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7 EXAMPLE: BAKING BREAD USING FOUR FACTORS EACH AT TWO LEVELS

7 EXAMPLE: BAKING BREAD USING FOUR FACTORS EACH AT TWO LEVELS

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Analysis of Covariance for Nonreplicated Experiments



17



Loaf Volume

2.5



*



2.0



Probit



1.0



0.0



A

AC



*



1.5



0.5



CD



*

**



*

**

*

*

**

0



1



C



*



2



3



4



5



*



6



7



8



ABS (Treatment Effects of Loaf Volume)



FIGURE 17.4 Half-normal plot of the treatment effects for the loaf volume data without the

covariate.



Loaf Volume

210



ac

* *abd * ad

* bc

* ab

*cd *d*acd

*bcd

*abcd

* bd



Loaf volume



200

190

180



c *

abc*



* a



170



* b



160



* (1)



150

10



20



30

Yeast Viability



FIGURE 17.5 Plot of the loaf volume and yeast viability data.



© 2002 by CRC Press LLC



40



18



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 17.8

PROC GLM Code to Fit the Full Factorial Effects Model

to the Loaf Volume and Yeast Viability Data

proc glm data=values;

model y x=a b c a*b a*c b*c a*b*c d a*d b*d

a*b*d c*d a*c*d b*c*d a*b*c*d/solution;

Parameter

Intercept

a

b

c

a*b

a*c

b*c

a*b*c

d

a*d

b*d

a*b*d

c*d

a*c*d

b*c*d

a*b*c*d



Estimate for Loaf Volume

189.4313

6.2688

–1.9063

4.9938

0.8562

–6.0688

–0.1938

0.8688

–0.0437

–1.9563

–0.5812

–0.1687

–7.0313

1.8063

0.4813

–1.9563



Estimate for Yeast Viability

23.1813

1.2313

–4.2438

2.6313

1.7313

–1.5688

1.3313

2.3563

–1.0313

–1.5563

–0.9563

–1.6563

–1.7313

1.0438

–1.3813

–5.0813



Note: Results are the estimates of the factorial effects.



of the variance for an analysis without the covariate is 24.2552 (analysis not shown).

The estimate of the slope is 0.4114 cm3 per unit of viability score. All of the terms

in the model have significance levels of 0.0015 or less, indicating the selected terms

are non-null effects. The PROC REG code in Table 17.10 was used to investigate

the possibility of additional non-null effects that were not identified by the scatter

plot in Figure 17.6. The options indicate that the first five terms are to be included

in the model and the other terms are to be considered as possible non-null effects

if one or more can be entered with a significance level of 0.15 or less and can stay

in the model if the resulting significance level is less than 0.05, using the variable

selection procedure stepwise. No other variables were included in the model, indicating that the non-null effects selected from the scatter plot are adequate to describe

the data set. This example shows that the half-normal probability plot can identify

the non-null effects to be used in the analysis of covariance, but using the covariate

in the analysis substantially reduces the estimate of the variance. This reduced

variance could be used to compare adjusted means computed for the important effects

in the model. The adjusted means are not computed for this example, but the process

is demonstrated in the last two examples in this chapter.



© 2002 by CRC Press LLC



Analysis of Covariance for Nonreplicated Experiments



19



Loaf Volume

8

Treat Effects For Loaf Volume



A



*



*C



4



*



*



* * **

*



0



*



*



* *



-4



* AC

* CD



-8

-6



-4



-2



0



2



4



Treat Effects For Yeast Viability



FIGURE 17.6 Plot of the treatment effects for loaf volume by treatment effects for yeast

viability.



17.8 EXAMPLE: HAMBURGER PATTIES WITH FOUR

FACTORS EACH AT TWO LEVELS

The data in Figure 17.7 displayed in a JMP® table are the acceptability scores of

hamburger patties made several different ways when the fat content of the batch of

hamburger from which a patty was made is considered as a possible covariate. The

different ways the hamburger patties were made are from a four-way factorial

treatment structure where A is cooking method, B is grind size, C is filler amount,

and D is patty size. The half-normal probability plot in Figure 17.8 indicates that

when the information about the fat content is ignored, ABD is the only important

factorial effect. The scatter plot in Figure 17.9 indicates that the acceptability scores

generally decrease when the level of fat increases, although the fat content is

confounded with the factorial effects. Figure 17.10 is the fit model screen used to

fit the full factorial effects model to the acceptance scores and the fat content data.

Figure 17.11 is a data table display of the estimated factorial effects for both variables, which also includes their scatter plot. The scatter plot in Figure 17.12 is an

enlargement of the scatter plot in Figure 17.11 where the effects not on the common

line are denoted. Thus the possible non-null effects are A, C, AC, and CD. The fit

model screen in Figure 17.13 contains the model specification to fit the fat content

as a covariate along with the factorial effects A, C, AC, and CD. The results of the

least squares fit of the model to the acceptance scores are in Figure 17.14. The

significance levels of the effects in the model are all less than 0.0021, indicating the



© 2002 by CRC Press LLC



20



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 17.9

PROC GLM Code to Fit the Final Model to the Loaf Volume

Data Using Yeast Viability as the Covariate

proc glm data=values; model y=a c a*c c*d

x/solution;

Source

Model

Error

Corrected Total



df

5

10

15



SS

2582.1152

92.7392

2674.8544



MS

516.4230

9.2739



FValue

55.69



ProbF

0.0000



Source

a

c

a*c

c*d

x



df

1

1

1

1

1



SS(III)

519.0192

220.9805

453.2654

610.4328

174.0677



MS

519.0192

220.9805

453.2654

610.4328

174.0677



FValue

55.97

23.83

48.88

65.82

18.77



ProbF

0.0000

0.0006

0.0000

0.0000

0.0015



Estimate

179.8953

5.7623

3.9113

–5.4234

–6.3191

0.4114



StdErr

2.3290

0.7703

0.8013

0.7758

0.7789

0.0950



tValue

77.24

7.48

4.88

–6.99

–8.11

4.33



Probt

0.0000

0.0000

0.0006

0.0000

0.0000

0.0015



Parameter

Intercept

a

c

a*c

c*d

x



TABLE 17.10

PROC REG Code to Use Stepwise Regression to Search for Additional

Non-Null Effects

proc reg data=check; model y =a c ac cd x b ab bc abc d ad bd abd

acd bcd abcd/selection=stepwise include=5 slstay=0.05 slentry=0.15;



selected effects are non-null. The estimate of the slope is –1.94 acceptance score

units per one unit of fat content. The evaluation of possible other non-null effects

is left as an exercise for the reader.



17.9 EXAMPLE: STRENGTH OF COMPOSITE

MATERIAL COUPONS WITH TWO COVARIATES

This example involves two covariates and thus demonstrates the process needed to

be followed to evaluate the effect of more than one covariate. The same process can

be used when it is suspected that the slopes are not equal for the levels of one of

the factors, as is demonstrated in Section 17.10. The data in Table 17.11 are strength

of a coupon made of composite material (Y), average thickness of a coupon (X),

© 2002 by CRC Press LLC



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