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7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH MISSING CELLS

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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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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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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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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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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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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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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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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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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



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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



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