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9 EXAMPLE: TEACHING METHODS WITH ONE COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT AND ONE COVARIATE MEASURED ON THE SMALL SIZE EXPERIMENTAL UNIT

9 EXAMPLE: TEACHING METHODS WITH ONE COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT AND ONE COVARIATE MEASURED ON THE SMALL SIZE EXPERIMENTAL UNIT

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Analysis of Covariance for Split-Plot and Strip-Plot Design Structures



37



experimental design being a one-way treatment structure in a completely randomized

design structure. The levels of sex were conceptually randomly assigned to the

students within each class; thus the student is the experimental unit for levels of

sex. The student experimental design is a one-way treatment structure in a randomized complete block design structure where there are multiple students of each sex

within each class or block. The data in Table 15.18 shows that there were 8 to

13 students of each sex within each classroom. Table 15.19 contains an analysis of

variance table for this split-plot design without using any of the covariate information. The teacher error term is based on 6 degrees of freedom and the student error

term is based on 181 degrees of freedom. Thus the approximate number of degrees

of freedom associated with the denominator for the various tests from the mixed

model can be between 2 and 183, depending on the number of parameters in the

covariate part of the model. Pre-tests were given to each student to measure the

initial knowledge of mathematics and the number of years of teaching experience

for each teacher were used as possible covariates. A model with factorial effects for

the intercepts, the slopes for years of experience and the slopes for pre-test scores

is in Equation 15.8. The results of fitting Model 15.8 to the teaching method and the

PROC MIXED code to do so are in Table 15.20. There are several terms in the covariate

part of the model with coefficients that are not significantly different from zero; thus

some model building needed to be carried out. The first term deleted was

pre*method*sex as it has the largest significance level — 0.9131. The additional terms

deleted using a backward or stepwise deletion process were pre*sex, yrs*method*sex,

and yrs*method. The resulting reduced model is



}

}



Sc ijkm = µ + ρi + θ Yrs ij + δ Pr e ijkm + κ i Pr e ijkm + a ij



teacher



+ τ k + (ρτ)ik + λ k Yrs ij + ε ijkm



student



(15.39)



The PROC MIXED code in Table 15.21 enables PROC MIXED to fit Model 15.39

to the teaching method data. The significance levels corresponding to pre*method

and yrs*sex are 0.0098 and 0.0494, respectively, indicating there is sufficient evidence to conclude that the slopes for years of experience are different for each sex

and the slopes for pre-test scores are different for the teaching methods. The full

rank model for the intercepts, the slopes for years of experience, and the slopes for

pre-test scores is

Scijkm = µ ik + κ i Pr e ijkm + λ k Yrsij + a ij + ε ijkm



(15.40)



The PROC MIXED code in Table 15.22 is used to fit the full rank model,

Equation 15.40. The REML estimates of the variance components and combined

estimates of the intercepts and slopes are in Table 15.22. The significance levels

corresponding to method*sex, pre*method, and yrs*sex are very small, indicating

there is sufficient evidence to conclude that all of the parameters are not equal to

zero. The adjusted means evaluated at four values of pre-test scores, 77.67 (mean

© 2002 by CRC Press LLC



38



TABLE 15.18

Test Scores for Male and Female Students Taught by One of Three Teaching Methods Where a Pre-Test

Score and Years of Teaching Experience are Possible Covariates

Teacher 1 with 7 years of exp

Method

I

I

I

I

I

I

I

I

I

I

I

I



ID

1

2

3

4

5

6

7

8

9

10

11

12



Score

82

80

75

77

80

82

77

75

79

77



Pre Sc

94

92

81

72

90

86

65

65

65

80



Female

Score

76

81

74

81

76

76

78

77

77

78

79

76



Pre Sc

69

79

79

79

70

77

80

63

84

70

83

87



Teacher 4 with 7 years of exp

Male

II

II

II



1

2

3



Score

80

75

78



© 2002 by CRC Press LLC



Pre Sc

90

76

84



Female

Score

79

85

85



Pre Sc

66

67

84



Teacher 2 with 4 years of exp

Male

Score

79

80

78

79

81

73

77

79

76

80



Pre Sc

67

74

92

72

75

60

85

76

81

85



Female

Score

77

78

78

77

77

76

79

79

77



Pre Sc

66

74

81

64

75

64

88

62

75



Teacher 5 with 8 years of exp

Male

Score

75

81

76



Pre Sc

81

80

74



Female

Score

79

82

86



Pre Sc

65

63

94



Teacher 3 with 4 years of exp

Male

Score

77

76

81

79

77

79

74

79

76

83

78

76



Pre Sc

67

93

79

72

62

82

62

88

63

91

62

93



Female

Score

79

76

71

76

72

79

77

74



Pre Sc

90

92

66

85

61

95

91

95



Teacher 6 with 3 years of exp

Male

Score

72

75

70



Pre Sc

89

63

80



Female

Score

76

76

74



Pre Sc

82

85

64



Analysis of Messy Data, Volume III: Analysis of Covariance



Male



4

5

6

7

8

9

10

11

12



76

78

71

70

78

78

77

74

76



76

94

85

87

89

91

90

63

86



85

80

79

81

80

81

79



76

68

66

91

84

90

90



Teacher 7 with 8 years of exp

Male

III

III

III

III

III

III

III

III

III

III

III

III

III



1

2

3

4

5

6

7

8

9

10

11

12

13



Score

85

81

84

91

88

86

82

81

82

81

80

83



Female

Score

89

94

85

84

85

84

91

88

88

89

88

84

83



Pre Sc

93

75

75

72

63

65

78

73

69

68

62

66

66



71

74

84

86

86

83

70

64



83

83

84

86

85

82

81



75

65

86

87

68

72

69



Teacher 8 with 6 years of exp

Male

Score

83

80

78

77

82

88

83

82

88

84

77



Pre Sc

63

64

61

65

74

89

83

87

77

75

65



Female

Score

84

84

88

84

85

84

88

89

90

85



Pre Sc

79

66

94

90

67

74

75

78

92

75



73

74

74

76

67

73

65



84

69

83

91

70

72

62



77

77

75

78

76

80

76



83

74

90

78

73

75

80



Teacher 9 with 6 years of exp

Male

Score

86

88

89

87

90

90

86

91

86

90



Pre Sc

66

85

86

86

94

91

83

90

74

89



Female

Score

99

99

89

95

90

91

92

92

92

90

94

90

92



Pre Sc

88

89

78

75

76

82

90

74

75

70

92

78

89



39



© 2002 by CRC Press LLC



Pre Sc

89

66

76

92

86

93

82

66

76

62

68

65



77

78

76

78

80

78

74

78



Analysis of Covariance for Split-Plot and Strip-Plot Design Structures



II

II

II

II

II

II

II

II

II



40



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 15.19

Analysis of Variance Table without Covariate

Information for the Teaching Method Study

Source



df



EMS



Method



2



σ + 21.293 σ a2 + φ 2 (method )



Error(teacher) = Class(method)



6



σ ε2 + 21.333 σ a2



Sex



1



σ ε2 + φ 2 (sex)



Method*sex



2



σ ε2 + φ 2 (method ∗ sex)



Error(student)



181



2

ε



σ ε2



TABLE15.20

PROC MIXED Code and Results for Fitting a Model with Two-Way Factorial

Effects for the Intercepts and Slopes for the Teaching Method Data Set

proc mixed data=score method=reml cl covtest;

class method teacher sex;

model score=method|sex pre pre*method pre*sex Pre*sex*method;

yrs yrs*method yrs*sex yrs*sex*method/ddfm=kr;

random teacher(method);

CovParm

Teacher(method)

Residual



Estimate

0.2385

5.5977



StdErr

0.4422

0.6039



ZValue

0.54

9.27



ProbZ

0.2948

0.0000



Effect

method

sex

method*sex

pre

pre*method

pre*sex

pre*method*sex

yrs

yrs*method

yrs*sex

yrs*method*sex



NumDF

2

1

2

1

2

1

2

1

2

1

2



DenDF

22.4

174.7

174.1

172.0

172.0

174.8

174.4

2.7

2.8

172.3

172.2



FValue

2.28

1.84

0.58

44.65

4.69

1.84

0.09

32.84

4.16

3.65

0.12



ProbF

0.1258

0.1763

0.5631

0.0000

0.0104

0.1766

0.9131

0.0134

0.1471

0.0577

0.8852



Alpha

0.05

0.05



Lower

0.0371

4.5798



Upper

32153.4762

6.9990



pre-test score), 70, 80, and 90, and the mean number of years of experience

(6.59 years) are in Table 15.23. The adjusted means evaluated at three values of

years of experience, 0, 5, and 10 years, and the mean pre-test score (77.67) are in

Table 15.24. The adjusted means evaluated at 0 years of experience correspond to

the expected mean test scores for new teachers. Since the slopes for years of

experience are a function of sex, the female and male students need to be compared

© 2002 by CRC Press LLC



Analysis of Covariance for Split-Plot and Strip-Plot Design Structures



41



TABLE 15.21

PROC MIXED Code and Results for Fitting a Model with Two-Way

Factorial Effects for the Intercepts and One-Way Effects for the Slopes

Corresponding to Pre-Test Scores and for Years of Teaching Experience

proc mixed data=score method=reml cl covtest;

class method teacher sex;

model score=method|sex pre pre*method

yrs yrs*sex/ddfm=kr;

random teacher(method);

CovParm

Teacher(method)

Residual



Estimate

0.9198

5.5054



StdErr

0.7554

0.5853



ZValue

1.22

9.41



ProbZ

0.1117

0.0000



Effect

method

sex

method*sex

pre

pre*method

yrs

yrs*sex



NumDF

2

1

2

1

2

1

1



DenDF

150.4

177.5

177.3

181.2

181.1

5.0

177.6



FValue

0.40

0.13

28.64

49.17

4.75

23.67

3.92



ProbF

0.6685

0.7212

0.0000

0.0000

0.0098

0.0047

0.0494



Alpha

0.05

0.05



Lower

0.2938

4.5169



Upper

13.1064

6.8600



at three or more values. Figures 15.19 and 15.20 contain plots of the teaching method

by sex means in Tables 15.23 and 15.24. The female and male regression lines within

a teaching method are parallel when graphed over the levels of pre-test scores. The

regression models for the three teaching methods are parallel within each level of

sex when graphed over the years of experience. Table 15.25 contains the pairwise

comparisons of the female and male mean test scores within each level of teaching

method for the for 0, 5, 10, and 6.59 years of experience and a pre-test score of

77.67. The means of the females and the means of the males are not significantly

different (p = 0.05) for teaching method I with 6.59 and 10 years of experience and

for teaching method III with 0 years of experience. The slopes for pre-test scores

are a function of teaching method, the teaching methods need to be compared at

three or more values of pre-test scores. Tables 15.26 and 15.27 are comparisons of

the means of the three teaching methods within females and within males evaluated

at 6.59 years of experience and pre-test scores of 70, 80, 90, and 77.67. The means

of teaching method III are significantly larger than the means of teaching method I

which are larger than the means of teaching method II for all comparisons. The

above are LSD type comparisons, but other multiple comparison techniques could

be used in making this set of comparisons.

This example involves two covariates where one is measured on the whole plot

or large experimental unit and one is measured on the small experimental unit, a

combination of the examples in Sections 15.7 and 15.8. As long as the computations

are carried out using a mixed models approach, there is really no difference in the

analysis of this split-plot design than carrying out an analysis of covariance with a

© 2002 by CRC Press LLC



42



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 15.22

PROC MIXED Code to Fit a Full Rank Means Model for the Intercepts

and Slopes

proc mixed data=score method=reml cl covtest;

class method teacher sex;

model score=method*sex pre*method yrs*sex/noint ddfm=kr solution;

random teacher(method);

CovParm

Teacher(method)

Residual



Estimate

0.9198

5.5054



StdErr

0.7554

0.5853



ZValue

1.22

9.41



ProbZ

0.1117

0.0000



Alpha

0.05

0.05



Lower

0.2938

4.5169



Upper

13.1064

6.8600



Effect

method*sex

method*sex

method*sex

method*sex

method*sex

method*sex

pre*method

pre*method

pre*method

yrs*sex

yrs*sex



method

I

I

II

II

III

III

I

II

III



sex

F

M

F

M

F

M



F

M



Estimate

64.607

67.829

66.854

63.521

63.809

62.608

0.086

0.089

0.202

1.108

0.750



StdErr

2.569

2.520

2.956

2.938

2.972

2.872

0.029

0.033

0.031

0.210

0.213



df

82.7

79.7

81.6

80.7

36.2

32.7

179.5

180.8

182.0

7.3

7.7



tValue

25.15

26.91

22.62

21.62

21.47

21.80

3.00

2.69

6.58

5.28

3.52



Probt

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

0.0031

0.0077

0.0000

0.0010

0.0084



Effect

method*sex

pre*method

yrs*sex



NumDF

6

3

2



DenDF

109.7

181.0

12.2



FValue

227.84

19.63

13.26



ProbF

0.0000

0.0000

0.0009



two-way treatment structure in a completely randomized or randomized complete

block design structure. The next example involves a strip-plot design with three sizes

of experimental units.



15.10 EXAMPLE: COMFORT STUDY IN A STRIP-PLOT

DESIGN WITH THREE SIZES OF EXPERIMENTAL

UNITS AND THREE COVARIATES

An environmental engineer designed an experiment to evaluate the affect of three

environmental conditions on the comfort of female and male human subjects. On a

given day, one female and one male were subjected to one of the three environments

where both persons were put into an environmental chamber at the same time. On

two other days, the same two persons were subjected to the other two environmental

conditions; thus each person is subjected to all three environmental conditions on

different days. After 1 hour in the environmental chamber, each person was given

a set of questions which were used to compute a comfort score where the larger

score indicates the person is warmer and the smaller score indicates the person is

© 2002 by CRC Press LLC



Analysis of Covariance for Split-Plot and Strip-Plot Design Structures



43



TABLE 15.23

LSMEANS Code to Provide Adjusted method*sex Means Evaluated

at the Mean Number of Years of Teaching Experience and Four

Values of Pre-Test Scores

lsmeans

lsmeans

lsmeans

lsmeans



Effect

method*sex

method*sex

method*sex

method*sex

method*sex

method*sex



method*sex/at

method*sex/at

method*sex/at

method*sex/at



method

I

I

II

II

III

III



sex

F

M

F

M

F

M



means diff;

(pre)=(70) diff;

(pre)=(80) diff;

(pre)=(90) diff;



Years

6.59

6.59

6.59

6.59

6.59

6.59



Pre=77.76

Estimate

78.57

79.43

81.04

75.35

86.82

83.26



Pre=70

Estimate

77.91

78.77

80.36

74.67

85.26

81.70



Pre=80

Estimate

78.77

79.63

81.25

75.56

87.29

83.73



Pre=90

Estimate

79.63

80.49

82.14

76.44

89.31

85.75



TABLE 15.24

LSMEANS Code to Provide Adjusted method*sex Means

Evaluated at the Mean Number of Pre-Test Score

and for Three Values of Years of Teaching Experience

lsmeans method*sex/at (yrs)=(0) diff;

lsmeans method*sex/at (yrs)=(5) diff;

lsmeans method*sex/at (yrs)=(10) diff;



Effect

method*sex

method*sex

method*sex

method*sex

method*sex

method*sex



method

I

I

II

II

III

III



sex

F

M

F

M

F

M



pre

77.67

77.67

77.67

77.67

77.67

77.67



yr=0

Estimate

71.27

74.49

73.74

70.41

79.52

78.32



yr=5

Estimate

76.81

78.24

79.28

74.16

85.06

82.07



yr=10

Estimate

82.35

81.99

84.83

77.91

90.60

85.82



cooler (Table 15.28). Figure 15.21 is a graphical representation of the randomization

process for this experiment. The picture depicts just 3 of the 12 blocks where 1 block

consists of 3 horizontal rectangles and 2 vertical rectangles. The two vertical rectangles represent two persons and the three horizontal rectangles represent the days

on which the two persons were observed within an environmental chamber set to

one of the environmental conditions. The randomization process is to randomly

assign sex of person to the two persons within each block (conceptually at least).

Thus the person is the experimental unit for sex of person. If day is ignored, the

person design is a one-way treatment structure (2 sexes) in a randomized complete

block design structure (12 blocks). The error term for persons is computed as the

© 2002 by CRC Press LLC



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