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9 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH UNEQUAL VARIANCES

9 EXAMPLE: TWO-WAY TREATMENT STRUCTURE WITH UNEQUAL VARIANCES

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18



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.11

Cutting Tool Data Ran at Different Feed Rates and Speeds with Roughness

the Dependent Variable and Hardness the Possible Covariate

Speed 100 rpm

Speed 200 rpm

Speed 400 rpm

Speed 800 rpm

Feed

Rate Roughness Hardness Roughness Hardness Roughness Hardness Roughness Hardness

0.01

50

61

65

59

84

64

111

50

0.01

53

65

55

44

104

70

142

52

0.01

56

57

59

52

73

50

147

47

0.01

41

43

55

46

89

55

135

44

0.01

44

46

62

56

89

58

134

62

0.01

43

51

63

59

87

55

148

53

0.01

42

41

67

66

84

58

162

68

0.01

48

53

59

48

83

56

139

65

0.02

0.02

0.02

0.02

0.02

0.02

0.02

0.02



64

61

64

62

65

58

72

63



54

58

60

57

60

50

67

63



81

81

70

61

67

71

79

69



65

53

43

48

43

44

69

40



108

118

136

109

101

104

93

104



41

67

69

69

61

62

44

62



192

152

190

166

167

152

139

188



68

49

46

46

65

45

64

60



0.04

0.04

0.04

0.04

0.04

0.04

0.04

0.04



97

79

86

90

92

74

92

82



62

48

55

54

59

42

57

44



103

105

121

101

107

103

102

108



61

53

69

50

61

57

57

63



123

137

153

137

111

131

137

155



41

41

64

58

50

46

42

68



197

216

190

187

212

211

207

220



59

46

55

58

63

47

40

64



0.08

0.08

0.08

0.08

0.08

0.08

0.08

0.08



141

142

132

119

147

119

145

147



66

61

49

41

69

42

57

63



158

154

166

159

164

171

162

163



56

49

59

55

65

68

64

68



192

195

180

187

210

174

204

198



69

57

52

56

62

44

63

48



279

293

266

303

281

256

289

287



55

52

52

69

61

41

58

47



in the hardness values of the bar stock used in the experiment; thus the hardness of

each piece of bar was measured to be used as a possible covariate. A model to

describe the linear relationship between roughness and hardness (plot the data to

see if this is an adequate assumption) is

© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



19



y ijk + α ij + βij x ijk + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,



(



ε ijk ~ N 0, σ ij2



(14.2)



)



where yijk denotes the roughness measure, xijk denotes the hardness values, and αij

and βij denote the intercept and slope of the simple linear regression model for ith

level of speed and the jth level of feed rate. The cell variances are displayed in

Table 14.12 and since all of the cells are based on eight observations Hartley’s Max-F

test can be used to test for the equality of variances. The Max-Fvalue is 457.98/1.21 =

378.50. The .01 percentage point from the Max-F distribution for 16 treatments and

seven degrees of freedom per variance is 30.5. Thus there is sufficient evidence to

conclude that the variances are not equal. The first PROC GLM code in Table 14.12

is used to fit Model 14.2 to the data and compute the residuals, and the second PROC

GLM code is used to compute the value of Levene’s statistic. The results are in the

middle of Table 14.12. Levene’s approach has been extended to the two-way treatment structure by fitting a factorial treatment structure to the absolute value of the

residuals instead of just a one-way treatment structure. The results show that there

are unequal variances (use the F-value for Model) and that the levels of rpm are

contributing the most to the unequal variances while the levels of feed and the

interaction between the levels of feed and the levels of rpm contribute less. The

LSMEANS statement was used to provide multiple comparisons among the rpm

means of the absolute values of the residuals using an LSD type of approach. The

lower part of Table 14.12 are the significance levels of the pairwise comparisons

between the rpm means. The means of 1 and 2 are not different, but all other

comparisons are significantly different. The means of the absolute values of the

residuals for the levels of feed are also presented, but they are not significantly

different. The results in Table 14.12 indicate the variances are unequal for the levels

of speed (denoted by rpm) and are not unequal for the other parts of the treatment

structure. One possible simplification is to group rpm levels of 100 and 200 and fit

a three variance model, but that was not considered here. The PROC MIXED code

in Table 14.13 fits Model 14.2 with unequal variances for all combinations of the

levels of feed and the levels of rpm as specified by

“repeated/group=feed*rpm;”

The results in Table 14.13 consist of the list of estimated variances for each cell (as

also shown in Table 14.12) and the information criteria. For this model, the value

of AIC is 780.6. Model 14.2 with unequal variances for each level of rpm is

y ijk + α ij + βij x ijk + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,



(



ε ijk ~ N 0, σ i2

© 2002 by CRC Press LLC



)



(14.3)



20



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.12

Treatment Combination Variances and PROC GLM Code

to Compute the Residuals and to Compute Levene’s Type

Tests of Equality of Variances for the Cutting Tool Data Set

proc glm data=res;

class feed rpm; model rough=feed|rpm rpm*hard*feed;

output out=res r=r;

data res; set res; absr=abs(r);

proc glm data=res;; class feed rpm; model

absr=feed|rpm;

lsmeans feed|rpm/pdiff;

Variances

rpm=100

rpm=200

rpm=400

rpm=800



Feed=.01

9.76

1.21

30.87

211.48



Feed=.02

6.29

29.49

135.55

457.98



Feed=.04

7.47

16.06

140.85

173.04



Feed=.08

21.61

10.69

108.85

127.78



df

15

112

127



SS

2182.3828

2406.1654

4588.5482



MS

145.4922

21.4836



FValue

6.77



ProbF

0.0000



df

3

3

9



SS(III)

152.8896

1783.3713

246.1220



MS

50.9632

594.4571

27.3469



FValue

2.37

27.67

1.27



ProbF

0.0742

0.0000

0.2594



Effect

Feed

Feed

Feed

Feed



Feed

0.01

0.02

0.04

0.08



LSMean

4.4171

7.4871

5.7339

5.6678



Effect

rpm

rpm

rpm

rpm



Rpm

100

200

400

800



LSMean

2.3131

2.4260

7.2930

11.2737



_1



_2

0.9225



_3

0.0000

0.0001



_4

0.0000

0.0000

0.0008



Source

Model

Error

Corrected Total

Source

Feed

rpm

Feed*rpm



RowName

1

2

3

4



0.9225

0.0000

0.0000



0.0001

0.0000



0.0008



which is fit by the PROC MIXED code in Table 14.14. The results consist of the

four estimated variances for the levels of rpm and the associated information criteria.

The value of the AIC is 777.7, indicating that the four variance model is fitting the

© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



21



TABLE 14.13

PROC MIXED Code to Fit a Model with Unequal Variances for Each

Feed*rpm Combination with the Results Including the Information Criteria

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard*feed*rpm/ddfm=satterth;

repeated/group=rpm*feed;

CovParm

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual

Residual



Group

Feed*rpm 0.01

Feed*rpm 0.01

Feed*rpm 0.01

Feed*rpm 0.01

Feed*rpm 0.02

Feed*rpm 0.02

Feed*rpm 0.02

Feed*rpm 0.02

Feed*rpm 0.04

Feed*rpm 0.04

Feed*rpm 0.04

Feed*rpm 0.04

Feed*rpm 0.08

Feed*rpm 0.08

Feed*rpm 0.08

Feed*rpm 0.08



Neg2LogLike

748.6

Effect

Feed

rpm

Feed*rpm

Hard*Feed*rpm



Estimate

9.76

1.21

30.87

211.48

6.29

29.49

135.55

457.98

7.47

16.06

140.85

173.04

21.61

10.69

108.85

127.78



StdErr

5.63

0.70

17.82

122.10

3.63

17.03

78.26

264.42

4.31

9.27

81.32

99.91

12.48

6.17

62.84

73.77



ZValue

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73

1.73



Lower

4.05

0.50

12.82

87.81

2.61

12.25

56.29

190.17

3.10

6.67

58.49

71.85

8.98

4.44

45.20

53.06



Upper

47.33

5.86

149.67

1025.47

30.49

143.01

657.31

2220.80

36.24

77.87

682.98

839.10

104.81

51.84

527.82

619.61



Parameters

16



AIC

780.6



AICC

787.5



HQIC

799.2



BIC

826.3



CAIC

842.3



NumDF

3

3

9

16



DenDF

24.6

28.7

18.9

6



FValue

15.66

20.68

1.18

17.67



ProbF

0.0000

0.0000

0.3592

0.0010



100

200

400

800

100

200

400

800

100

200

400

800

100

200

400

800



variance structure of the data as well as the sixteen variance structure. Also included

in Table 14.14 is the statistic to test the slopes equal to zero hypothesis which is the

F statistic corresponding to Hard*Feed*rpm. The resulting significance level is

0.0000, indicating there is sufficient evidence to believe that the slopes are not all

equal to zero.

The PROC MIXED code in Table 14.15 fits the model



(



) (



)



y ijk = µ + τ i + γ j + δ ij + λ + ρi + φ j + θij x ijk + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,



(



)



ε ijk ~ N 0, σ i2 ,

© 2002 by CRC Press LLC



(14.4)



22



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.14

PROC MIXED Code to Fit a Model with Unequal Variances for Each

Level of rpm for the Cutting Tool Study

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard*rpm*feed/ddfm=satterth;

repeated/group=rpm;

CovParm

Residual

Residual

Residual

Residual

Neg2LogLike

769.7

Effect

Feed

rpm

Feed*rpm

Hard*Feed*rpm



Group

rpm 100

rpm 200

rpm 400

rpm 800



Estimate

11.28

14.36

104.03

242.57



StdErr

3.26

4.15

30.03

70.02



ZValue

3.46

3.46

3.46

3.46



Lower

6.88

8.76

63.43

147.89



Upper

21.84

27.80

201.33

469.45



Parameters

4



AIC

777.7



AICC

778.2



HQIC

782.4



BIC

789.1



CAIC

793.1



NumDF

3

3

9

16



DenDF

52.6

34.3

35.9

24



FValue

10.77

18.58

0.85

12.11



ProbF

0.0000

0.0000

0.5795

0.0000



TABLE 14.15

PROC MIXED Code to Fit a Model with Unequal Variances for Each

Level of rpm and a Factorial Structure for the Covariate Part of the Model

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard hard*feed hard*rpm hard*feed*rpm/ddfm=satterth;

repeated/group=rpm;

CovParm

Residual

Residual

Residual

Residual



Group

rpm 100

rpm 200

rpm 400

rpm 800



Estimate

11.28

14.36

104.03

242.57



StdErr

3.26

4.15

30.03

70.02



ZValue

3.46

3.46

3.46

3.46



Effect

Feed

rpm

Feed*rpm

Hard

Hard*Feed

Hard*rpm

Hard*Feed*rpm



NumDF

3

3

9

1

3

3

9



DenDF

52.6

34.3

35.9

52.4

51.5

34.0

35.5



FValue

10.77

18.58

0.85

45.04

0.71

0.79

0.54



ProbF

0.0000

0.0000

0.5795

0.0000

0.5485

0.5082

0.8349



© 2002 by CRC Press LLC



Lower

6.88

8.76

63.43

147.89



Upper

21.84

27.80

201.33

469.45



Analysis of Covariance Models with Heterogeneous Errors



23



TABLE 14.16

PROC MIXED Code to Fit a Model with Unequal Variances

for Each Level of rpm and a Factorial Structure for the

Covariate Part of the Model without the Hard*Feed*rpm Term

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard hard*feed

hard*rpm/ddfm=satterth;

repeated/group=rpm;

CovParm

Residual

Residual

Residual

Residual



Group

rpm 100

rpm 200

rpm 400

rpm 800



Estimate

10.91

13.95

96.17

237.74



StdErr

3.07

3.88

26.22

64.89



ZValue

3.56

3.60

3.67

3.66



Effect

Feed

rpm

Feed*rpm

Hard

Hard*Feed

Hard*rpm



NumDF

3

3

9

1

3

3



DenDF

67.6

37.7

37.3

53.6

60.2

37.4



FValue

40.30

20.62

8.18

48.36

2.72

0.80



ProbF

0.0000

0.0000

0.0000

0.0000

0.0522

0.5010



Lower

6.73

8.64

60.07

148.43



Upper

20.70

26.24

178.40

441.38



in which unequal variances for the levels of rpm and the slopes and intercepts of

Model 14.2 are expressed as factorial effects. The significance level corresponding

to Hard*Feed*rpm is 0.8349, indicating that there is not a significant interaction

effect on the slopes. A reduced model is



(



) (



)



y ijk = µ + τ i + γ j + δ ij + λ + ρi + φ j x ijk + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,



(



(14.5)



)



ε ijk ~ N 0, σ i2 ,

where αij = µ + τi + γj + δij and βij = λ + ρi + φj.

Table 14.16 contains the PROC MIXED code to fit the reduced model obtained

excluding the Hard*Feed*rpm term and the results indicate that the slopes are not

unequal levels of rpm since the significance level corresponding to Hard*rpm is

0.5010. A reduced model without the Hard*rpm and Hard*Feed*rpm terms is



(



) (



)



y ijk = µ + τ i + γ j + δ ij + λ + φ j x ijk + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,



(



)



ε ijk ~ N 0, σ i2 ,

© 2002 by CRC Press LLC



(14.6)



TABLE 14.17

PROC MIXED Code to Fit a Model with Unequal Variances for

Each Level of rpm and a Factorial Structure for the Covariate

Part of the Model without the rpm and Hard*feed*rpm Term

proc mixed data=twoway cl covtest ic;

class feed rpm;

model rough=feed|rpm hard hard*feed/ddfm=satterth;

repeated/group=rpm;

CovParm

Residual

Residual

Residual

Residual



Group

rpm 100

rpm 200

rpm 400

rpm 800



Estimate

10.92

14.27

93.88

232.83



StdErr

3.06

3.95

25.15

62.46



ZValue

3.57

3.62

3.73

3.73



Effect

Feed

rpm

Feed*rpm

Hard

Hard*Feed



NumDF

3

3

9

1

3



DenDF

67.9

38.5

38.7

64.4

60.4



FValue

39.89

767.09

8.18

174.68

3.91



ProbF

0.0000

0.0000

0.0000

0.0000

0.0129



Lower

6.75

8.86

59.06

146.41



Upper

20.67

26.74

172.01

427.00



where αij = µ + τi + γj + δij and βj = λ + φj. This reduced model is fit by the PROC

MIXED code in Table 14.17. The model could also be expressed as



(



) ( )



y ijk = µ + τ i + γ j + δ ij + β j x ijk + ε ijk ,

i = 1, 2, 3, 4, j = 1, 2, 3, 4, k = 1, 2, …, 8,



(



(14.7)



)



ε ijk ~ N 0, σ i2 ,

The results indicate that there is sufficient information to conclude that the slopes

for the levels of Feed are unequal, since the significance level corresponding to

Hard*Feed is 0.0129. There is an indication that there is an interaction effect on the

intercepts; thus the LSMEANS statements in Table 14.18 were used to provide

adjusted means or predicted values from the regression models at three values of

hardness, i.e., at Hard = 55.45 (the mean), 40, and 70. The estimated values from

Hard = 40 and 70 were used to provide plots of the predicted regression models and

they are displayed in Figures 14.2 through 14.9. Multiple comparisons using the

simulate adjustment were carried out for making comparisons of the levels of feed

within each level of rpm and comparing the levels of rpm within each level of feed.

The means of all levels of rpm with each level of feed were significantly different

(not displayed here). Table 14.19 contains all of the comparisons of the level of feed

within each level of rpm computed at each of the three values of Hard. The columns

Adjp(40), Adjp(mn), and Adjp(70) are the adjusted significance levels using the

© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



25



TABLE 14.18

Adjusted Means for the Feed*rpm Combinations Computed

at Three Values of Hardness

lsmeans feed*rpm/diff at means adjust=simulate;

lsmeans feed*rpm/diff at hard=40 adjust=simulate;

lsmeans feed*rpm/diff at hard=70 adjust=simulate;

Hard=55.45

Feed

0.01

0.01

0.01

0.01

0.02

0.02

0.02

0.02

0.04

0.04

0.04

0.04

0.08

0.08

0.08

0.08



rpm

100

200

400

800

100

200

400

800

100

200

400

800

100

200

400

800



Estimate

49.05

61.61

85.01

139.94

61.96

74.91

107.07

168.29

89.08

103.13

139.33

206.33

135.99

157.44

191.64

282.75



Hard=40



StdErr

1.23

1.35

3.44

5.39

1.22

1.43

3.45

5.39

1.22

1.40

3.47

5.40

1.17

1.42

3.43

5.40



Estimate

40.12

52.68

76.08

131.01

53.85

66.80

98.95

160.18

74.98

89.03

125.24

192.23

121.64

143.09

177.29

268.40



StdErr

1.79

2.03

3.98

5.65

2.31

1.75

4.00

5.64

1.99

2.75

3.71

5.68

1.95

2.41

3.78

5.57



Hard=70

Estimate

57.45

70.01

93.41

148.34

69.60

82.55

114.70

175.93

102.35

116.40

152.61

219.60

149.50

170.95

205.15

296.26



StdErr

2.31

2.25

3.67

5.64

1.69

2.46

3.61

5.62

2.50

1.95

4.18

5.77

1.80

1.63

3.67

5.61



Roughness



Roughness models for levels of RPM

Feed= 0.01

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



Hardness

100



200



FIGURE 14.2 RPM models at Feed = 0.10.

© 2002 by CRC Press LLC



400



800



70.00



75.00



26



Analysis of Messy Data, Volume III: Analysis of Covariance



Roughness



Roughness models for levels of RPM

Feed= 0.02

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



70.00



75.00



70.00



75.00



Hardness

100



200



400



800



FIGURE 14.3 RPM models at Feed = 0.02.



Roughness



Roughness models for levels of RPM

Feed= 0.04

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



Hardness

100



200



400



800



FIGURE 14.4 RPM models at Feed = 0.04.



simulate method (Westfall, 1996) for comparing the feed means within each level

of rpm at Hard = 40, 55.45, and 70, respectively. All feed means within each level

of rpm are significantly different (0.05) except for 0.01 and 0.02 at rpm = 800 and

Hard = 70, which has a significance level of 0.0586. The final part of the analysis

was to investigate how the factorial effects change the relationships between the

© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



27



Roughness



Roughness models for levels of RPM

Feed= 0.08

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



70.00



75.00



70.00



75.00



Hardness

100



200



400



800



FIGURE 14.5 RPM models at Feed = 0.08.



Roughness



Roughness models for levels of Feed

rpm= 100

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



Hardness

0.01



0.02



0.04



0.08



FIGURE 14.6 Feed models at rpm = 100.



models evaluated at different values of hardness. Table 14.20 contains the PROC

MIXED code to fit Model 14.7 to the data using three different adjusted values of

the covariate, Hard40 = Hard – 40, Hardmn = Hard – 55.45, and Hard70 = Hard –

70. The factorial effects in the analyses in Table 14.20 are measuring the effect of

the levels of Feed, levels of rpm, and their interaction on the models evaluated at

© 2002 by CRC Press LLC



28



Analysis of Messy Data, Volume III: Analysis of Covariance



Roughness



Roughness models for levels of Feed

rpm= 200

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



70.00



75.00



70.00



75.00



Hardness

0.01



0.02



0.04



0.08



FIGURE 14.7 Feed models at rpm = 200.



Roughness



Roughness models for levels of Feed

rpm= 400

300

270

240

210

180

150

120

90

60

30

0

35.00



40.00



45.00



50.00



55.00



60.00



65.00



Hardness

0.01



0.02



0.04



0.08



FIGURE 14.8 Feed models at rpm = 400.



the three levels of Hard. All of the factorial effects are highly significant, indicating

that comparisons among the cell means is a reasonable approach to evaluating the

effect of the levels of feed and the levels of rpm.



© 2002 by CRC Press LLC



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