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8 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH UNEQUAL VARIANCES

8 EXAMPLE: ONE-WAY TREATMENT STRUCTURE WITH UNEQUAL VARIANCES

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Analysis of Covariance Models with Heterogeneous Errors



11



TABLE 14.2

Variances for Each of the Study Methods

Using Unequal Slopes Model with Pre

as the Covariate

proc sort data=oneway; by method;

proc reg; model post=pre; by method;

Method

1

2

3

4

5



df

6

4

6

5

5



MS

15.6484

0.2266

16.5795

17.4375

0.7613



TABLE 14.3

PROC GLM Code to Carry Out Levene’s Test of Equality

of Variances

proc glm data = oneway; class method;

model post=method pre*method/ solution;

output out=prebymethod r=r;

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

proc glm data = prebymethod;

class method;

model absr=method; lsmeans method/pdiff;

Source

Model

Error

Corrected Total



df

4

31

35



SS

48.1557

92.4285

140.5842



MS

12.0389

2.9816



FValue

4.04



ProbF

0.0095



Source

Method



df

4



SS

48.1557



MS

12.0389



FValue

4.04



ProbF

0.0095



Method

1

2

3

4

5



LSMean

2.8598

0.3488

3.0895

2.6136

0.5984

_2

0.0113



_3

0.7920

0.0062



_4

0.7848

0.0249

0.5982



_5

0.0167

0.7967

0.0090

0.0367



RowName

1

2

3

4

5



© 2002 by CRC Press LLC



_1

0.0113

0.7920

0.7848

0.0167



0.0062

0.0249

0.7967



0.5982

0.0090



0.0367



12



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.4

PROC GLM Code to Provide Tests for Equality

of Variances between Groups (Code) and within

Groups of Study Methods

proc glm data = prebymethod;

class code method;

model absr=code method(code);

lsmean code/pdiff;

Source

Model

Error

Corrected Total



df

4

31

35



SS

48.1557

92.4285

140.5842



MS

12.0389

2.9816



FValue

4.04



ProbF

0.0095



Source

Code

Method(code)



df

1

3



SS(III)

46.8294

1.0468



MS

46.8294

0.3489



FValue

15.71

0.12



ProbF

0.0004

0.9494



Code

1

2



Method



LSMean

2.8543

0.4736



ProbtDiff

0.0004



Effect

Code

Code



statement was used to provide the means of the absolute values of the residuals for

each of the study methods in an attempt to possibly group the methods as to the

magnitude of the variances. The bottom part of Table 14.3 contains the significance

levels for the pairwise comparisons of the study method means of the absolute values

of residuals. The approximate multiple comparison process indicates that methods

2 and 5 have smaller variances than methods 1, 3, and 4. A new variable, code, was

constructed where code=1 for methods 1, 3, and 4 and code=2 for methods 2 and

5. The PROC GLM code in Table 14.4 provides another analysis of the absolute

values of the residuals where the effects in the model are code and method(code).

The Fvalue corresponding to source code provides a Levene’s type test of the

hypothesis of the equality of the two code variances. The significance level is 0.0004,

indicating there is sufficient evidence to conclude that the two variances are not

equal. The Fvalue corresponding to the source method(code) provides a test of the

hypothesis that the three variances for methods 1, 3, and 4 are equal and the two

variances for methods 2 and 5 are equal. The significance level is 0.9494, indicating

there is not sufficient evidence to conclude the variances within each value of code

are unequal. Thus, the simplest model for the variances involves two variances one

for each level of code. Table 14.5 contains the PROC GLM code used to compute

the mean square residual for each code. The variance for code =1 is 16.5033 which

is based on 17 degrees of freedom and for code=2 is 0.5237 which is based on

9 degrees of freedom. These estimates of the variances can be computed by pooling

the respective variances from Table 14.2.

Another method that can be used to determine the adequate form of the variance

part of the model is to use some information criteria (Littell et al., 1996) that is



© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



13



TABLE 14.5

Variances for Each Group of Study

Methods (Code)

proc sort data=oneway; by code;

proc glm data=oneway; class method;

model post=method pre*method; by code;

Code

1

2



df

17

9



MS

16.5033

0.5237



TABLE 14.6

PROC MIXED Code to Fit Equal Variance Model with Unequal Slopes

to the Study Method Data

proc mixed data = oneway cl covtest ic;

class method;

model post=method pre pre*method/ddfm=satterth solution;

CovParm

Residual

Neg2LogLike

175.3

Effect

method

pre

pre*method



Estimate

10.9719



StdErr

3.0430



ZValue

3.61



ProbZ

0.0002



Alpha

0.05



Lower

6.8046



Parameters

1



AIC

177.3



AICC

177.5



HQIC

177.6



BIC

178.5



CAIC

179.5



NumDF

4

1

4



DenDF

26

26

26



FValue

1.77

15.92

2.04



ProbF

0.1642

0.0005

0.1184



Upper

20.6061



available in PROC MIXED. For the demonstration here, the value of AIC was used

where the smaller the AIC value, the better the variance structure. Table 14.6 contains

the PROC MIXED code to fit Model 14.1 where the variance structure consists of

assuming all variances are equal. The value of AIC is 177.3. Table 14.7 contains the

PROC MIXED code to fit a variance model with unequal variances for each of the

methods. The statement “repeated/group=method;” specifies that the residuals have a different variance for each level of method. This model does not involve

repeated measures, but the repeated statement is being used. In this case just think

of the repeated statement as the residual statement since it is the variances of the

residual part of the model that are being specified. The value of AIC is 163.3. The

estimates of the variances in Table 14.7 are the same as those in Table 14.2. Finally,

the PROC MIXED code in Table 14.8 fits the model with different variances for

each level of code. The value of AIC is 158.8. Based on the assumption that the

variance structure with the smaller AIC is more adequate, the structure with different

variances for each level of code would be selected. When the variance structures of



© 2002 by CRC Press LLC



14



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 14.7

PROC MIXED Code to Fit the Unequal Variance Model for Each Study Method

with Unequal Slopes

proc mixed data = oneway cl covtest ic;

class method;

model post=method pre pre*method/ddfm=satterth solution;

repeated/group=method;

CovParm

Residual

Residual

Residual

Residual

Residual

Neg2LogLike

153.3

Effect

method

pre

pre*method



Group

Method 1

Method 2

Method 3

Method 4

Method 5



Estimate

15.6484

0.2266

16.5795

17.4375

0.7613



StdErr

9.0346

0.1602

9.5722

11.0285

0.4815



ZValue

1.73

1.41

1.73

1.58

1.58



ProbZ

0.0416

0.0786

0.0416

0.0569

0.0569



Alpha

0.05

0.05

0.05

0.05

0.05



Parameters

5



AIC

163.3



AICC

166.3



HQIC

166.1



BIC

171.3



CAIC

176.3



NumDF

4

1

4



DenDF

8.6

17.1

8.8



FValue

22.12

20.15

2.32



ProbF

0.0001

0.0003

0.1371



Lower

6.4979

0.0814

6.8845

6.7943

0.2966



Upper

75.8805

1.8713

80.3958

104.8923

4.5796



TABLE 14.8

PROC MIXED Code to Fit the Unequal Variance Model for Each Group

of Study Methods with Unequal Study Method Slopes

proc mixed data = oneway cl covtest ic;

class method;

model post=method pre pre*method/ddfm=satterth solution;

repeated/group=code;

CovParm

Residual

Residual

Neg2LogLike

154.8

Effect

method

pre

pre*method



Group

Group 1

Group 2



Estimate

16.5033

0.5237



StdErr

5.6606

0.2469



ZValue

2.92

2.12



ProbZ

0.0018

0.0169



Alpha

0.05

0.05



Parameters

2



AIC

158.8



AICC

159.4



HQIC

159.9



BIC

162.0



CAIC

164.0



NumDF

4

1

4



DenDF

15.2

18.0

14.6



FValue

21.75

20.11

2.51



ProbF

0.0000

0.0003

0.0870



Lower

9.2927

0.2478



Upper

37.0899

1.7454



models become more and more complicated, a Levene’s type test statistic will not

necessarily exist and so an information criteria can be used to select an adequate

covariance structure.

© 2002 by CRC Press LLC



Analysis of Covariance Models with Heterogeneous Errors



15



For this model, one could use Bartlett’s test and/or Hartley’s test (modified for

unequal sample sizes) to test the equal variance hypothesis. The Levene’s type test

is easy to compute when your software has the ability to compute the residuals,

store them, and then analyze the absolute value of the residuals. PROC GLM has

the ability to provide several different tests of homogeneity of variance when the

model is a one-way treatment structure in a CRD design structure. Some of those

methods use other functions of the residuals which can be easily adapted for the

analysis of covariance model.

Based on the above selected covariance structure, the next step is to investigate

the form of the covariate part of the model. (Note: There is the covariate part of the

model that has to do with the covariates and the covariance part of the model that

has to do with the variances and covariances of the data.) Using the fixed effects

analysis from Table 14.8, the significance level corresponding to pre*method is

0.0870. The conclusion is that there is not sufficient evidence to conclude the slopes

are unequal, so an equal slopes model was selected to continue the analysis (a plot

of the residuals should be carried out for each method before this type of conclusion

is reached). The PROC MIXED code in Table 14.9 is used to fit a common slope

model with a different variance for each level of code. (Note that code is included



TABLE 14.9

PROC MIXED Code to Fit the Unequal Variance Model for Each Group

of Study Methods with Equal Study Method Slopes

proc mixed data = oneway cl covtest ic;

class method code;

model post=method pre/ddfm=satterth solution;

repeated/group=code;

CovParm

Residual

Residual



Group

Group 1

Group 2



Estimate

21.0000

0.5555



StdErr

6.7071

0.2521



ZValue

3.13

2.20



ProbZ

0.0009

0.0138



Alpha

0.05

0.05



Parameters

2



AIC

160.4



AICC

160.9



HQIC

161.6



BIC

163.6



CAIC

165.6



Effect

method

pre



NumDF

4

1



DenDF

16.9

10.8



FValue

166.97

30.02



ProbF

0.0000

0.0002



Effect

Intercept

method

method

method

method

method

pre



Method



Estimate

67.3975

–11.8149

–11.4306

–5.8980

–1.3898

0.0000

0.1616



StdErr

0.8367

1.6551

0.4495

1.6455

1.7650



df

10.66

21.31

9.865

20.86

21.11



tValue

80.55

–7.14

–25.43

–3.58

–0.79



Probt

0.0000

0.0000

0.0000

0.0018

0.4398



0.0295



10.78



5.48



0.0002



Neg2LogLike

156.4



© 2002 by CRC Press LLC



1

2

3

4

5



Lower

12.2351

0.2689



Upper

44.1750

1.7473



16



Analysis of Messy Data, Volume III: Analysis of Covariance



in the class statement, but it does not have to be in the model statement.) Since the

covariate part of the model has been simplified, the estimates of the two variances

are a little larger. The estimate of the slope is 0.1616, indicating that the post exam

score increased 0.1616 points for each additional point on the pre exam. The F-value

corresponding to source method provides a test of the equal intercepts hypothesis,

but since the lines are parallel, it is also a test of the equal models evaluated at some

value of pre hypothesis. In this case, the significance level is 0.000, indicating there

is sufficient evidence to conclude that the models are not equal. The LSMEAN

statements in Table 14.10 provide estimated values for the post scores at three values



TABLE 14.10

LSMEAN Statements to Provide Adjusted Means at Three Values

of Pre and Pairwise Comparison of the Means at Pre = 23.5,

the Mean of the Pre Scores

lsmeans method/pdiff at means;

lsmeans method/pdiff at pre=10;

lsmeans method/pdiff at pre=40;

Method

1

2

3

4

5

1

2

3

4

5

1

2

3

4

5



Pre

23.5

23.5

23.5

23.5

23.5

10.0

10.0

10.0

10.0

10.0

40.0

40.0

40.0

40.0

40.0



Estimate

59.3800

59.7642

65.2968

69.8051

71.1949

57.1985

57.5828

63.1153

67.6236

69.0134

62.0462

62.4305

67.9631

72.4714

73.8612



StdErr

1.6228

0.3143

1.6272

1.7346

0.2972

1.6488

0.4412

1.7108

1.7584

0.5677

1.7205

0.6419

1.6546

1.8270

0.4826



df

19.7

9.8

19.9

19.7

9.8

21.0

10.3

23.7

20.8

10.5

24.1

10.5

21.2

23.6

10.4



tValue

36.59

190.17

40.13

40.24

239.54

34.69

130.52

36.89

38.46

121.56

36.06

97.25

41.08

39.67

153.06



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

0.0000



Method

1

1

1

1

2

2

2

3

3

4



_Method

2

3

4

5

3

4

5

4

5

5



Pre

23.5

23.5

23.5

23.5

23.5

23.5

23.5

23.5

23.5

23.5



Estimate

–0.3843

–5.9169

–10.4251

–11.8149

–5.5326

–10.0409

–11.4306

–4.5083

–5.8980

–1.3898



StdErr

1.6486

2.3042

2.3717

1.6551

1.6644

1.7586

0.4495

2.3844

1.6455

1.7650



df

21.0

20.0

19.6

21.3

21.7

20.8

9.9

20.0

20.9

21.1



tValue

–0.23

–2.57

–4.40

–7.14

–3.32

–5.71

–25.43

–1.89

–3.58

–0.79



© 2002 by CRC Press LLC



Probt

0.8179

0.0183

0.0003

0.0000

0.0031

0.0000

0.0000

0.0732

0.0018

0.4398



Analysis of Covariance Models with Heterogeneous Errors



17



Exam Scores by Pretest Scores for Study Methods

80



Score on Exam



75

70

65

60

55

50

0.00



10.00



20.00



30.00



40.00



50.00



Score on Pretest

1



2



3



4



5



FIGURE 14.1 Plot of the regression lines for each of the five study methods using the parallel

lines model.



of the pre scores, 23.4 (mean), 10, and 40. The adjusted means at 23.4 can be used

to report the results and the adjusted means at 10 and 40 can be used to provide a

graph of the estimated regression lines as displayed in Figure 14.1. The bottom part

of Table 14.10 provides LSD type pairwise comparisons between the study method

means where (1,2), (3,4), and (4,5) methods are not significantly different and all

other comparisons are significantly different (p < 0.05).

The analysis of this model is a preview of the process to be used in later chapters.

The process is to first determine an adequate set of regression models that describes

the mean of the dependent variable as a function of the covariates. Then use that

covariate model to study the relationship among the variances of the treatment

combinations. Once an adequate covariance structure is selected, go back to the

regression or covariate part of the model and simplify it as much as possible using

strategies described in previous chapters.



14.9 EXAMPLE: TWO-WAY TREATMENT STRUCTURE

WITH UNEQUAL VARIANCES

The data in Table 14.11 are from an experiment designed to evaluate the effect of

the speed in rpm of a circular bar of steel and the feed rate into a cutting tool on

the roughness of the surface finish of the final turned product, where the depth of

cut was set to 0.02 in. The treatment structure is a two-way with four levels of speed

(rpm) and four levels of feed rate feed. The design structure is a completely randomized design with eight replications per treatment combination. There is variation



© 2002 by CRC Press LLC



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



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