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10 EXAMPLE: COMFORT STUDY IN A STRIP-PLOT DESIGN WITH THREE SIZES OF EXPERIMENTAL UNITS AND THREE COVARIATES

10 EXAMPLE: COMFORT STUDY IN A STRIP-PLOT DESIGN WITH THREE SIZES OF EXPERIMENTAL UNITS AND THREE COVARIATES

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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



44



Analysis of Messy Data, Volume III: Analysis of Covariance



Teaching Methods vs. Pre Test Score

at 6.67 Years of Experience

90



Test Score



85



*



*



*



80



75



70

65



70



75



80



85



90



95



Pre Test Score

Method l Female

Method l Male



* * *



Method ll Female

Method ll Male



Method lll Female

Method lll Male



FIGURE 15.19 Graph of least squares means for method by sex combinations against pretest score evaluated at years of experience of 6.59.



Teaching Methods vs. Years of Experience

at Pre Test Score = 77.67

95



Test Score



90

85



*



80



*



75



*

70

0



1



2



3



4



5



6



7



8



9



10



Years of Experience

Method l Female

Method l Male



* * *



Method ll Female

Method ll Male



Method lll Female

Method lll Male



FIGURE 15.20 Graph of least squares means for method by sex combinations against years

of experience evaluated at pre-test score of 77.67.

© 2002 by CRC Press LLC



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



45



TABLE 15.25

Pairwise Comparisons of Female and Male Means within Each

Teaching Method Evaluated at a Pre-Test Score of 77.67 and 0, 5,

10, and 6.59 Years of Teaching Experience

method

I

II

III

I

II

III

I

II

III

I

II

III



sex

F

F

F

F

F

F

F

F

F

F

F

F



_sex

M

M

M

M

M

M

M

M

M

M

M

M



pre

77.67

77.67

77.67

77.67

77.67

77.67

77.67

77.67

77.67

77.67

77.67

77.67



years

0.00

0.00

0.00

5.00

5.00

5.00

10.00

10.00

10.00

6.59

6.59

6.59



Estimate

–3.222

3.333

1.201

–1.430

5.124

2.992

0.361

6.916

4.784

–0.862

5.692

3.560



StdErr

1.102

1.242

1.615

0.605

0.625

0.832

1.075

0.937

0.641

0.663

0.607

0.652



df

178

177

177

178

177

177

178

177

178

178

177

177



tValue

–2.92

2.68

0.74

–2.36

8.19

3.60

0.34

7.38

7.46

–1.30

9.38

5.46



Probt

0.0039

0.0080

0.4583

0.0192

0.0000

0.0004

0.7372

0.0000

0.0000

0.1950

0.0000

0.0000



TABLE 15.26

Pairwise Comparisons of the Teaching Method Means for Female Students

Evaluated at 6.59 Years of Teaching Experience and Pre-Test Scores of 70,

80, 90, and 77.67

sex

F

F

F

F

F

F

F

F

F

F

F

F



method

I

I

II

I

I

II

I

I

II

I

I

II



_method

II

III

III

II

III

III

II

III

III

II

III

III



pre

70.00

70.00

70.00

80.00

80.00

80.00

90.00

90.00

90.00

77.67

77.67

77.67



years

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59



Estimate

–2.451

–7.354

–4.903

–2.481

–8.519

–6.038

–2.510

–9.683

–7.174

–2.474

–8.248

–5.774



StdErr

1.060

1.231

1.133

1.020

1.210

1.109

1.154

1.326

1.254

1.013

1.202

1.099



df

9.1

7.9

8.3

7.8

7.4

7.6

12.5

10.5

12.2

7.6

7.2

7.4



tValue

–2.31

–5.97

–4.33

–2.43

–7.04

–5.44

–2.18

–7.30

–5.72

–2.44

–6.86

–5.25



Probt

0.0459

0.0003

0.0023

0.0419

0.0002

0.0007

0.0494

0.0000

0.0001

0.0420

0.0002

0.0010



sex*block interaction. The two persons assigned to a given block will be subjected

to the three environmental conditions on three different days. Next, randomly assign

the levels of environment to the days within each block so that one environmental

condition is used on a given day. Thus, day is the experimental unit for the levels

of environment and the day design (ignoring persons) is a one-way treatment structure (three levels of environment) in a randomized complete block design structure

where the day error is computed by the environment*day interaction. The time period

© 2002 by CRC Press LLC



46



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 15.27

Pairwise Comparisons of the Teaching Method Means for Male Students

Evaluated at 6.59 Years of Teaching Experience and Pre-Test Scores of 70,

80, 90, and 77.67

sex

M

M

M

M

M

M

M

M

M

M

M

M



method

I

I

II

I

I

II

I

I

II

I

I

II



_method

II

III

III

II

III

III

II

III

III

II

III

III



pre

70.00

70.00

70.00

80.00

80.00

80.00

90.00

90.00

90.00

77.67

77.67

77.67



years

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59

6.59



Estimate

4.103

–2.932

–7.035

4.074

–4.096

–8.170

4.045

–5.261

–9.305

4.081

–3.825

–7.906



StdErr

1.066

1.231

1.148

1.006

1.213

1.090

1.123

1.331

1.206

1.003

1.204

1.088



df

9.3

7.9

8.7

7.4

7.5

7.1

11.3

10.7

10.5

7.3

7.3

7.1



tValue

3.85

–2.38

–6.13

4.05

–3.38

–7.49

3.60

–3.95

–7.72

4.07

–3.18

–7.27



Probt

0.0037

0.0447

0.0002

0.0044

0.0107

0.0001

0.0040

0.0024

0.0000

0.0044

0.0148

0.0002



a person spends within an environment is the experimental unit for the interaction

between the levels of sex and the levels of environment. Therefore, this experiment

involves three different sizes of experimental units. Table 15.29 contains the analysis

of variance table corresponding to this design using the model



}

+ ρ +p }



Sc ijk = µ + b i

j



ij



block

person



+ δ k + d ik } day

+ (ρδ) jk + ε ijk



}



(15.41)



day ∗ person



The person error term is based on 11 degrees of freedom, the day error term is based

on 22 degrees of freedom, and the day*person error term is based on 22 degrees of

freedom. The approximate denominator degrees of freedom associated with the tests

from the mixed model analysis can be from 2 to 55 depending on the number of

covariance parameters in the model.

Three variables were measured to be considered as possible covariates for this

analysis: the age of each person was determined, which is a covariate measured on

the person experimental unit; the outdoor temperature (°F) at 10 a.m. in the morning

of the day two persons were subjected to an environmental condition, which is a

covariate measured on the day experimental unit; and finally, the baseline skin

temperature (°F) of each person was measured at the beginning of the test, which

is a covariate measured on the day*person experimental unit. A model with factorial

effects for the intercepts and for the slopes of each of the three covariates is

© 2002 by CRC Press LLC



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



47



TABLE 15.28

Comfort Scores for Males and Females Exposed to Three Different

Environments Where the Person’s Age, the Temperature of the Day,

and Baseline Skin Temperature are Possible Covariates. Prefix F_

is for females and M_ is for males.

Block

1

1

1



Env

I

II

III



temp

53

52

54



F_score

10

10

13



F_age

24

24

24



F_baseSkT

86

88

94



M_score

8.9

9.4

11.0



M_age

23

23

23



M_baseSkT

90

92

89



2

2

2



I

II

III



54

54

53



8

9

11



26

26

26



87

90

88



8.4

8.5

10.0



25

25

25



91

87

92



3

3

3



I

II

III



51

54

54



9

10

12



18

18

18



90

91

90



8.0

8.9

10.0



20

20

20



89

90

90



4

4

4



I

II

III



58

59

64



8

9

10



26

26

26



92

92

88



8.6

9.0

11.2



25

25

25



90

87

93



5

5

5



I

II

III



64

67

69



8

9

11



24

24

24



91

91

90



8.6

9.9

11.0



18

18

18



93

91

94



6

6

6



I

II

III



69

69

67



8

9

10



25

25

25



93

94

89



7.8

9.3

9.8



23

23

23



93

93

94



7

7

7



I

II

III



66

65

65



8

9

11



19

19

19



90

88

93



7.1

8.5

9.2



24

24

24



91

86

94



8

8

8



I

II

III



63

64

61



8

10

11



20

20

20



94

90

93



8.3

9.8

10.5



24

24

24



87

89

87



9

9

9



I

II

III



59

56

57



8

9

11



20

20

20



93

94

90



7.6

8.3

9.9



22

22

22



93

90

91



10

10

10



I

II

III



54

54

51



9

10

11



21

21

21



89

93

93



8.4

9.1

9.8



22

22

22



92

91

88



11

11

11



I

II

III



51

53

51



8

10

11



21

21

21



88

92

86



7.0

8.8

9.4



21

21

21



93

88

91



12

12

12



I

II

III



51

47

51



8

10

11



19

19

19



94

87

90



7.4

8.5

9.6



19

19

19



86

91

92



© 2002 by CRC Press LLC



48



Analysis of Messy Data, Volume III: Analysis of Covariance



SEX



Day by person

is experimental

unit for sex*env



F



M



RANDOMIZE



Randomization Scheme

for First Three Blocks of

Comfort Study



Day is

experimental unit

for environment



RANDOMIZE



lll

ll

l



Environment



Block 1



Person is

experimental unit

for sex



Block 2



Block 3



FIGURE 15.21 Graphical representation of the randomization scheme for the comfort study

where the vertical rectangles represent persons and the horizontal rectangles represent days.



TABLE 15.29

Analysis of Variance Table without Covariate Information

for the Comfort Study in a Strip-Plot Design

Source

Block

Sex

Error(person) = Sex*Block

Environment

Error(day) = Env*Block

Env*Sex

Error(day*person)



© 2002 by CRC Press LLC



df

11

1

11

2

22

2

22



EMS

σ + 2 σ + 3 σ p2 + 6 σ b2

2

ε



2

d



σ ε2 + 3 σ p2 + φ 2 (sex)

σ ε2 + 3 σ p2

σ ε2 + 3 σ d2 + φ 2 (Env)

σ ε2 + 3 σ d2

σ ε2 + φ 2 (Env ∗ sex)

σ ε2



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



49



TABLE 15.30

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

Factorial Effects Model for the Intercepts and for the Slopes for Age,

Day Temperature, and Baseline Skin Temperature

proc mixed method=reml cl covtest data=comfort;

class block env sex;

model score=env|sex age age*sex age*env age*sex*env

temp sex*temp temp*env temp*sex*env

baseSkt baseskt*sex baseskt*env baseskt*sex*env/solution ddfm=kr;

random block block*sex block*env;

CovParm

Block

Block*sex

Block*Env

Residual



Estimate

0.1957

0.0834

0.1007

0.0126



StdErr

0.1357

0.0430

0.0384

0.0060



ZValue

1.44

1.94

2.62

2.10



ProbZ

0.0747

0.0261

0.0044

0.0179



Effect

Env

sex

Env*sex

age

age*sex

age*Env

age*Env*sex

temp

temp*sex

temp*Env

temp*Env*sex

baseSkT

baseSkT*sex

baseSkT*Env

baseSkT*Env*sex



NumDF

2

1

2

1

1

2

2

1

1

2

2

1

1

2

2



DenDF

20.9

19.7

16.3

14.3

10.2

13.9

10.4

16.9

20.1

17.6

12.1

10.9

14.4

13.2

16.5



FValue

1.03

9.53

1.60

1.45

4.28

1.80

0.10

0.29

10.77

0.43

2.33

13.84

0.62

0.57

1.57



ProbF

0.3746

0.0059

0.2320

0.2481

0.0648

0.2011

0.9022

0.5990

0.0037

0.6580

0.1391

0.0034

0.4423

0.5792

0.2385



Alpha

0.05

0.05

0.05

0.05



Lower

0.0713

0.0373

0.0537

0.0059



} block

+ p } person

+ d } day



Upper

1.5227

0.3224

0.2530

0.0425



Sc ijk = µ + τ age ij + α temp ik + β baseSkTijk + b i

+ ρ j + γ j age ij + ψ j temp ik + θ j baseSkTijk

+ δ k + ζ k age ij + δ k temp ik + φ k baseSkTijk



ij



(15.42)



ik



+ (ρδ) jk + η jk age ij + ν jk temp ik + ω jk baseSkTijk + ε ijk



}



day ∗ person



The PROC MIXED code in Table 15.30 is used to fit Model 15.42 to the comfort

study data set. The REML estimates of the variance components are in the second

part of Table 15.30 and the lower part of the table contains the tests statistics for



© 2002 by CRC Press LLC



50



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 15.31

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

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

for the Slopes of Day Temperature, and a Common Slope for Baseline

Skin Temperature

proc mixed method=reml cl covtest data=comfort;

class block env sex;

model score=env|sex temp sex*temp baseSkt/ddfm=kr;

random block block*sex block*env;

CovParm

Block

Block*sex

Block*Env

Residual



Estimate

0.1445

0.1511

0.0730

0.0186



StdErr

0.1237

0.0682

0.0264

0.0058



ZValue

1.17

2.22

2.77

3.18



ProbZ

0.1213

0.0133

0.0028

0.0007



Effect

Env

sex

Env*sex

temp

temp*sex

baseSkT



NumDF

2

1

2

1

1

1



DenDF

19.9

29.4

20.4

18.6

29.8

23.0



FValue

215.51

13.32

42.09

0.13

10.54

10.68



ProbF

0.0000

0.0010

0.0000

0.7263

0.0029

0.0034



Alpha

0.05

0.05

0.05

0.05



Lower

0.0446

0.0734

0.0401

0.0109



Upper

2.4677

0.4714

0.1729

0.0386



the various factorial effects in the model. Several of the terms corresponding to the

covariate part of the model have significance levels that are larger than 0.05, indicating that some of the model’s parameters are not significantly different from zero.

So, some model building is in order. Using the stepwise deletion process, the first

term to be deleted was age*env*sex (p = 0.9022). Continuing the process, the following terms were deleted in order: age*sex*env, baseskt*sex*env, baseskt*env,

age*env, temp*sex*env, temp*env, age*sex, and age. The model, after reducing the

covariate part by stepwise deletion, is



}

+p }



Sc ijk = µ + α temp ik + β baseSkTijk + b i

+ ρ j + ψ j temp ik



ij



block

person



+ δ k + d ik } day

+ (ρδ) jk + ε ijk



}



(15.43)



day ∗ person



The PROC MIXED code in Table 15.31 fits the reduced model to the data set. The

REML estimates of the variance components in Table 15.31 are not too different

than the REML estimates variance components for the full model (from Table 15.30).



© 2002 by CRC Press LLC



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



51



TABLE 15.32

PROC MIXED Code to Fit the Full Rank Means Model

for the Intercepts and Temperature Slopes

proc mixed method=reml cl covtest data=comfort;

class block env sex;

model score=env*sex

baseSkt sex*temp/noint solution

ddfm=kr;

random block block*sex block*env;

CovParm

Block

Block*sex

Block*Env

Residual



Estimate

0.1445

0.1511

0.0730

0.0186



StdErr

0.1237

0.0682

0.0264

0.0058



ZValue

1.17

2.22

2.77

3.18



ProbZ

0.1213

0.0133

0.0028

0.0007



Alpha

0.05

0.05

0.05

0.05



Lower

0.0446

0.0734

0.0401

0.0109



Upper

2.4677

0.4714

0.1729

0.0386



Effect

Env*sex

Env*sex

Env*sex

Env*sex

Env*sex

Env*sex

baseSkT

temp*sex

temp*sex



Env

I

I

II

II

III

III



sex

F

M

F

M

F

M

F

M



Estimate

6.4634

2.8864

7.6986

3.9108

9.2540

4.9631

0.0331

–0.0222

0.0368



StdErr

1.5620

1.5464

1.5649

1.5420

1.5666

1.5555

0.0101

0.0224

0.0224



DF

42.2

41.3

42.2

41.1

42.0

41.3

23.0

26.1

26.2



tValue

4.14

1.87

4.92

2.54

5.91

3.19

3.27

–0.99

1.64



Probt

0.0002

0.0691

0.0000

0.0151

0.0000

0.0027

0.0034

0.3291

0.1125



Effect

Env*sex

baseSkT

temp*sex



NumDF

6

1

2



DenDF

37.3

23.0

29.9



FValue

86.53

10.30

5.22



ProbF

0.0000

0.0039

0.0114



The tests for the remaining terms of the covariate part of the model have significance

levels less than 0.01. The exception is the coefficient for temp, but since there is a

significant difference between the slopes for males and for females, temp was

retained in the model. A means model with a two-way means model effects representation for the intercepts, a one-way means model effects representation for the

temp slopes for sex, and a common slope for baseSkt is

Sc ijk = µ ik + β baseSkTijk + ψ j temp ik + b i + p ij + d ik + ε ijk



(15.44)



The PROC MIXED code and results from fitting Model 15.44 to the comfort data

are in Table 15.32. The estimate of the slopes for temp are –0.0222 and 0.0368 for

females and males, respectively. Neither temperature slope is significantly different

from zero (significance levels of 0.3291 and 0.1125), but the two slopes are significantly different from each other; thus temperature with different slopes for the two

sexes was included in the model. The REML estimates of the variance components



© 2002 by CRC Press LLC



52



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 15.33

LSMEANS Statements to Provide Adjusted Means for the

Environment by Sex Combinations Evaluated at the Baseline

Skin Temperature of 90.54 and Three Day Temperatures

lsmeans env*sex/diff at temp=50;

lsmeans env*sex/diff at temp=60;

lsmeans env*sex/diff at temp=70;

Env

I

I

II

II

III

III

I

I

II

II

III

III

I

I

II

II

III

III



sex

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M

F

M



baseSkT

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54



temp

50.00

50.00

50.00

50.00

50.00

50.00

60.00

60.00

60.00

60.00

60.00

60.00

70.00

70.00

70.00

70.00

70.00

70.00



Estimate

8.35

7.72

9.58

8.74

11.14

9.80

8.12

8.09

9.36

9.11

10.91

10.16

7.90

8.45

9.14

9.48

10.69

10.53



StdErr

0.25

0.25

0.25

0.25

0.25

0.25

0.19

0.19

0.19

0.19

0.18

0.18

0.33

0.33

0.33

0.33

0.32

0.32



DF

23

23

23

23

23

23

21

21

21

21

20

21

24

24

24

24

24

24



tValue

33.44

30.92

38.20

34.74

43.69

38.45

43.55

43.34

50.29

48.95

59.10

54.96

24.12

25.78

28.02

29.09

33.28

32.70



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

0.0000

0.0000

0.0000



in Table 15.32 are identical to those in Table 15.31, indicating that the two models

are fitting the data identically. Since the regression models are different for the two

levels of sex, the models need to be evaluated at three or more values of temp.

Table 15.33 contains the LSMEANS statements and results for computing the

adjusted means at the mean baseSkt value (90.54°F) and at temp values of 50, 60,

and 70°F. Pairwise comparisons using the LSD approach between the female means

and the male means within each environment at temp values of 50, 60, and 70°F

are in Table 15.34. Females are significantly warmer than males for environments I,

II, and III at temp = 50°F and for environment III at 60°F. Table 15.35 contains the

pairwise comparisons of the levels of environment within each sex evaluated at just

one value of temp, 60°F. Persons exposed to environment I are significantly cooler

than those exposed to environment II which are significantly cooler than those in

environment III. Figure 15.22 is a plot of the adjusted means in Table 15.33 over

the values of temp. The three environment models within each sex are parallel, but

models from different levels of sex are not parallel.

Figure 15.23 contains the JMP® data table with the comfort data set. The variables block, env, and sex are declared to be nominal and the other variables are

declared to be continuous. The fit model table is in Figure 15.24. The terms block,

© 2002 by CRC Press LLC



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



53



TABLE 15.34

Comparisons of Female and Male Means within Each Environment Evaluated

at the Baseline Skin Temperature of 90.54 and Three Day Temperatures

Env

I

I

I

II

II

II

III

III

III



sex

F

F

F

F

F

F

F

F

F



_sex

M

M

M

M

M

M

M

M

M



baseSkT

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54

90.54



temp

50

60

70

50

60

70

50

60

70



Estimate

0.63

0.04

–0.55

0.84

0.25

–0.34

1.34

0.75

0.16



StdErr

0.22

0.17

0.28

0.22

0.17

0.28

0.22

0.17

0.27



df

18.3

13.0

22.8

18.5

13.1

22.8

18.7

12.9

22.6



tValue

2.86

0.21

–1.98

3.79

1.43

–1.23

6.00

4.36

0.58



Probt

0.0103

0.8355

0.0595

0.0013

0.1763

0.2301

0.0000

0.0008

0.5646



TABLE 15.35

Pairwise Comparisons of Environment Means within Males

and Females Evaluated at the Baseline Skin Temperature of 90.54

and Three Day Temperatures

sex

F

F

F

M

M

M



Env

I

I

II

I

I

II



_Env

II

III

III

II

III

III



baseSkT

90.54

90.54

90.54

90.54

90.54

90.54



temp

60

60

60

60

60

60



Estimate

–1.24

–2.79

–1.56

–1.02

–2.08

–1.05



StdErr

0.12

0.12

0.12

0.12

0.12

0.12



df

24.3

24.4

24.4

24.6

24.5

25.1



tValue

–9.99

–22.54

–12.56

–8.26

–16.76

–8.43



Probt

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000



sex*block, and env*block are declared to be random effects by using the attributes

selection menu. The remaining terms in the model are the factorial effects for the

intercepts, the one-way effects model for the slopes of temp, and a common slope

for baseSkt, the same model fit by PROC MIXED to produce the results in

Table 15.31. Deselect the center polynomials option from the model specification

menu of the JMP® fit model screen. The analysis of variance table with the error

mean square is in Figure 15.25. This error term provides an estimate of the day*person variance component. The F-ratio corresponding to the source “MODEL” is not

interpretable since this model sum of squares combines all of the fixed effects and

all of the random effects in one test. Figure 15.26 contains the REML estimates of

the variance components and tests for the fixed and random effects in the model.

The results in Figure 15.26 are a little different than the PROC MIXED results in

Table 15.31, but differences are mainly due to different approximations to the denominator degrees of freedom. Adjusted means could be obtained from JMP® by using

the custom test menu, as was done in Chapter 3, but those results were not included

here.

© 2002 by CRC Press LLC



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