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7 EXAMPLE: COMPARING RESPONSE SURFACE MODELS

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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

22

Analysis of Messy Data, Volume III: Analysis of Covariance

TABLE 4.18

Code and Analysis of Variance Table to Test Each Set of the

Slopes are Equal to Zero

PROC GLM DATA=RESPSUR; CLASS FORMULA;

MODEL FORCE=FORMULA CONC1*FORMULA CONC2*FORMULA

C11*FORMULA C22*FORMULA C12*FORMULA;

Source

Model

Error

Corrected Total

df

11

12

23

SS

456.55

9.17

465.72

MS

41.50

0.76

FValue

54.29

ProbF

0.0000

Source

FORMULA

CONC1*FORMULA

CONC2*FORMULA

C11*FORMULA

C22*FORMULA

C12*FORMULA

df

1

2

2

2

2

2

SS (Type III)

13.16

56.97

17.20

201.94

30.97

111.68

MS

13.16

28.49

8.60

100.97

15.49

55.84

FValue

17.21

37.26

11.25

132.08

20.26

73.04

ProbF

0.0013

0.0000

0.0018

0.0000

0.0001

0.0000

include Formula, Conc1, and Conc2 in the class statement and then use Formula*Conc1*Conc2 in the model statement (see Table 4.17). This process constructs

a means type model where there is one mean for each combination of Formula by

Conc1 by Conc2. For this example there are nine combinations for each formula.

The resulting Error Sum of Squares measures the variability of those experimental

units treated alike, which in this case are those with levels (0,0) of (Conc1,Conc2)

for each formula. There are four center points for each formulation; thus there are

six degrees of freedom for pure error with three degrees of freedom coming from

each formula. Next, Model 4.13 is fit to the data to (1) determine if the quadratic

response surface model adequately describes the data from each formulation, (2) to

obtain estimates of the parameters of the models, and (3) to test the hypotheses that

the sets of slopes are equal to zero, i.e., Hos: βs1 = βs2 = 0 vs. Has: (not Hos), s = 1,

2, …, 5. The SAS® system code to fit Model 4.13 is in Table 4.18 along with the

results. The terms C11, C22, and C12 denote CONC1*CONC1, CONC2*CONC2,

and CONC1*CONC2, respectively. The sum of squares due to lack of fit is computed

by subtracting the sum of squares for pure error (see Table 4.17) from the current

model error sum of squares (Table 4.18). The computations are SS Lack of Fit =

9.17 – 7.04 = 2.13, based on 6 = 12 – 6 degrees of freedom. The value of the statistic

to test the lack of the ability of the model to fit the cell means is F = (2.13/6)/(7.04/6) =

0.303, indicating there is no evidence of lack of fit. The F statistics in Table 4.18

provide tests of the following hypotheses: CONC1*FORMULA tests H0: β11 = β12 =

0 vs. Ha:(not H0:) with significance level 0.0000, CONC2*FORMULA tests H0:

β21 = β22 = 0 vs. Ha:(not H0:) with significance level 0.0018, C11*FORMULA tests

H0: β31 = β32 = 0 vs. Ha:(not H0:) with significance level 0.0000, C22*FORMULA

tests H0: β41 = β42 = 0 vs. Ha:(not H0:) with significance level 0.0001, and C12*FORMULA tests H0: β51 = β52 = 0 vs. Ha:(not H0:) with significance level 0.0000. Each

© 2002 by CRC Press LLC

Multiple Covariants on One-Way Treatment Structure

23

TABLE 4.19

Code and Analysis of Variance Table for Testing

the Parallelism Hypothesis for Each Covariate

PROC GLM DATA=RESPSUR; CLASS FORMULA;

MODEL FORCE = CONC1 CONC2 C11 C22 C12

FORMULA CONC1*FORMULA CONC2*FORMULA C11*FORMULA

C22*FORMULA C12*FORMULA;

Source

Model

Error

Corrected Total

df

11

12

23

SS

456.55

9.17

465.72

MS

41.50

0.76

FValue

54.29

ProbF

0.0000

Source

CONC1

CONC2

C11

C22

C12

FORMULA

CONC1*FORMULA

CONC2*FORMULA

C11*FORMULA

C22*FORMULA

C12*FORMULA

df

1

1

1

1

1

1

1

1

1

1

1

SS (Type III)

56.91

16.61

201.29

26.01

105.42

13.16

0.06

0.60

0.65

4.96

6.27

MS

56.91

16.61

201.29

26.01

105.42

13.16

0.06

0.60

0.65

4.96

6.27

FValue

74.44

21.72

263.31

34.03

137.89

17.21

0.08

0.78

0.85

6.49

8.20

ProbF

0.0000

0.0006

0.0000

0.0001

0.0000

0.0013

0.7764

0.3941

0.3754

0.0256

0.0143

of these sum of squares is based on two degrees of freedom since the hypothesis

corresponds to specifying two parameters are equal to zero. The results of these

F statistics indicate the respective sets of slopes are not all equal to zero.

The next step in the analysis is to check for parallelism in the direction of each

covariate. The SAS® system code for testing for the parallelism hypotheses for each

of the covariates in the model is in Table 4.19. The model includes each of the

individual terms as well as the interactions, e.g., CONC1 and CONC1*FORMULA.

When both terms are included in the model, the resulting F statistics provide tests

that the slopes for the two levels of formula are equal. In this case, CONC1*FORMULA tests H0: β11 = β12 vs. Ha:(not H0:) with significance level 0.7764,

CONC2*FORMULA tests H0: β21 = β22 vs. Ha:(not H0:) with significance level

0.3941, C11*FORMULA tests H0: β31 = β32 vs. Ha:(not H0:) with significance level

0.3754, C22*FORMULA tests H0: β41 = β42 vs. Ha:(not H0:) with significance level

0.0256, and C12*FORMULA tests H0: β51 = β52 vs. Ha:(not H0:) with significance

level 0.0143. Each of these sum of squares is based on one degree of freedom since

they correspond to specifying the difference between two parameters is equal to

zero. The information in Table 4.19 indicates that (1) β41 ≠ β42 (C22*FORMULA)

and 2) β51 ≠ β52 (C12*FORMULA).

The sum of squares corresponding to each specific line in Table 4.19 provides

a conditional sum of squares due to that effect given the other terms in the model.

© 2002 by CRC Press LLC

24

Analysis of Messy Data, Volume III: Analysis of Covariance

TABLE 4.20

Code and Analysis of Variance Table for the Final Model Used

to Compare the Response Surfaces for the Soybean Data

PROC GLM DATA=RESPSUR; CLASS FORMULA;

MODEL FORCE = CONC1 CONC2 C11 FORMULA C22*FORMULA

C12*FORMULA/SOLUTION NOINT P;

Source

Model

Error

Uncorrected Total

df

9

15

24

SS (Type III)

3022.87

10.48

3033.35

MS

335.87

0.70

FValue

480.56

ProbF

0.0000

Source

CONC1

CONC2

C11

FORMULA

C22*FORMULA

C12*FORMULA

df

1

1

1

2

2

2

SS

56.91

16.61

201.29

1570.24

31.95

111.68

MS

56.91

16.61

201.29

785.12

15.98

55.84

FValue

81.42

23.76

288.01

1123.34

22.86

79.90

ProbF

0.0000

0.0002

0.0000

0.0000

0.0000

0.0000

Thus one needs to be careful when deleting more than one term from the model at

a time. If several terms are deleted at once, the fit of the resulting model needs to

be checked, i.e., make sure that the MS Residual does not increase too much, etc.

For this particular problem, a stepwise deletion process was used to simplify the

model which resulted in

y ijkm = µ i + β1 x1 j + β2 x 2 k + β3 x12j + β 4 i x 22k + β5i x1 j x 2 k + ε ijkm

where

i =

j =

k =

m=

(4.14)

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.

The SAS® system code for fitting Model 4.14 to the pill data is in Table 4.20.

The analysis of variance table is in Table 4.20 where the mean square error is

0.70 and it is based on 15 degrees of freedom. The estimates of the parameters of

the model are in Table 4.21. Table 4.22 contains SAS® system code for computing

a test for lack of fit for Model 4.14. The process is to define two new variables, C1 =

CONC1 and C2 = CONC2. Include C1 and C2 in the model in place of CONC1

and CONC2, but include CONC1 and CONC2 in the Class statement. Finally include

CONC1*CONC2*FORMULA in the model. The Type III sum of squares for

CONC1*CONC2*FORMULA (which is the same here as the Type I sum of squares

since CONC1*CONC2*FORMULA is the last term in the model) is the sum of

squares due to deviations of the model from the cells’ means given all of the other

© 2002 by CRC Press LLC

Multiple Covariants on One-Way Treatment Structure

25

TABLE 4.21

Parameter Estimates for the Final Model

Parameter

CONC1

CONC2

C11

FORMULA U3Y

FORMULA X2Z

C22*FORMULA U3Y

C22*FORMULA X2Z

C12*FORMULA U3Y

C12*FORMULA X2Z

Estimate

1.886

1.019

–3.966

12.835

15.040

–2.093

–0.758

4.515

2.745

StdErr

0.209

0.209

0.234

0.374

0.374

0.327

0.327

0.418

0.418

tValue

9.02

4.87

–16.97

34.33

40.23

–6.40

–2.32

10.80

6.57

Probt

0.0000

0.0002

0.0000

0.0000

0.0000

0.0000

0.0350

0.0000

0.0000

TABLE 4.22

Code and Analysis of Variance Table to Provide a Test of Lack

of Fit for the Final Model

PROC GLM DATA=RESPSUR; CLASS FORMULA CONC1 CONC2;

MODEL FORCE = C1 C2 C11 FORMULA C22*FORMULA C12*FORMULA

FORMULA*CONC1*CONC2;

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

C1

C2

C11

FORMULA

C22*FORMULA

C12*FORMULA

FORMULA*CONC1*CONC2

df

0

0

0

1

0

0

9

SS (Type III)

0.00

0.00

0.00

8.41

0.00

0.00

3.45

MS

FValue

ProbF

8.41

7.17

0.0366

0.38

0.33

0.9356

terms are in the model. Thus, the sum of squares corresponding to

CONC1*CONC2*FORMULA is the sum of squares due to the lack of the model

fitting the cell means. The lack of fit sum of squares is 3.45 and the value of the F

statistic is 0.33 with significance level being 0.9356, indicating there is no evidence

that the model fails to fit the data.

Using the estimates of the parameters in Table 4.20, the equations for estimating

the the two response surfaces are

yˆ U 3Y = 12.835 + 1.886 X1 + 1.019 X 2 − 3.966 X12 − 2.093 X 22 + 4.515 X1X 2

© 2002 by CRC Press LLC

26

Analysis of Messy Data, Volume III: Analysis of Covariance

and

yˆ X 2 Z = 15.040 + 1.886 X1 + 1.019 X 2 − 3.966 X12 − 0.758 X 22 + 2.745 X1X 2 .

These response surfaces for the two formulations have common slopes in the X1,

X2, and X21 directions and different slopes in X22 and X1X2 directions, where X1 =

CONC1 and X2 = CONC2.

The combination of the binders at which the response surface estimates the

maximum strength is determined by differentiating the model with respect to X1 and

X2, equating the derivatives to zero, and solving the resulting equations. The two

sets of equations are:

U3Y: − 2(3.966) X1 + 4.515 X 2 = −1.886

4.515 X1 − 2(2.093) X 2 = −1.019

and

X2 Z: − 2(3.966) X1 + 2.745 X 2 = −1.886

2.745 X1 − 2(0.758) X 2 = −1.019 .

The concentrations at which the models estimate the maximum strength occurs are

(1) for U3Y (CONC1 = 0.975, CONC2 = 1.295) and for Z2X (CONC1 = 1.260,

CONC2 = 2.953). The maximum for X2Z occurs outside of the range of experimentation, so one needs to be careful in assessing the usefulness of the estimate.

Table 4.23 contains the predicted values (PFORCE) for each point in the design

space and the two formulations. The maximum observed response means occur at

TABLE 4.23

Predicted Values of Force for the Two Formulations

at Each of the Design Points in the Experiment

U3Y

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

© 2002 by CRC Press LLC

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

PFORCE

14.20

10.09

1.39

2.24

8.39

7.21

3.13

7.57

12.84

12.84

12.84

12.84

X2Z

STE

0.58

0.58

0.58

0.52

0.58

0.58

0.58

0.52

0.37

0.37

0.37

0.37

PFORCE

15.97

14.96

6.70

4.44

10.16

12.08

8.44

9.78

15.04

15.04

15.04

15.04

STE

0.58

0.58

0.58

0.52

0.58

0.58

0.58

0.52

0.37

0.37

0.37

0.37

Multiple Covariants on One-Way Treatment Structure

27

FIGURE 4.11 Predicted regression surface for formula U3Y over the range of concentrations.

FIGURE 4.12 Predicted regression surface for formula X2Z over the range of concentrations.

(CONC1, CONC2) = (1, 1) for each of the formulations. The estimated standard

errors associated with the predicted values are also included (STE). Remember these

are estimated standard errors for the mean of the model and not estimated standard

deviations associated with an individual measurement at a given combination of

CONC1 and CONC2. The major conclusion that can be made by looking the

predicted values is that X2Z produces stronger pills at each of the observed combinations of the two binders.

Figures 4.11 and 4.12 are plots of the predicted response surfaces for each of the

two formulations. To make further comparisons between the two response surfaces, a

set of estimate statements were generated to construct a grid over the range of (CONC1,

CONC2) so that estimates of the differences of the two models could be evaluated

(along with the estimated standard errors). The difference between the two models is

Diff( U 3Y − X 2 Z ) = (µ1 − µ 2 ) + (β 41 − β 42 ) X 22 + (β 51 − β 52 ) X1X 2

© 2002 by CRC Press LLC

28

Analysis of Messy Data, Volume III: Analysis of Covariance

FIGURE 4.13 Surface of the difference between the two treatments’ response surfaces where

the dark areas are regions where the surfaces are not significantly different.

FIGURE 4.14 Contour plot for the surface of the differences between the two treatments’

response surfaces.

which is a comparison that involves three linearly independent parameters (combinations of parameters) (µ1 – µ2), (Β41 – β42), and (β51 – β52). To compare the response

surfaces over the grid, a Scheffe approach was utilized where a value D was computed

as D = 3F .05,3,15 . If the computed difference divided by its estimated standard error

at a grid point was less in absolute value than D, then it was declared that the two

formulations produce similar mean responses at that point. Figure 4.13 is a graph of

the difference of the two response surfaces, where the large symbols indicate grid

points where the response is not significantly different from zero (points where the

two formulations produce similar mean responses). Figure 4.14 is a contour plot of

the differences with contours in the regions where the differences are closest to zero.

© 2002 by CRC Press LLC

Multiple Covariants on One-Way Treatment Structure

29

REFERENCE

Special Issue on the Analysis of Covariance, Biometrics, September 1957, and Biometrics,

September 1982.

EXERCISES

EXERCISE 4.1: The average daily gain ADG during 3 to 9 months of age of

3 breeds of calves was studied. It was thought that ADG is partly an inherited trait;

were determined when they were growing during 3 to 9 months of age and were

used as possible covariates. The data are in the following table. Use the analysis of

covariance strategy to construct an appropriate model and then use the model to

carry out comparisons of the three breeds.

Average Daily Gain Data for Exercise 4.1

Breed 1

Breed 2

Breed 3

© 2002 by CRC Press LLC

2.80

3.36

3.12

2.75

2.82

3.08

2.49

1.83

2.54

2.53

2.87

2.19

3.03

2.60

3.84

3.93

4.17

3.96

4.31

4.45

4.02

4.45

4.38

2.1

2.8

3.0

2.0

2.3

2.8

1.2

1.4

1.8

2.3

1.9

2.3

1.2

1.8

3.0

3.1

3.9

3.3

2.6

3.0

2.6

3.0

3.7

2.4

2.2

1.7

2.1

1.8

2.0

2.1

1.0

1.7

2.0

2.5

1.1

3.0

2.3

2.3

2.0

2.3

2.2

3.2

3.2

2.7

3.3

3.1

30

Analysis of Messy Data, Volume III: Analysis of Covariance

EXERCISE 4.2: Carry out an analysis of covariance for the following data set by

determining the appropriate model and then making the needed treatment comparisons. Y is the response variable and X and Z are the covariates.

Data for Exercise 4.2

Treatment A

X

90.5

79.4

94.7

75.4

95.9

79.6

83.7

96.1

72.0

95.6

© 2002 by CRC Press LLC

Y

23.3

7.0

11.8

21.2

14.8

14.5

16.6

19.5

20.3

21.8

Z

13.4

35.1

34.1

14.0

31.4

27.1

26.5

24.1

15.4

19.6

Treatment B

X

90.1

74.2

93.6

93.0

90.9

85.7

88.2

96.9

89.6

82.2

83.9

73.1

Y

15.7

21.8

14.9

9.7

23.3

11.4

11.1

18.5

12.4

18.1

22.0

20.7

Z

39.9

32.6

41.1

43.7

34.1

41.1

42.5

41.3

42.3

37.0

33.5

33.1

Treatment C

X

75.0

86.6

87.7

91.5

95.2

85.9

79.8

91.2

91.8

Y

14.3

10.3

11.3

4.0

16.4

23.6

14.8

6.1

22.3

Z

48.3

49.1

50.9

42.9

55.8

59.8

51.1

45.6

61.0

C0317ch05 frame Page 123 Monday, June 25, 2001 10:08 PM

5

Two-Way Treatment

Structure and Analysis

of Covariance in a

Completely Randomized

Design Structure

5.1 INTRODUCTION

The main difference between the analysis of a one-way treatment structure and that

of a two-way treatment structure is that the sums of squares for intercepts and slopes

can be partitioned into row treatment effects, column treatment effects, and row

treatment by column treatment interaction effects. One of the objectives of a good

analysis is to determine if a different slope is needed for each treatment combination

and, if not, to determine whether a different slope is needed for each row treatment

and whether a different slope is needed for each column treatment. Once an adequate

model has been determined in terms of the slope parameters, the analysis is completed

by making comparisons of interest between the planes or lines at various selected

values of the covariate. If there are unequal slopes for the levels of the row treatments

or column treatments or both, the analysis of the intercepts provides a comparison

of the regression surfaces at the value of the covariate equal to zero. Great care must

be used in the interpretation of results when there are unequal slopes in two-way

and higher order treatment structures. Several examples involving one covariate are

used to demonstrate these concepts.

5.2 THE MODEL

The cell means model for a two-way treatment structure in a completely randomized

design structure with one covariate is

y ijk = α ij + βijx ijk + ε ijk

i = 1, 2, …, s, j = 1, 2, …, t, k = 1, 2, …, n ij

(5.1)

where the ijth cell consists of the ith level of the row treatment in combination with

the jth level of the column treatment, αij denotes the intercept for the ijth cell, and βij

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