496
0.620
7.216
26.00
49.44
28.15
90.62
24.32
25.64
4.54
52.93
and for depth. Some of the F-values are small, but most are large. Table 5.23 contains
a selected set of contrast statements to compare the models at BHN = 210. These
contrast statements are identical to those in Table 5.20, except that both TRT and
BHN*TRT are included. The coefficients of BHN*TRT depend on the value of
BHN, unlike the contrasts in Table 5.21. The results of the contrasts for BHN = 210
and 280 are in Table 5.24. There is evidence that the depth models are different at
BHN = 280 and the speed models are different at BHN = 210. Table 5.25 contains
the PROC GLM code to fit an effects model for both the intercepts and the slopes.
Table 5.25 contains the Type I sums of squares, while the Type II, Type III, and
Type IV sums of squares are in Tables 5.26 through 5.28. There are four different
sums of squares for Speed, Depth, and BHN*Speed. There are three different sums
of squares for BHN and BHN*Depth. There are two different sums of squares for
Speed*Depth, and there is one sum of square for BHN*Speed*Depth. When there
is more than one sum of squares for a given effect, then more than one hypothesis
is being tested about that effect. Chapter 14 of Milliken and Johnson (1992) presents
a detailed discussion of the hypotheses being tested. The general form of the estimable function is in Table 5.29. The most important aspect of this table is that the
general form of the estimable function for the intercepts is identical to the general
form of the estimable function for the slopes and thus would be the general form
of the estimable function for the models evaluated at a specific value of BHN. Since
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Two-Way Treatment Structure and Analysis of Covariance
151
TABLE 5.23
Selected Contrasts Statements to Evaluate Both Types of Estimable Functions
for Comparing the Models at BNH = 210
* COMPARISONS OF MODELS AS IN CHAPTER 13 OF AMD I;
CONTRAST ‘Inter of Models at BHN=210-EX 13.2’ TRT 1 -1 -1 0 1 0
0 BHN*TRT 210 -210 -210 0 210 0 0,
TRT 0 0 1 -1 0 -1 1 BHN*TRT 0 0 210 -210 0 -210 210;
CONTRAST ‘Model TEST at BHN=210 - EX 13.3’
TRT 1 1 -1 0 -1 0 0 BHN*TRT 210 210 -210 0 -210 0 0, TRT 0 0
1 1 0 -1 -1 BHN*TRT 0 0 210 210 0 -210 -210;
* MODEL COMPARISONS at BHN=210 like FOR CHAPTER 14 for Depth;
CONTRAST ‘(Model-210 11+21-13-23)/2’ TRT .5 -.5 .5 0 -.5 0 0
BHN*TRT 105 -105 105 0 -105 0 0;
CONTRAST ‘(Model 210 21+31-22-32)/2’ TRT 0 0 .5 -.5 0 .5 -.5
BHN*TRT 0 0 105 -105 0 105 -105;
CONTRAST ‘Model 210 11-13’ TRT 1 -1 0 0 0 0 0 BHN*TRT 210 -210
0 0 0 0 0; 0 0 0 0 -210 0, TRT 0 0 .5 .5 0 -.5 -.5 BHN*TRT 0
0 105 105 0 -105 -105;
TABLE 5.24
The Sums of Squares Corresponding to the Two Sets of Estimable
Functions for the Models at BHN Values of 210 and 280
BHN = 210
BHN = 280
Inter of Models -EX 13.2
Model TEST - EX 13.3
(Model- 11+21-13-23)/2
(Model 21+31-22-32)/2
Model 11-13
Model 21-22
Model 21-23
Model 22-23
Model 31-32
SS Depth Type IV
df
2
2
1
1
1
1
1
1
1
2
SS
0.502
38.936
0.478
0.083
0.725
0.012
0.002
0.025
0.314
0.478
MS
0.251
19.468
0.478
0.083
0.725
0.012
0.002
0.025
0.314
0.239
FValue
1.84
142.80
3.51
0.61
5.31
0.09
0.01
0.18
2.31
1.75
SS
0.341
0.358
25.848
1.667
11.116
0.950
14.752
3.719
0.718
25.851
MS
0.170
0.179
25.848
1.667
11.116
0.950
14.752
3.719
0.718
12.925
FValue
1.25
1.31
189.60
12.23
81.54
6.97
108.21
27.28
5.27
94.81
(Model 11+13-21-23)/2
(Model 21+22-31-32)/2
Model 11-21
Model 11-31
Model 21-31
Model 22-32
Model 13-23
SS Speed Type IV
1
1
1
1
1
1
1
2
10.154
19.951
7.710
35.347
9.389
10.596
3.090
43.709
10.154
19.951
7.710
35.347
9.389
10.596
3.090
21.855
74.48
146.35
56.56
259.28
68.87
77.72
22.67
160.31
0.009
0.327
0.288
0.528
0.099
0.231
0.095
0.613
0.009
0.327
0.288
0.528
0.099
0.231
0.095
0.306
0.07
2.40
2.11
3.87
0.73
1.69
0.70
2.25
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Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.25
PROC GLM Code to Fit the Effects Model for Both the
Intercepts and the Slope with the Type I Sums
of Squares
proc glm data=common; class Speed Depth;
model Rough=Speed Depth Speed*Depth BHN BHN*Speed
BHN*Depth BHN*Speed*Depth /solution ss1 ss2 ss3
ss4 e e1 e2 e3 e4;
Source
Model
Error
Corrected Total
df
13
31
44
SS
110.088
4.226
114.314
MS
8.468
0.136
FValue
62.12
ProbF
0.0000
Source
Speed
Depth
Speed*Depth
BHN
BHN*Speed
BHN*Depth
BHN*Speed*Depth
df
2
2
2
1
2
2
2
SS (Type I)
33.648
34.911
1.685
0.094
32.375
6.839
0.536
MS
16.824
17.455
0.843
0.094
16.188
3.419
0.268
FValue
123.41
128.04
6.18
0.69
118.74
25.08
1.97
ProbF
0.0000
0.0000
0.0055
0.4127
0.0000
0.0000
0.1572
MS
10.172
2.305
0.274
0.094
7.994
3.419
0.268
FValue
74.61
16.91
2.01
0.69
58.64
25.08
1.97
ProbF
0.0000
0.0000
0.1514
0.4127
0.0000
0.0000
0.1572
TABLE 5.26
Type II Sums of Squares
Source
Speed
Depth
Speed*Depth
BHN
BHN*Speed
BHN*Depth
BHN*Speed*Depth
df
2
2
2
1
2
2
2
SS (Type II)
20.344
4.609
0.547
0.094
15.989
6.839
0.536
these two parameters have identical forms of the general estimable function, then
the general form of the estimable function of the models evaluated at a specified
value of BHN has the same structure.
There is no evidence that there is an interaction between the levels of speed and
the levels of depth for the intercepts and covariate part of the model since the significance levels corresponding to speed*depth is 0.01514 and to BHN*speed*depth is
0.1572.
The three-way interaction term was removed and a model with unequal slopes
for speeds and unequal slopes for depths and an additive effects for the intercepts
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Two-Way Treatment Structure and Analysis of Covariance
153
TABLE 5.27
Type III Sums of Squares
Source
Speed
Depth
Speed*Depth
BHN
BHN*Speed
BHN*Depth
BHN*Speed*Depth
df
2
2
2
1
2
2
2
SS (Type III)
20.067
4.192
0.547
0.011
15.742
6.421
0.536
MS
10.034
2.096
0.274
0.011
7.871
3.211
0.268
FValue
73.60
15.38
2.01
0.08
57.73
23.55
1.97
ProbF
0.0000
0.0000
0.1514
0.7778
0.0000
0.0000
0.1572
TABLE 5.28
Type IV Sums of Squares
Source
Speed
Depth
Speed*Depth
BHN
BHN*Speed
BHN*Depth
BHN*Speed*Depth
df
2
2
2
1
2
2
2
SS (Type IV)
18.346
4.484
0.547
0.001
14.433
6.701
0.536
MS
9.173
2.242
0.274
0.001
7.216
3.351
0.268
FValue
67.28
16.45
2.01
0.01
52.93
24.58
1.97
ProbF
0.0000
0.0000
0.1514
0.9208
0.0000
0.0000
0.1572
was used to describe the data. The effects model with additive effects for speed and
depth for both the slopes and intercepts can be expressed as
(
) (
)
Yijk = µ + α i + γ j + β + ρi + φ j X ij + ε ijk .
An assumption of no interaction is assuming there is no interaction among the speeds
and depths, whether the cell was observed– or not. The slope for speed 1 is computed as and the slope
for βs1 = β + ρ1 + φ. and the slope for depth 1 is computed
–
–
as βd1 = β + ρ. + φ1. The PROC GLM code in Table 5.30 fits the above model to
the data and the test statistics indicate there are significant speed and depth effects
for both the intercepts and the slopes.Table 5.31 contains the estimate statements
needed to compte each of the slopes for the speed models and for the depth models.
Table 5.32 uses the LSMEANS statement evaluated at BHN = 0 (the origin) to
provide the estimates of the intercepts for each of the models.
These slope and intercept estimates were used to provide graphs of the speed
models and the depth models as shown in Figures 5.11 and 5.12. The least squares
means for the speed and depth models evaluated at BHN = 210, 280, and 242.6 are
displayed in Table 5.33. Pairwise comparisons of the speed models at each of the
three values of BHN are in Table 5.34, while Table 5.35 contains the pairwise
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Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.29
General Form of the Estimable Function for the Intercepts and for the Slopes
Effect
Intercept
Intercept
Speed 100 (m/min)
Speed 150 (m/min)
Speed 200 (m/min)
Depth 1 mm
Depth 2 mm
Depth 3 mm
Speed*Depth 100 (m/min) 1 mm
Speed*Depth 100 (m/min) 3 mm
Speed*Depth 150 (m/min) 1 mm
Speed*Depth 150 (m/min) 2 mm
Speed*Depth 150 (m/min) 3 mm
Speed*Depth 200 (m/min) 1 mm
Speed*Depth 200 (m/min) 2 mm
Slopes
BHN
BHN*Speed 100 (m/min)
BHN*Speed 150 (m/min)
BHN*Speed 200 (m/min)
BHN*Depth 1 mm
BHN*Depth 2 mm
BHN*Depth 3 mm
BHN*Speed*Depth 100 (m/min) 1
BHN*Speed*Depth 100 (m/min) 3
BHN*Speed*Depth 150 (m/min) 1
BHN*Speed*Depth 150 (m/min) 2
BHN*Speed*Depth 150 (m/min) 3
BHN*Speed*Depth 200 (m/min) 1
Coefficients
L1
L2
L3
L1 – L2 – L3
L5
L6
L1 – L5 – L6
L8
L2 – L8
L10
–L1 + L2 + L3 + L5 + L6 – L8 – L10
L1 – L2 – L5 – L6 + L8
L5 – L8 – L10
L1 – L2 – L3 – L5 + L8 + L10
mm
mm
mm
mm
mm
mm
L15
L16
L17
L15 – L16 – L17
L19
L20
L15 – L19 – L20
L22
L16 – L22
L24
–L15 + L16 + L17 + L19 + L20 – L22 – L24
L15 – L16 – L19 – L20 + L22
L19 – L22 – L24
comparisons of the levels of depth. The three speed models are significantly different
at BHN = 210 and 242.6, but are not significantly different at BHN = 280. The depth
models are significantly different at BHN 242.6 and 280, but are not significantly
different at BHN = 210. The graphs in Figures 5.11 and 5.12 help understand the
above interpretations. The model used to describe this data set consisted of slopes
that were simplified in form to be a function of the levels of speed and of the levels
of depth, but not of the interaction between the levels of the two factors. Simplifying
the slopes is part of the proposed analysis of covariance strategy, but the model also
includes a simplified form for the intercepts. The intercepts were expressed as an
additive function of the levels of speed and of the levels of depth. This is a dangerous
process. The test for interaction between the levels of speed and the levels of depth
for both slopes and intercepts concerns only those cells where there are data and
cannot address the occurrence of interaction involving the cells with no data.
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Two-Way Treatment Structure and Analysis of Covariance
155
TABLE 5.30
PROC GLM Code to Fit the Reduced Model
to the Roughness Data
proc glm data=common; class Speed Depth;
model Rough=Speed Depth BHN BHN*Speed
BHN*Depth /ss1 ss2 ss3 ss4 e e1 e2 e3 e4;
Source
Model
Error
Corrected Total
df
9
35
44
SS
109.479
4.835
114.314
MS
12.164
0.138
FValue
88.05
ProbF
0.0000
Source
Speed
Depth
BHN
BHN*Speed
BHN*Depth
df
2
2
1
2
2
SS (Type III)
20.300
4.709
0.035
15.922
7.073
MS
10.150
2.354
0.035
7.961
3.536
FValue
73.47
17.04
0.26
57.63
25.60
ProbF
0.0000
0.0000
0.6166
0.0000
0.0000
TABLE 5.31
Estimate Statement to Provide Estimates of the Slopes
for the Speed Models and the Depth Models
Estimate ‘Slope speed 1’ BHN
BHN*Depth 1 1 1/divisor=3;
Estimate ‘Slope speed 2’ BHN
BHN*Depth 1 1 1/divisor=3;
Estimate ‘Slope speed 3’ BHN
BHN*Depth 1 1 1/divisor=3;
Estimate ‘Slope depth 1’ BHN
BHN*Speed 1 1 1/divisor=3;
Estimate ‘Slope depth 2’ BHN
BHN*Speed 1 1 1/divisor=3;
Estimate ‘Slope depth 3’ BHN
BHN*Speed 1 1 1/divisor=3;
Label
SLOPE
SLOPE
SLOPE
SLOPE
SLOPE
SLOPE
SPEED 1
SPEED 2
SPEED 3
DEPTH 1
DEPTH 2
DEPTH 3
Estimate
0.0482
0.0090
–0.0525
–0.0197
–0.0080
0.0324
1 BHN*Speed 3 0 0
3 BHN*Speed 0 3 0
3 BHN*Speed 0 0 3
3 BHN*Depth 3 0 0
3 BHN*Depth 0 3 0
3 BHN*Depth 0 0 3
StdErr
0.0062
0.0047
0.0062
0.0043
0.0070
0.0061
df
35
35
35
35
35
35
tValue
7.75
1.90
–8.52
–4.64
–1.14
5.29
Probt
0.0000
0.0661
0.0000
0.0000
0.2602
0.0000
Generally, when there are missing cells, a global statement of the nonexistence
of interaction cannot be made. Only a decision about the nonsignificance of interaction among the cells that have data can result. The reason the form of the intercepts
was simplified is so that least squares means could be computed. When there are
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Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.32
Estimates of the Intercepts for the Speed
Models and the Depth Models from the
LSMEANS Statement
lsmeans speed depth/ diff at BHN=0;
Speed
100 (m/min)
150 (m/min)
200 (m/min)
Depth
1 mm
2 mm
3 mm
Estimate
–8.164
2.672
19.269
8.758
6.650
–1.632
StdErr
1.534
1.140
1.481
1.037
1.681
1.483
Models For Depths
Roughness
9
6
3
0
210
220
230
240
250
260
270
280
Hardness (BHN)
1 mm
2 mm
3 mm
FIGURE 5.11 Graph of the depth regression models.
missing cells and the data suggest there is no interaction between the levels of the
two factors, then one possible step is to simplify the form of the intercepts.
The researcher must be comfortable with the assumption across all of the cells
(even those with no data), i.e., does the assumption make sense? If the no interaction
assumption is reasonable, then the model can be simplified. When the model is
simplified, the degrees of freedom for interaction are pooled with those of the degrees
of freedom for error. Pooling degrees of freedom from interaction can have the effect
of contaminating the estimate of the variance as well as increasing the number of
degrees of freedom to more than were purchased when the experiment was designed.
If no cells are empty, then it is not recommended to reduce the form of the intercepts.
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Two-Way Treatment Structure and Analysis of Covariance
157
TABLE 5.33
Least Squaes Means for the Levels of Speed and the Levels of Depth
for BHN= 210, 242.6 and 280
lsmeans speed depth/ diff at BHN=210 e;
lsmeans speed depth/ diff at BHN=280 e;
lsmeans speed depth/ diff at means e;
BHN = 210
Speed
Speed
Speed
100 (m/min)
150 (m/min)
200 (m/min)
Estimate
1.963
4.559
8.251
Depth
Depth
Depth
1 mm
2 mm
3 mm
Estimate
4.616
4.975
5.181
BHN = 280
BHN = 242.6
StdErr
0.252
0.166
0.217
Estimate
5.339
5.188
4.579
StdErr
0.244
0.207
0.274
Estimate
3.535
4.852
6.541
StdErr
0.121
0.086
0.118
StdErr
0.164
0.243
0.225
Estimate
3.236
4.417
7.452
StdErr
0.179
0.298
0.267
Estimate
3.974
4.715
6.239
StdErr
0.085
0.117
0.120
TABLE 5.34
Comparisons of the Speed Models at BHN=210, 242.6, and 280
Speed
100 (m/min)
100 (m/min)
150 (m/min)
100 (m/min)
100 (m/min)
150 (m/min)
100 (m/min)
100 (m/min)
150 (m/min)
_Speed
150 (m/min)
200 (m/min)
200 (m/min)
150 (m/min)
200 (m/min)
200 (m/min)
150 (m/min)
200 (m/min)
200 (m/min)
BHN
210.0
210.0
210.0
280.0
280.0
280.0
242.6
242.6
242.6
Estimate
–2.596
–6.289
–3.693
0.151
0.760
0.609
–1.317
–3.006
–1.689
StdErr
0.296
0.339
0.285
0.297
0.409
0.357
0.147
0.177
0.149
tValue
–8.78
–18.57
–12.97
0.51
1.86
1.70
–8.97
–16.96
–11.35
Probt
0.0000
0.0000
0.0000
0.6153
0.0716
0.0973
0.0000
0.0000
0.0000
Table 5.36 contains the estimate statements needed to compute the estimates of
the cell means at a BHN value of 210. Only the estimate statements for the levels
of speed for depth 1 are included because the other statements can be constructed
using the pattern of the statements provided. The body of the table contains the
estimates of the cell means, using the additive model, evaluated at BHN = 210. The
speed least squares means were computed by averaging over the levels of the depth
estimated cell means. For example, the least squares mean for speed 100 (m/min)
is (1.6550 + 2.0139 + 2.2199)/3 = 1.9603. This is identical to the least squares mean
in Table 5.33. The least squares means for depth are computed by averaging over
the levels of speed estimated cell means. This process of computing cell means
illustrates the importance of making sure the no interaction assumption is reasonable
© 2002 by CRC Press LLC
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158
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 5.35
Comparisons of the Depth Models at BHN = 210,
242.6, and 280
Depth
1 mm
1 mm
2 mm
1 mm
1 mm
2 mm
1 mm
1 mm
2 mm
_Depth
2 mm
3 mm
3 mm
2 mm
3 mm
3 mm
2 mm
3 mm
3 mm
BHN
210.0
210.0
210.0
280.0
280.0
280.0
242.6
242.6
242.6
Estimate
–0.359
–0.565
–0.206
–1.181
–4.216
–3.035
–0.742
–2.265
–1.524
StdErr
0.291
0.283
0.337
0.366
0.305
0.448
0.149
0.145
0.176
tValue
–1.23
–2.00
–0.61
–3.23
–13.84
–6.78
–4.98
–15.58
–8.67
Probt
0.2255
0.0537
0.5454
0.0027
0.0000
0.0000
0.0000
0.0000
0.0000
260
270
Models For Speeds
Roughness
9
6
3
0
210
220
230
240
250
280
Hardness (BHN)
100 (m/min)
150 (m/min)
200 (m/min)
FIGURE 5.12 Graph of the speed regression models.
as the estimated cell means used in computing the depth and speed least squares
means are computed using a model with additive effects for the intercepts and slopes.
If the no interaction assumption is not appropriate for all of the cells in the study
(including the empty cells), then the resulting depth and speed marginal or least
squares means are not meaningful.
5.8 EXTENSIONS
When the treatment structure involves more than two factors and/or two or more
covariates, the effects models for the treatment structure need to be fit to the
© 2002 by CRC Press LLC