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