6 EXAMPLE: MILLING FLOUR USING THREE FACTORS EACH AT TWO LEVELS
Tải bản đầy đủ - 0trang
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