PracticeProblems_FractionalFactorialDesigns
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A 16-run fractional factorial experiment in nine factors was conducted by Chrysler Motors Engineering and described in the article “Sheet Molded Compound Process Improvement,” by P.I. Hsieh and D.E. Goodwin (
Fourth
Symposium on Taguchi Methods
, American Supplier Institute, Dearborn, MI, 1986, pp. 13-21). The purpose was to reduce the number of defects in the finish of sheet-molded grill opening panels. The design, and the resulting number of defects, c
, observed on each run, is shown in Table P8.10. This is a resolution III fraction with generators E=BD, F=BCD, G=AC, H=ACD
, and J=AB.
Table P8.10
Run A B C D E F G H J c c
F&T’s Modification 1 - - - - + - + - + 56 7.48 7.52 2 + - - - + - - + - 17 4.12 4.18 3 - + - - - + + - - 2 1.41 1.57 4 + + - - - + - + + 4 2.00 2.12 5 - - + - + + - + + 3 1.73 1.87 6 + - + - + + + - - 4 2.00 2.12 7 - + + - - - - + - 50 7.07 7.12 8 + + + - - - + - + 2 1.41 1.57 9 - - - + - + + + + 1 1.00 1.21 10 + - - + - + - - - 0 0.00 0.50 11 - + - + + - + + - 3 1.73 1.87 12 + + - + + - - - + 12 3.46 3.54 13 - - + + - - - - + 3 1.73 1.87 14 + - + + - - + + - 4 2.00 2.12 15 - + + + + + - - - 0 0.00 0.50 16 + + + + + + + + + 0 0.00 0.50 (a)
Find the defining relation and the alias relationships in this design. I = ABJ = ACG = BDE = CEF = DGH = FHJ = ABFH = ACDH = ADEJ = AEFG = BCDF = BCGJ = BEGH = CEHJ = DFGJ = ABCEH = ABDFG = ACDFJ = ADEFH = AEGHJ = BCDHJ = BCFGH = BEFGJ = CDEGJ = ABCDEG = ABCEFJ = ABDGHJ = ACFGHJ = BDEFHJ = CDEFGH = ABCDEFGHJ (b)
Estimate the factor effects and use a normal probability plot to tentatively identify the important factors. Residual
Normal % probability
Normal plot of residuals
-0.189813
-0.106313
-0.0228125
0.0606875
0.144187
1
5
10
20
30
50
70
80
90
95
99
2
2
Predicted
Residuals
Residuals vs. Predicted
-0.189813
-0.106313
-0.0228125
0.0606875
0.144187
1.08
1.18
1.27
1.37
1.47
145
The effects are shown below in the Design Expert
output. The normal probability plot of effects identifies factors A
, D
, F
, and interactions AD
, AF
, BC
, BG
as important. Design Expert Output
Term Effect SumSqr % Contribtn Model Intercept Model A -9.375 351.562 7.75573 Model B -1.875 14.0625 0.310229 Model C -3.625 52.5625 1.15957 Model D -14.375 826.562 18.2346 Error E 3.625 52.5625 1.15957 Model F -16.625 1105.56 24.3895 Model G -2.125 18.0625 0.398472 Error H 0.375 0.5625 0.0124092 Error J 0.125 0.0625 0.0013788 Model AD 11.625 540.563 11.9252 Error AE 2.125 18.0625 0.398472 Model AF 9.875 390.063 8.60507 Error AH 1.375 7.5625 0.166834 Model BC 11.375 517.563 11.4178 Model BG -12.625 637.562 14.0651 Lenth's ME 13.9775 Lenth's SME 28.3764 (c)
Fit an appropriate model using the factors identified in part (b) above. The analysis of variance and corresponding model is shown below. Factors B
, C
, and G
are included for hierarchal purposes. Design Expert Output
Response:
c
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Mean
F
Source
Squares
DF
Square
Value
Prob > F
Model 4454.13 10 445.41 28.26 0.0009 significant A
351.56
1
351.56
22.30
0.0052
B
14.06
1
14.06
0.89
0.3883
C
52.56
1
52.56
3.33
0.1274
D
826.56
1
826.56
52.44
0.0008
F
1105.56
1
1105.56
70.14
0.0004
G
18.06
1
18.06
1.15
0.3333
AD
540.56
1
540.56
34.29
0.0021
DESIGN-EXPERT Plot
c
A: A
B: B
C: C
D: D
E: E
F: F
G: G
H: H
J: J
N o rm a l p lo t
N o rm a l % p ro b a b ility
E ffe c t
-1 6 .6 2
-9 .5 6
-2 .5 0
4 .5 6
1 1 .6 2
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
A
D
F
AD
AF
B C
B G
146
AF
390.06
1
390.06
24.75
0.0042
BC
517.56
1
517.56
32.84
0.0023
BG
637.56
1
637.56
40.45
0.0014
Residual 78.81 5 15.76 Cor Total 4532.94 15 The Model F-value of 28.26 implies the model is significant. There is only a 0.09% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. 3.97 R-Squared 0.9826 Mean 10.06 Adj R-Squared 0.9478 C.V. 39.46 Pred R-Squared 0.8220 PRESS 807.04 Adeq Precision 17.771 Coefficient
Standard
95% CI
95% CI
Factor
Estimate
DF
Error
Low
High
VIF
Intercept 10.06 1 0.99 7.51 12.61 A-A -4.69 1 0.99 -7.24 -2.14 1.00 B-B -0.94 1 0.99 -3.49 1.61 1.00 C-C -1.81 1 0.99 -4.36 0.74 1.00 D-D -7.19 1 0.99 -9.74 -4.64 1.00 F-F -8.31 1 0.99 -10.86 -5.76 1.00 G-G -1.06 1 0.99 -3.61 1.49 1.00 AD 5.81 1 0.99 3.26 8.36 1.00 AF 4.94 1 0.99 2.39 7.49 1.00 BC 5.69 1 0.99 3.14 8.24 1.00 BG -6.31 1 0.99 -8.86 -3.76 1.00 Final Equation in Terms of Coded Factors:
c = +10.06 -4.69 * A -0.94 * B -1.81 * C -7.19 * D -8.31 * F -1.06 * G +5.81 * A * D +4.94 * A * F +5.69 * B * C -6.31 * B * G Final Equation in Terms of Actual Factors:
c = +10.06250 -4.68750 * A -0.93750 * B -1.81250 * C -7.18750 * D -8.31250 * F -1.06250 * G +5.81250 * A * D +4.93750 * A * F +5.68750 * B * C -6.31250 * B * G (d)
Plot the residuals from this model versus the predicted number of defects. Also, prepare a normal probability plot of the residuals. Comment on the adequacy of these plots. 147
There is a significant problem with inequality of variance. This is likely caused by the response variable being a count. A transformation may be appropriate. (e)
In part (d) you should have noticed an indication that the variance of the response is not constant (considering that the response is a count, you should have expected this). The previous table also shows a transformation on c, the square root, that is a widely used variance stabilizing transformation for count data (refer to the discussion of variance stabilizing transformations in Chapter 3). Repeat parts (a) through (d) using the transformed response and comment on your results. Specifically, are the residual plots improved? Design Expert Output Term Effect SumSqr % Contribtn Model Intercept Error A -0.895 3.2041 4.2936 Model B -0.3725 0.555025 0.743752 Error C -0.6575 1.72922 2.31722 Model D -2.1625 18.7056 25.0662 Error E 0.4875 0.950625 1.27387 Model F -2.6075 27.1962 36.4439 Model G -0.385 0.5929 0.794506 Error H 0.27 0.2916 0.390754 Error J 0.06 0.0144 0.0192965 Error AD 1.145 5.2441 7.02727 Error AE 0.555 1.2321 1.65106 Error AF 0.86 2.9584 3.96436 Error AH 0.0425 0.007225 0.00968175 Error BC 0.6275 1.57502 2.11059 Model BG -1.61 10.3684 13.894 Lenth's ME 2.27978 Lenth's SME 4.62829 The analysis of the data with the square root transformation identifies only D
, F
, the BG
interaction as being significant. The original analysis identified factor A
and several two factor interactions as being significant. Residual
Normal % probability
Normal plot of residuals
-3.8125
-1.90625
0
1.90625
3.8125
1
5
10
20
30
50
70
80
90
95
99
Predicted
Residuals
Residuals vs. Predicted
-3.8125
-1.90625
0
1.90625
3.8125
-3.81
10.81
25.44
40.06
54.69
148
Design Expert Output Response:
sqrt
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of
Mean
F
Source
Squares
DF
Square
Value
Prob > F
Model 57.42 5 11.48 6.67 0.0056 significant B
0.56
1
0.56
0.32
0.5826
D
18.71
1
18.71
10.87
0.0081
F
27.20
1
27.20
15.81
0.0026
G
0.59
1
0.59
0.34
0.5702
BG
10.37
1
10.37
6.03
0.0340
Residual 17.21 10 1.72 Cor Total 74.62 15 The Model F-value of 6.67 implies the model is significant. There is only a 0.56% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. 1.31 R-Squared 0.7694 Mean 2.32 Adj R-Squared 0.6541 C.V. 56.51 Pred R-Squared 0.4097 PRESS 44.05 Adeq Precision 8.422 Coefficient
Standard
95% CI
95% CI
Factor
Estimate
DF
Error
Low
High
VIF
Intercept 2.32 1 0.33 1.59 3.05 B-B -0.19 1 0.33 -0.92 0.54 1.00 D-D -1.08 1 0.33 -1.81 -0.35 1.00 F-F -1.30 1 0.33 -2.03 -0.57 1.00 G-G -0.19 1 0.33 -0.92 0.54 1.00 BG -0.80 1 0.33 -1.54 -0.074 1.00 Final Equation in Terms of Coded Factors:
sqrt = +2.32 -0.19 * B -1.08 * D -1.30 * F -0.19 * G -0.80 * B * G Final Equation in Terms of Actual Factors:
sqrt = DESIGN-EXPERT Plot
sqrt
A: A
B: B
C: C
D: D
E: E
F: F
G: G
H: H
J: J
N o rm a l p lo t
N o rm a l % p ro b a b ility
E ffe c t
-2 .6 1
-1 .6 7
-0 .7 3
0 .2 1
1 .1 4
1
5
1 0
2 0
3 0
5 0
7 0
8 0
9 0
9 5
9 9
D
F
B G
149
+2.32125 -0.18625 * B -1.08125 * D -1.30375 * F -0.19250 * G -0.80500 * B * G The residual plots are acceptable; although, there appears to be a slight “u” shape to the residuals versus predicted plot. (f)
There is a modification to the square root transformation proposed by Freeman and Tukey (“Transformations Related to the Angular and the Square Root,” Annals of Mathematical Statistics
, Vol. 21, 1950, pp. 607-611) that improves its performance. F&T’s modification to the square root transformation is: [
]
1
2
1
+
+
c
c
Rework parts (a) through (d) using this transformation and comment on the results. (For an interesting discussion and analysis of this experiment, refer to “Analysis of Factorial Experiments with Defects or Defectives as the Response,” by S. Bisgaard and H.T. Fuller, Quality Engineering
, Vol. 7, 1994-5, pp. 429-443.) Design Expert Output Term Effect SumSqr % Contribtn Model Intercept Error A -0.86 2.9584 4.38512 Model B -0.325 0.4225 0.626255 Error C -0.605 1.4641 2.17018 Model D -1.995 15.9201 23.5977 Error E 0.5025 1.01002 1.49712 Model F -2.425 23.5225 34.8664 Model G -0.4025 0.648025 0.960541 Error H 0.225 0.2025 0.300158 Error J 0.0275 0.003025 0.00448383 Error AD 1.1625 5.40562 8.01254 Error AE 0.505 1.0201 1.51205 Error AF 0.8825 3.11523 4.61757 Error AH 0.0725 0.021025 0.0311645 Error BC 0.7525 2.26503 3.35735 Residual
Normal % probability
Normal plot of residuals
-2.1125
-1.09062
-0.06875
0.953125
1.975
1
5
10
20
30
50
70
80
90
95
99
Predicted
Residuals
Residuals vs. Predicted
-2.1125
-1.09062
-0.06875
0.953125
1.975
-1.25
0.44
2.13
3.83
5.52
150
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