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6 EXAMPLE: MODELS THAT ARE QUADRATIC FUNCTIONS OF THE COVARIATE

6 EXAMPLE: MODELS THAT ARE QUADRATIC FUNCTIONS OF THE COVARIATE

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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



Multiple Covariants on One-Way Treatment Structure



15



LSMeans Differences Tukey HSD

Alpha= 0.050



Q= 3.10773

LSMean[i]



Mean[i]-Mean[i] 1



2



3



4



5



6



7



8



Std Err Dif

Lower CL Dif

Upper CL Dif



1



0 4.33834 7.90393 6.25222 3.03412 8.65252 5.11993 -0.5678

0 2.61267 -2.6116 2.62354 2.60354 2.6249 2.66077 2.62879

0 -3.7811 -0.2122 -1.901 -5.057 0.49501 -3.149 -8.7374

0 12.4578 16.0201 14.4055 11.1252 16.81 13.3889 7.60183

0 3.56559 1.91388 -1.3042 4.31418 0.7816 -4.9061

-4.3383

0 2.60555 2.61271 2.61438 2.62435 2.62288 2.61434

2.61267

0 -4.5318 -6.2057 -9.429 -3.8416 -7.3696 -13.031

-12.458

0 11.663 10.0335 6.82058 12.47 8.93281 3.21858

3.78114



2



3



-7.9039 -3.5656

-2.6116 2.60555



0 -1.6517 -4.8698 0.74859 -2.784 -8.4717

0 2.62599 2.61022 2.63655 2.63413 2.62904



-11.663

4.53178

-1.9139

2.61271

-10.034

6.20574



0 -9.8126 -12.982 -7.4451 -10.97 -16.642

0 6.50918 3.24206 8.9423 5.40219 -0.6013

1.6517

0 -3.2181 2.40029 -1.1323

-6.82

0 2.63143 2.60998 2.62741 2.60371

2.62599

0 -11.396 -5.7108 -9.2976 -14.912

-6.5092

0 4.95969 10.5114 7.03302 1.27165

9.81259



-16.02

0.21223

-6.2522

2.62354

-14.405

1.90103



4



LSMeans Differences Tukey HSD

LSMean[i]

Mean[i]-Mean[i] 1

Std Err Dif



2



3



4



5



6



7



8



Lower CL Dif

LSMean[



Upper CL Dif



5



6



7



8



0 5.61839 2.08581 -3.6019

-3.0341 1.30422 4.86981 3.2181

0 2.63483 2.66455 2.63645

2.60354 2.61438 2.61022 2.63143

-11.125 -6.8206 -3.2421 -4.9597

0

-2.57 -6.1949 -11.795

5.05698 9.42901 12.9817 11.3959

0 13.8068 10.3665 4.59149

-8.6525 -4.3142 -0.7486 -2.4003 -5.6184

0 -3.5326 -9.2203

2.6248 2.62435 2.63655 2.60998 2.63483

0 2.65408 2.6166

-16.81 -12.47 -8.9423 -10.511 -13.807

0 -11.781 -17.353

0 4.71558 -1.088

-0.495 3.8416 7.44512 5.71084 2.56997

-5.1199 -0.7816 2.78399 1.13229 -2.0858 3.53258

0 -5.6877

0 2.623

2.66077 2.62288 2.63413 2.62741 2.66455 2.65408

-13.389 -8.9328 -5.4022 -7.033 -10.367 -4.7156

0 -13.839

0 2.4639

3.14904 7.36961 10.9702 9.2976 6.1949 11.7807

0

0.56776 4.9061 8.47169 6.81999 3.60188 9.22028 5.6877

0

2.62879 2.61434 2.62904 2.60371 2.63645 2.6166 2.623

-7.6018 -3.2186 0.30133 -1.2716 -4.5915 1.08795 -2.4639

0

0

8.73736 13.0308 16.642 14.9116 11.7953 17.3526 13.8393



FIGURE 4.9 Comparisons of the herbicide least squares means using Tukey’s HSD multiple

comparison procedure.



At this point it has already been established that there is a relationship between Yield

and X and X2, i.e., the corresponding slopes are different than zero (analysis not

shown). The next step in the analysis of covariance is to decide whether common

© 2002 by CRC Press LLC



16



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 4.9

Yields for Three Treatments and the Values

of the Covariate

Treatment 1

Yield

3.7

4.7

5.3

5.5

5.7

3.8

1.0

3.3

5.3

5.3

6.1

5.8



X

2.2

3.5

4.1

4.1

7.4

3.0

0.5

2.4

8.4

8.5

8.0

6.3



Treatment 2

Yield

7.2

2.5

4.3

6.9

8.0

3.7

5.2

8.0

6.9

4.3

6.8

7.6



X

5.8

9.9

1.2

6.0

5.5

9.7

8.6

5.0

2.6

8.4

3.0

5.2



Treatment 3

Yield

5.7

6.1

6.4

9.6

9.4

8.2

10.2

6.0

9.4

5.0

7.5

5.3



X

8.8

9.0

8.8

4.1

2.4

4.1

4.4

8.9

5.3

9.5

1.1

0.1



linear slopes and/or common quadratic slopes are possible simplifications of the

model. The first hypotheses to be tested are

H 01 : β11 = β12 = β13 vs. H: ( not H 01 ) and

H 02 : β 21 = β 22 = β 23 vs. H: ( not H 02 ) .

Table 4.10 contains the SAS® system code statements necessary to test H01 and H02

as well as the results. The significance levels for the two tests are 0.6463 for X*TRT

and 0.1364 for X*X*TRT. Thus, given there are unequal slopes for the X2 terms,

then the model does not have to be unequal slopes for the X terms, or given there

are unequal slopes for the X terms, then the model does not have to be unequal

slopes for the X2 terms. Since the significance level for X*TRT is larger than that

for X*X*TRT, X*TRT was deleted. The resulting model has a common slope for

the X term and unequal slopes for the X2 term. Table 4.11 contains the SAS® system

code for fitting the above model to provide a test of the equality of the quadratic

term slopes. The results are in the lower part of Table 4.11. The significance level

corresponding to X*X*TRT is 0.0000, indicating that the slopes are not equal for

the three treatments in the X2 direction.

The final model with a common slope for the X direction and unequal slopes

in the X2 direction is

y ij = α i + β1 X ij + β 2 i X ij2 + ε ij , i = 1, 2, 3 and j = 1, 2, …, 12.



(4.12)



The code in Table 4.12 fits Model 4.12 and also contains the analysis of variance

table. Since the model used the NOINT option, the F statistic corresponding to TRT

© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



17



TABLE 4.10

Code to Fit a Model with Unequal Slopes in Both

the X and X2 Directions and Provide Tests

for Equality of Each of the Two Sets of Slopes

PROC GLM DATA=QUAD; CLASS TRT;

MODEL YIELD= TRT X X*TRT X*X X*X*TRT/SS3;

Source

Model

Error

Corr Total



df

8

27

35



SS

140.22

6.53

146.75



MS

17.53

0.24



FValue

72.53



ProbF

0.0000



Source

TRT

X

X*TRT

X*X

X*X*TRT



df

2

1

2

1

2



SS (Type III)

16.00

47.03

0.21

50.30

1.04



MS

8.00

47.03

0.11

50.30

0.52



FValue

33.11

194.58

0.44

208.14

2.15



ProbF

0.0000

0.0000

0.6463

0.0000

0.1364



TABLE 4.11

Code to Fit a Model with a Common Slope

in the X Direction and Unequal Slopes in the

X2 Direction and Test Equality of the X2 Slopes

PROC GLM DATA=QUAD; CLASS TRT;

MODEL YIELD= TRT X X*X X*X*TRT /SS3;

Source

Model

Error

Corr Total



df

6

29

35



SS

140.01

6.74

146.75



MS

23.33

0.23



FValue

100.41



ProbF

0.0000



Source

TRT

X

X*X

X*X*TRT



df

2

1

1

2



SS (Type III)

73.63

49.50

53.86

17.67



MS

36.81

49.50

53.86

8.83



FValue

158.41

212.98

231.76

38.01



ProbF

0.0000

0.0000

0.0000

0.0000



provides a test H0: α1 = α2 = α3=0 vs. Ha:(not H0). The F statistic corresponding to

X provides a test of H0: β1 = 0 vs. Ha:(not H0) and the F statistic corresponding to

X*X*TRT provides a test of H0: β1 = β2 = β3 = 0 vs. Ha:(not H0). The significance

levels are very small, indicating there is sufficient evidence to reject the null hypotheses. The estimates of the parameters of the model are in Table 4.13.

Table 4.14 contains the code to provide estimates of the regression lines at several

values of X. The first statement, LSMEANS TRT/STDERR PDIFF, provides

adjusted means that are not interpretable since they are computed by evaluating the

© 2002 by CRC Press LLC



18



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 4.12

Code to Fit Model 4.12 with Analysis of Variance

Table

PROC GLM DATA=QUAD; CLASS TRT;

MODEL YIELD=TRT X X*X*TRT /NOINT SOLUTION SS3;

Source

Model

Error

Uncor Tot



df

7

29

36



SS

1432.41

6.74

1439.15



MS

204.63

0.23



FValue

880.49



ProbF

0.0000



Source

TRT

X

X*X*TRT



df

3

1

3



SS (Type III)

96.98

49.50

91.60



MS

32.33

49.50

30.53



FValue

139.10

212.98

131.38



ProbF

0.0000

0.0000

0.0000



TABLE 4.13

Estimates of the Parameters of Model 4.12

Parameter

TRT 1

TRT 2

TRT 3

X

X*X*TRT 1

X*X*TRT 2

X*X*TRT 3



Estimate

–0.0102

2.9889

5.3598

1.8520

–0.1432

–0.1894

–0.2000



StdErr

0.3224

0.3885

0.3163

0.1269

0.0133

0.0112

0.0122



tValue

–0.03

7.69

16.94

14.59

–10.75

–16.88

–16.45



Probt

0.9751

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000



model at the average value of X (5.4389) and the average value of X2 (37.9478),

which does not correspond to any points on the regression models since the square

of the mean of X is 5.43892 = 29.5816. The estimate statements in Table 4.15 are

included to demonstrate the computation of the LSMEANS. The last estimate statement in Table 4.15 is used to provide the computation for Treatment 1 as used by

LSMEANS TRT/STDERR PDIFF. The average value of X is 5.4388889 and the

average value of X2 is 37.9477778, while the square of the average value of X is

29.581512665. The last statement uses 37.9477778 for X2 in the computation to

provide a value of 4.630. That is the same result corresponding to LSMEAN in

Table 4.14 for Treatment 1. Thus the usual LSMEAN statement provides incorrect

adjusted values. The third estimate statement in Table 4.15 uses the average value

of X and the square of the average value of X in the computations, providing 5.828

as the adjusted mean for Treatment 1. The last two LSMEAN statements in

Table 4.14, LSMEANS TRT/PDIFF at MEANS and LSMEANS TRT/PDIFF at

X=5.4388889, use the correct computations and provide the adjusted mean for

Treatment 1 of 5.828. The other two LSMEAN statements are used to obtain adjusted

means at a large value of X (X = 9) and a small value of X (X = 1). The first two

© 2002 by CRC Press LLC



Multiple Covariants on One-Way Treatment Structure



19



TABLE 4.14

Code and Results for Computing Adjusted Means

LSMEANS

LSMEANS

LSMEANS

LSMEANS

LSMEANS



TRT/STDERR PDIFF;

TRT/PDIFF AT X=1;

TRT/PDIFF AT X=9;

TRT/PDIFF AT MEANS;

TRT/PDIFF AT X=5.4388889;



LSMEAN

Incorrect

X=1



X=9



MEANS



X=5.4388889



TRT

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3



LSMEAN

4.63

5.88

7.84

1.70

4.65

7.01

5.06

4.32

5.83

5.83

7.46

9.52

5.83

7.46

9.52



RowName

1

2

3

1

2

3

1

2

3

1

2

3

1

2

3



_1

0.0000

0.0000

0.0000

0.0000

0.0590

0.0493

0.0000

0.0000

0.0000

0.0000



_2

0.0000

0.0000

0.0000

0.0000

0.0590

0.0000

0.0000

0.0000

0.0000



_3

0.0000

0.0000

0.0000

0.0000

0.0493

0.0000

0.0000

0.0000

0.0000

0.0000



0.0000



TABLE 4.15

Estimate Statements and Results Demonstrating

the Computation of the LSMEANS

ESTIMATE ‘TRT 1 AT X=1’ TRT 1 0 0 X 1 X*X*TRT 1 0 0;

ESTIMATE ‘TRT 1 AT X=9’ TRT 1 0 0 X 9 X*X*TRT 81 0 0;

ESTIMATE ‘TRT 1 AT X=5.4388889’ TRT 1 0 0 X 5.4388889

X*X*TRT 29.581512665 0 0;

ESTIMATE ‘TRT 1 LSM’ TRT 1 0 0 X 5.4388889 X*X*TRT

37.9477778 0 0;

Parameter

TRT 1 AT X=1

TRT 1 AT X=9

TRT 1 AT X=5.4388889

TRT 1 LSM



Estimate

1.699

5.062

5.828

4.630



StdErr

0.244

0.312

0.163

0.145



tValue

6.95

16.21

35.74

31.83



Probt

0.0000

0.0000

0.0000

0.0000



estimate statements in Table 4.15 are also used to demonstrate the computations of

the adjusted means at X = 1 and X = 9. Pairwise comparisons of the treatment means

for each set of adjusted means are in the columns labeled _1, _2, and _3 of Table 4.14.

The graph in Figure 4.10 displays the data and estimated regression models for the

three treatments with vertical lines at X = 1, X = 5.4388889 (the mean of X), and

© 2002 by CRC Press LLC



Analysis of Messy Data, Volume III: Analysis of Covariance



Yield



20



12

11

10

9

8

7

6

5

4

3

2

1

0



o

o



o

o



* *



o



*



++ +



+



o



o



** *



+



**



o

+ + ++ ooo

* o



+



*



*

*



+

0



1



2



3



4



5



6



7



8



* * *



Data

Model



9



10



Concentration of X

+ ++



Data

Model



TRT 1

TRT 2



ooo



Model

Data



TRT 1

TRT 3



TRT 2

TRT 3



FIGURE 4.10 Graph of the quadratic regression models with the data points for Example 4.5.



X = 9. The pairwise comparison significance levels in Table 4.14 indicate the three

regression lines are significantly different at X = 1 and at the mean of X, while at

X = 9, Treatment 1 is not signicantly different from the other two treatments while

Treatment 3 provides a significantly higher response than Treatment 2.



4.7 EXAMPLE: COMPARING RESPONSE SURFACE MODELS

A drug company had two formulation processes with which to make a certain type

of pill and they wanted to determine the concentrations of two binders which produce

the stronger pills. For each formulation, a treatment structure with design points

from a two factor rotatable central composite design with four center points in a

completely randomized design structure was used to collect information to study

the relationship between strength and the concentrations of the two binders. The

force (lb/in.) required to fracture the pill is the dependent measure. The data for the

two formulations, coded U3Y and X2Z, the force, and the coded concentrations of

the two binders (CONC1, CONC2) are in Table 4.16. The objectives of the experiment are (1) for each formulation determine an adequate model to describe the data

and estimate the combination of the binders that will produce a pill with maximum

strength and (2) to compare the response surfaces via analysis of covariance.

A quadratic response surface model was selected to describe the responses for

each of the formulations and is expressed as

y ijkm = µ i + β1i x1 j + β 2 i x 2 k + β 3i x ij2 + β 4 i x 22k + β 5i x ij x 2 k + ε ijkm



© 2002 by CRC Press LLC



(4.13)



Multiple Covariants on One-Way Treatment Structure



21



TABLE 4.16

Strength of Pills Made by Two Formulations

with Concentrations of Two Binders

CONC1

1.00

0.00

–1.00

–1.41

–1.00

0.00

1.00

1.41

0.00

0.00

0.00

0.00



CONC2

1.00

1.41

1.00

0.00

–1.00

–1.41

–1.00

0.00

0.00

0.00

0.00

0.00



U3Y Force

14.14

10.64

1.47

2.28

8.53

6.57

3.14

8.16

13.58

11.73

13.07

12.24



X2Z Force

16.02

14.07

7.20

3.85

10.30

12.78

8.14

9.45

13.98

15.66

16.83

14.41



TABLE 4.17

Analysis of Variance Table to Provide the Estimate of Pure

Error for the Response Surface Models Used to Compare

Pill Strength

PROC GLM DATA=RESPSUR; CLASS FORMULA CONC1 CONC2;

MODEL FORCE = FORMULA*CONC1*CONC2;



where

i =

j =

k =

m=



Source

Model

Error

Corrected Total



df

17

6

23



SS

458.69

7.04

465.72



MS

26.98

1.17



FValue

23.01



ProbF

0.0004



Source

Formula*Conc1*Conc2



df

17



SS(Type III)

458.69



MS

26.98



FValue

23.01



ProbF

0.0004



1, 2 for the two formulations

1, 2, …, 5, for the levels of x1 (CONC1),

1, 2, …, 5, for the levels of x2 (CONC2),

1 or 4 for the replications per formulation.



Since there are four replications of the center point for each formulation, the

variation between these observations within a formulation provides an estimate of

pure error (or a model free estimate of the variance). The estimate of the pure error

is σˆ 2PE = 1.17, which is the Mean Square Error from the analysis of variance in

Table 4.17. To compute the estimate of 2PE using PROC GLM of the SASđ system,



â 2002 by CRC Press LLC



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