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6 EXAMPLE: MILLING FLOUR USING THREE FACTORS EACH AT TWO LEVELS

6 EXAMPLE: MILLING FLOUR USING THREE FACTORS EACH AT TWO LEVELS

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12



Analysis of Messy Data, Volume III: Analysis of Covariance



Flour Milling

2.5



Probit



2.0



*C



1.5

1.0

0.5

0.0



*



*

0



*



2



*



*



*



4



6



8



10



12



14



ABS(Y)



FIGURE 17.1 Half-normal plot of the milled flour data without the covariate.



TABLE 17.2

PROC REG Code to Fit Model with Main Effect for C

with Analysis of Variance and Parameter Estimates

proc reg data=values;

model y=c;

Source

Model

Error

Corrected Total



df

1

6

7



SS

1255.0050

754.2700

2009.2750



MS

1255.0050

125.7117



FValue

9.98



ProbF

0.0196



Variable

Intercept

c



df

1

1



Estimate

136.6750

–12.5250



StdErr

3.9641

3.9641



tValue

34.48

–3.16



Probt

0.0000

0.0196



fit the full three-way factorial effects model to the pounds of flour data (y) and the

percent small kernels data (x). The estimates of the factorial effects for each of the

variables are in lower part of Table 17.4. Figure 17.3 is a scatter plot of the factorial

effects of pounds of flour milled by the factorial effects of the percent undersized

kernels. The observation from Figure 17.3 is that B and AC are possible non-null

effects, while C is not a possible non-null effect since its point is on the line drawn

through the other null effects points. Thus, B and AC are identified as possible nonnull effects and the PROC GLM code in Table 17.5 fits a model with the two non-null

effects and the covariate x (percent of undersized kernels) to the data set. The estimate

of the variance for this model is 7.0329 and the significance levels corresponding to

© 2002 by CRC Press LLC



Analysis of Covariance for Nonreplicated Experiments



13



TABLE 17.3

PROC REG Code to Fit Model with Main Effect

of C and Covariate with Analysis of Variance Table

and Parameter Estimates

proc reg data=values;

model y=c x;

Source

Model

Error

Corrected Total



df

2

5

7



SS

1255.3943

753.8807

2009.2750



MS

627.6972

150.7761



FValue

4.16



ProbF

0.0862



Variable

Intercept

c

x



df

1

1

1



Estimate

134.9436

–12.3450

0.0382



StdErr

34.3479

5.6027

0.7515



tValue

3.93

–2.20

0.05



Probt

0.0111

0.0788

0.9614



Flour Milling

160



Pounds of Flour



* (1)

* ac



140



*



120



c



*



*



b



*



bc



a



* ab



* abc

100

30



40



50



60



Percent Undersized Kernels



FIGURE 17.2 Plot of pounds of flour by percent undersized kernels.



the three terms in the model are all less than 0.0004, indicating all terms are needed

in the model. The estimate of the slope is 2.1798, indicating the model predicts that

there is a 2.1798-lb increase in the amount of flour milled for each 1% increase in

the percent of undersized kernels. A stepwise regression process was used to determine if there are any other possible non-null effects. Using the stepwise method, no

other variables would be entered in the model with a significance level of 0.15 or

less. Thus the use of this analysis of covariance process discovered the important

effects in the model are B and AC instead of C, the effect that would be used if the

© 2002 by CRC Press LLC



14



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 17.4

PROC GLM Code to Compute the Estimates of the

Effects for the Response Variable and the Covariate

proc glm data=values;

model y x=a b c a*b a*c b*c a*b*c/solution;

Parameter

Intercept

a

b

c

a*b

a*c

b*c

a*b*c



Estimate from y

136.6750

–0.0750

–5.0750

–12.5250

–3.4750

3.0750

–4.1250

–5.4750



Estimate from x

45.3375

–0.4875

2.6375

–4.7125

–1.6875

–3.7875

–1.5125

–2.5875



Flour Milling



Treat Effects For Y



7



*



AC



0



*

*



-7



-14



**



*B



*C

-5



-4



-3



-2



-1



0



1



2



3



Treat Effects For X



FIGURE 17.3 Plot of the factorial effects of pounds of flour (y) by the effects of percent

undersized kernels (x).



covariate information were to be ignored. Table 17.6 consists of the PROC GLM

code where C is included with the non-null effects and the covariate. The significance

level corresponding to C is 0.1766, indicating that C is not likely to be a non-null

effect. The estimate of the variance using the covariate (7.0329) is considerably less

than the estimate of the variance from the model without the covariate (125.7117).

Thus the covariate is important in describing the mean of the number of pounds

milled in the presence of the factorial effects.

© 2002 by CRC Press LLC



Analysis of Covariance for Nonreplicated Experiments



15



TABLE 17.5

PROC GLM Code to Fit the Model with the Main Effect

of B, the A*C Interaction Term, and the Covariate with

the Analysis of Variance Table and Parameter Estimates

proc glm data=values; model y=b a*c x/solution;

Source

Model

Error

Corrected Total



df

3

4

7



SS

1981.1435

28.1315

2009.2750



MS

660.3812

7.0329



FValue

93.90



ProbF

0.0004



Source

b

a*c

x



df

1

1

1



SS(III)

901.2185

881.7723

1699.4535



MS

901.2185

881.7723

1699.4535



FValue

128.14

125.38

241.64



ProbF

0.0003

0.0004

0.0001



Estimate

23.8063

–11.6411

12.5041

2.4895



StdErr

7.3211

1.0284

1.1167

0.1602



tValue

3.25

–11.32

11.20

15.54



Probt

0.0313

0.0003

0.0004

0.0001



Parameter

Intercept

b

a*c

x



TABLE 17.6

PROC GLM Code to Fit the Model with the Main Effects of

B and C, the A*C Interaction Term, and the Covariate with

the Analysis of Variance Table and Parameter Estimates

proc glm data=values; model y=b c a*c x/solution;

Source

Model

Error

Corrected Total



df

4

3

7



SS

1995.4365

13.8385

2009.2750



MS

498.8591

4.6128



FValue

108.15



ProbF

0.0014



Source

b

c

c*a

x



df

1

1

1

1



SS(III)

594.5822

14.2930

469.2961

458.7415



MS

594.5822

14.2930

469.2961

458.7415



FValue

128.90

3.10

101.74

99.45



ProbF

0.0015

0.1766

0.0021

0.0021



Estimate

37.8477

–10.8243

–2.2526

11.3310

2.1798



StdErr

9.9391

0.9534

1.2797

1.1234

0.2186



tValue

3.81

–11.35

–1.76

10.09

9.97



Probt

0.0318

0.0015

0.1766

0.0021

0.0021



Parameter

Intercept

b

c

c*a

x



© 2002 by CRC Press LLC



16



Analysis of Messy Data, Volume III: Analysis of Covariance



TABLE 17.7

Data on Loaf Volume from a Four-Way

Factorial Treatment Structure with One

Covariate, Yeast Viability

a

–1

–1

–1

–1

–1

–1

–1

–1

1

1

1

1

1

1

1

1



b

–1

–1

–1

–1

1

1

1

1

–1

–1

–1

–1

1

1

1

1



c

–1

–1

1

1

–1

–1

1

1

–1

–1

1

1

–1

–1

1

1



d

–1

1

–1

1

–1

1

–1

1

–1

1

–1

1

–1

1

–1

1



Loaf Volume

160.2

187.4

207.0

189.1

162.5

178.3

195.3

185.5

196.0

201.0

200.0

190.0

191.1

199.0

203.7

184.8



Yeast Viability

16.4

29.0

39.8

26.5

12.5

13.1

17.0

21.3

35.3

23.8

18.5

30.1

15.2

19.1

39.0

14.3



17.7 EXAMPLE: BAKING BREAD USING FOUR

FACTORS EACH AT TWO LEVELS

The data in Table 17.7 are from a baking experiment where loaf volume (y) in cm3

is the response variable and yeast viability (x) of dough used to make a loaf of bread

is a covariate. The treatment structure consists of four factors each at two levels

where a, b, c, and d denote mixing time, rising time, mixing temperature, and rising

temperature, respectively. The half-normal probability plot of the factorial effects

computed from loaf volume is in Figure 17.4. The half-normal probability plot

indicates that A, C, AC, and CD are the important factorial effects in the data. The

scatter plot in Figure 17.5 provides a visual suggestion that there is an increase in

the loaf volume as the level of yeast viability increases, although this conclusion is

made with the knowledge that the factorial effects are confounded with the level of

yeast viability. The full four-way factorial model is fit to the loaf volume and viability

data by using the PROC GLM code in Table 17.8. The lower part of Table 17.8

contains the estimates of the factorial effects. The scatter plot of the factorial effects

of loaf volume and the factorial effects of yeast viability is in Figure 17.6. The

observation from the scatter plot is that A, C, AC, and CD are possible non-null

effects, the same as identified by the half-normal probability plot in Figure 17.4.

Table 17.9 contains the PROC GLM code to fit the model with the terms A, C, AC,

CD, and yeast viability (x). The estimate of the variance is 9.2739, while the estimate



© 2002 by CRC Press LLC



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



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