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5 EXAMPLE: EFFECT OF HERBICIDES ON THE YIELD OF SOYBEANS — THREE COVARIATES

5 EXAMPLE: EFFECT OF HERBICIDES ON THE YIELD OF SOYBEANS — THREE COVARIATES

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8



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 4.5

PROC GLM Code, Analysis of Variance Table, and Parameter

Estimates for Model 4.8 for the Golf Ball Data

proc glm data=golf; class shaft;

model dist=shaft ht*shaft wt/noint solution;

Source

Model

Error

Uncorrected Total



df

7

23

30



SS

1231072.35

2211.65

1233284.00



MS

175867.48

96.16



FValue

1828.93



ProbF

0.0000



Source

shaft

ht*shaft

wt



df

3

3

1



SS

7340.41

16184.70

1056.79



MS

2446.80

5394.90

1056.79



FValue

25.45

56.10

10.99



ProbF

0.0000

0.0000

0.0030



Estimate

–598.142

–319.281

–154.622

10.220

6.377

3.743

0.334



StdErr

72.661

81.128

75.572

0.932

1.053

0.954

0.101



tValue

–8.23

–3.94

–2.05

10.97

6.06

3.92

3.32



Probt

0.0000

0.0007

0.0524

0.0000

0.0000

0.0007

0.0030



Parameter

shaft graphite

shaft steel1

shaft steel2

ht*shaft graphite

ht*shaft steel1

ht*shaft steel2

wt



TABLE 4.6

PROC GLM Code to Compute the Adjusted Means for wt =

192.6 lb and ht = 68, 73, and 78 in for the Golf Ball Data

lsmeans shaft/ pdiff at (ht wt)=(68 192.6) stderr;

lsmeans shaft/ pdiff at (ht wt)=(73 192.6) stderr;

lsmeans shaft/ pdiff at (ht wt)=(78 192.6) stderr;

Height

68



73



78



shaft

graphite

steel1

steel2

graphite

steel1

steel2

graphite

steel1

steel2



LSMEAN

161.120

178.642

164.191

212.221

210.527

182.906

263.322

242.412

201.621



StdErr

5.798

5.559

5.255

3.139

3.172

3.159

5.429

6.680

6.152



LSMEAN Number

1

2

3

1

2

3

1

2

3



Three covariates were measured on each plot: the percent silt, the percent clay, and

the amount of organic matter. The first step in the process is to determine if a linear

regression hyper-plane describes the data for each herbicide. After plotting the data

© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



9



TABLE 4.7

p-Values for Pair Wise Comparisons among

the Shaft Means at Three Values of Height

Height

68



73



78



Row Name

1

2

3

1

2

3

1

2

3



_1

0.0385

0.6959

0.7092

0.0000

0.0234

0.0000



_2

0.0385

0.0702

0.7092

0.0000

0.0234



_3

0.6959

0.0702

0.0000

0.0000

0.0000

0.0001



0.0001



FIGURE 4.1 Graph of the estimated regression planes for each type of shaft with points of

comparisons denoted by “o”.



(which is not shown), the model with different slopes for each herbicide-covariate

combination was fit to the data. The model with unequal slopes for each herbicidecovariate combination is

y ij = α i + β1i SILTij + β 2 i CLAYij + β 3i OM ij + ε ij

i = 1, 2, …, 8, and j = 1, 2, …, 12.



(4.9)



Figure 4.2 contains the JMP® fit model window with the model specification. Select

Yield to be the dependent variable. Include Herb, Silt, Clay, OM, Herb*Silt,

© 2002 by CRC Press LLC



10



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 4.8

Yields of Soybeans Treated with Eight Different Herbicides

from Plots with Covariates of Silt, Clay, and om

Herb

1

1

1

1

1

1

1

1

1

1

1

1



Yield

27.2

27.5

17.5

31.9

25

24

39.1

30.2

29.7

25

18.2

54.5



Silt

6

12

1

25

17

18

26

25

26

12

13

28



Clay

14

29

9

32

39

22

20

12

36

9

35

4



om

1.42

1.21

2.28

1.86

0.84

2.54

1.12

1.64

2.36

1.42

1.67

0.63



Herb

5

5

5

5

5

5

5

5

5

5

5

5



Yield

25.2

27.3

30.8

17.5

32.3

15.9

35

32

27.3

33.7

20

21.5



Silt

12

17

24

6

19

16

25

21

29

22

2

25



Clay

20

20

11

33

23

15

16

25

17

23

24

29



om

2.13

2.02

0.85

0.72

0.96

0.84

1.3

2.45

2.51

0.97

1.29

2.47



2

2

2

2

2

2

2

2

2

2

2

2



23.7

18.3

27.1

24.6

17.2

22.1

24.4

39.3

26.5

24.4

35.9

21



18

19

20

9

1

24

20

14

10

19

23

18



9

38

11

29

34

28

1

6

21

37

2

6



1.07

2.11

1.36

1.65

2.21

2.08

1.98

0.78

0.82

2.13

1.31

1.44



6

6

6

6

6

6

6

6

6

6

6

6



11.5

17.5

26.9

13

35.9

9

19.3

28.8

20.2

13.6

18.3

13.5



21

3

1

17

17

18

1

17

23

28

14

11



36

29

24

35

5

36

13

30

9

24

16

8



2.31

2

1.29

2.31

2.48

1.51

1.38

1.01

2.23

2.04

0.75

2.43



3

3

3

3

3

3

3

3

3

3

3

3



24.1

18

28.6

31.6

25.1

20.6

12.1

24.1

10.3

16.6

34.1

23.5



19

6

14

28

28

1

26

21

6

25

29

11



10

31

3

7

3

22

30

33

26

35

2

17



2.05

1.32

1.12

1.92

0.61

1.86

0.99

2.42

2.38

1.65

1.06

1.31



7

7

7

7

7

7

7

7

7

7

7

7



33

26.3

33.3

26.1

23.7

26.1

12.5

27.7

15.9

25.7

23.7

26.9



19

19

5

26

6

6

11

20

10

14

3

28



20

12

5

12

20

7

23

19

12

16

15

12



2.55

0.63

1.51

2.51

2.19

0.72

1.64

1.89

1.16

0.7

2.57

0.65



4

4

4

4

4

4



24.6

16.2

31.4

31.1

23.9

28.5



22

4

22

24

12

6



31

14

24

3

4

5



1.54

0.87

0.81

2.09

1.2

2.41



8

8

8

8

8

8



35

24.1

26.9

35.7

24

9.2



22

8

13

11

23

6



7

11

20

12

36

31



0.92

2.51

1.87

1.25

1.93

1.36



© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



11



TABLE 4.8 (continued)

Yields of Soybeans Treated with Eight Different Herbicides

from Plots with Covariates of Silt, Clay, and om

Herb

4

4

4

4

4

4



Yield

11.8

20

24.1

9.3

15.7

28.1



Silt

22

22

26

1

2

1



Clay

19

17

33

38

20

34



om

2.44

2.15

0.89

1.95

2.34

1.22



Herb

8

8

8

8

8

8



Yield

41

23.1

38.8

22.9

40.5

26.8



Silt

22

1

20

12

17

3



Clay

19

6

18

33

18

25



om

0.64

2.22

0.85

2.34

2.36

0.96



Note: om denotes organic matter.



FIGURE 4.2 JMP® fit window to fit unequal slopes in each of the directions of the three

covariates.



Herb*Clay, and Herb*OM as model effects and click on the “Run Model” button

to fit the model. Using this model, test the hypotheses that the herbicide slopes for

each of the covariates are equal, i.e., test

H 01 : β11 = β12 = … = β18 vs. H a1 : ( not H 01 ) ,

H 02 : β 21 = β 22 = … = β 28 vs. H a 2 : ( not H 02 ) , and

H 03 : β 31 = β 32 = … = β 38 vs. H a 3 : ( not H 03 ) .

The analysis of variance table for the model specified in Figure 4.2 is displayed in

Figure 4.3. The effect SILT*HERB tests the slopes in the SILT direction are equal,

© 2002 by CRC Press LLC



12



Analysis of Messy Data, Volume III: Analysis of Covariance



FIGURE 4.3 Analysis table for the unequal slopes model in each direction of the three

covariates.



the effect CLAY*HERB tests the slopes in CLAY direction are equal, and

OM*HERB tests the slopes are equal in the OM direction. The results of the analysis

in Figure 4.3 indicate there is not sufficient evidence to believe that the slopes are

not equal in each of the covariate directions where the significance levels are 0.1039

for SILT, 0.7124 for CLAY, and 0.2927 for OM. So the next step is to fit a common

slope model to the data. (A stepwise deletion process was carried out where

Herb*Clay was removed first, then Herb*OM, and finally Herb*Silt. Never delete

more than one term from the model at a time while simplifying the covariate part

of the model.)

The common slope model is

y ij = α i + β1 SILTij + β 2 CLAYij + β 3 OM ij + ε ij



(4.10)



and the JMP® Fit Model window with the model specification is displayed in

Figure 4.4. The results of fitting Model 4.10 to the data are in Figures 4.5 and 4.6.

The estimates of the slopes are βˆ 1 = 0.2946, βˆ 2 = –0.2762, and βˆ 3 = –2.2436.

The analysis has established that parallel hyper-planes adequately describe the

data; thus the differences between the treatments are represented by the distances

between the hyper-planes. The adjusted means are in Figure 4.7 which are the

herbicide hyper-planes evaluated at the average values of the covariates, 15.5833,

19.5208, and 1.6116 for SILT, CLAY, and OM, respectively. The adjusted mean for

herbicide 1 is computed by using the estimates in Figure 4.5 as

29.1604 = 29.2354 + 4.3417 + 0.2946 ∗ 15.5833 − 0.2762 ∗ 19.5208 − 2.2436 ∗ 1.6116.

Figure 4.8 is a plot of the least squares means or adjusted means with 95% confidence

intervals. Figure 4.9 contains all pairwise comparisons of the herbicide means. The

cells of the figure contain the difference of the means, the estimated standard error

of the difference, and 95% Tukey simultaneous confidence intervals about the differences. Using the confidence intervals about the differences, the mean of herbicide 8

is significantly different from the means of herbicides 3 and 6, and the mean of

herbicide 1 is significantly different from the mean of herbicide 6. There are no

other differences.

© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



13



FIGURE 4.4 JMP® model specification window for common slopes in each of the directions

of the three covariates.



FIGURE 4.5 Analysis of variance table for the common slopes model in each of the directions

of the three covariates.



4.6 EXAMPLE: MODELS THAT ARE QUADRATIC

FUNCTIONS OF THE COVARIATE

The yield of a process depends on the quantity of X that is present during the

development time. Two additives were added to determine if they could alter the

yield of the process, i.e., increase the yield. The data in Table 4.9 are the yields for

three treatments where TRT = 1 is the control (without additive) TRT = 2 and 3 are

the two additives and X is the covariate corresponding to each experimental unit.

The design structure is a completely randomized design.

On plotting the data, one sees that the relationship between yield and X is

quadratic. The analysis of covariance model used to describe this data is linear and

quadratic in X with the possibility of different intercepts and slopes for the three

treatments. The model used to describe the data is

y ij = α i + β1i X ij + β 2 i X ij2 + ε ij , i = 1, 2, 3 and j = 1, 2, …, 12.

© 2002 by CRC Press LLC



(4.11)



14



Analysis of Messy Data, Volume III: Analysis of Covariance



Parameter Estimates

Term

Estimate Std Error t Ratio Prob> t

Intercept

HERB[1]

HERB[2]

HERB[3]

HERB[4]

HERB[5]

HERB[6]

HERB[7]

SILT

CLAY

OM



29.235454

4.3416627

0.0033234

-3.562265

-1.910561

1.307541

-4.310853

-0.778272

0.2946062

-0.276215

-2.243604



2.483561

1.734624

1.723398

1.732177

1.7287

1.74023

1.741336

1.757427

0.079012

0.062854

1.067268



11.77

2.50

0.00

-2.06

-1.11

0.75

-2.48

-0.44

3.73

-4.39

-2.10



<.0001

0.0142

0.9985

0.0428

0.2722

0.4545

0.0153

0.6590

0.0003

<.0001

0.0385



FIGURE 4.6 Estimates of the parameters for the model with common slopes in each of the

three covariates directions.



Least Squares Means Table

Level

1

2

3

4

5

6

7

8



Least Sq Mean

29.160413

24.822073

21.256485

22.908189

26.126291

20.507897

24.040478

29.728174



Std Error



Mean



1.8526300

1.8421232

1.8503389

1.8470845

1.8578805

1.8589159

1.8739979

1.8501630



29.1500

25.3750

22.3917

22.0583

26.5417

18.9583

25.0750

29.0000



FIGURE 4.7 Least squares means for each of the herbicides.



FIGURE 4.8 Plot of the herbicide least squares means.

© 2002 by CRC Press LLC



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