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Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY

Chapter 8 Two-Level Fractional Factorial Designs

Solutions 8.1. Suppose that in the chemical process development experiment in Problem 6.11, it was only possible to run a one-half fraction of the 24 design. Construct the design and perform the statistical analysis, using the data from replicate 1. The required design is a 24-1 with I=ABCD. A – + – + – + – + Design Expert Output Term Effect Model Intercept Model A -12 Model B -1 Model C 4 Model D -1 Model AB 6 Model AC -1 Model AD -5 Error BC Aliased Error BD Aliased Error CD Aliased Error ABC Aliased Error ABD Aliased Error ACD Aliased Error BCD Aliased Error ABCD Aliased Lenth's ME Lenth's SME54.0516

B – – + + – – + +

C – – – – + + + +

SumSqr 288 2 32 2 72 2 50

22.5856

D=ABC – + + – + – – +

(1) ad bd ab cd ac bc abcd

% Contribtn 64.2857 0.446429 7.14286 0.446429 16.0714 0.446429 11.1607

90 72 87 83 99 81 88 80

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY

The largest effect is A. The next largest effects are the AB and AD interactions. A plausible tentative model would be A, AB and AD, along with B and D to preserve hierarchy. Design Expert Output Response: yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean Source Squares DF Square Model 414.00 5 82.80 A 288.00 1 288.00 B 2.00 1 2.00 D 2.00 1 2.00 AB 72.00 1 72.00 AD 50.00 1 50.00 Residual 34.00 2 17.00 Cor Total 448.00 7

F Value 4.87 16.94 0.12 0.12 4.24 2.94

Prob > F 0.1791 0.0543 0.7643 0.7643 0.1758 0.2285

not significant

The "Model F-value" of 4.87 implies the model is not significant relative to the noise. There is a 17.91 % chance that a "Model F-value" this large could occur due to noise. Std. Dev. Mean C.V. PRESS

Factor Intercept A-A B-B D-D AB AD

4.12 85.00 4.85 544.00 Coefficient Estimate 85.00 -6.00 -0.50 -0.50 3.00 -2.50

R-Squared Adj R-Squared Pred R-Squared Adeq Precision

DF 1 1 1 1 1 1

Standard Error 1.46 1.46 1.46 1.46 1.46 1.46

Final Equation in Terms of Coded Factors: yield +85.00 -6.00 -0.50 -0.50 +3 00

= *A *B *D *A*B

0.9241 0.7344 -0.2143 6.441

95% CI Low 78.73 -12.27 -6.77 -6.77 -3.27 -8.77

95% CI High 91.27 0.27 5.77 5.77 9.27 3.77

VIF 1.00 1.00 1.00 1.00 1.00

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY

Final Equation in Terms of Actual Factors: yield +85.00000 -6.00000 -0.50000 -0.50000 +3.00000 -2.50000

= *A *B *D *A*B *A*D

The Design-Expert output indicates that we really only need the main effect of factor A. The updated analysis is shown below: Design Expert Output Response: yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean Source Squares DF Square Model 288.00 1 288.00 A 288.00 1 288.00 Residual 160.00 6 26.67 Cor Total 448.00 7

F Value 10.80 10.80

Prob > F 0.0167 0.0167

significant

The Model F-value of 10.80 implies the model is significant. There is only a 1.67% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. Mean C.V. PRESS

Factor Intercept A-A

5.16 85.00 6.08 284.44 Coefficient Estimate 85.00 -6.00

R-Squared Adj R-Squared Pred R-Squared Adeq Precision Standard Error 1.83 1.83

DF 1 1

0.6429 0.5833 0.3651 4.648

95% CI Low 80.53 -10.47

95% CI High 89.47 -1.53

VIF 1.00

Final Equation in Terms of Coded Factors: yield +85.00 -6.00

= *A

Final Equation in Terms of Actual Factors: yield +85.00000 -6.00000

= *A

8.2. Suppose that in Problem 6.19, only a one-half fraction of the 24 design could be run. Construct the design and perform the analysis, using the data from replicate I. The required design is a 24-1 with I=ABCD. A – + – + – +

B – – + + – –

C – – – – + +

D=ABC – + + – + –

(1) ad bd ab cd ac

7.037 16.867 13.876 17.273 11.846 4 368

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY

Design Expert Output Term Effect Effects Require Intercept Model A 3.0105 Model B 4.011 Model C -3.4565 Model D 5.051 Model AB 1.8345 Model AC -3.603 Model AD 0.3885 Lenth's ME 19.5168 Lenth's SME 46.7074

SumSqr

% Contribtn

18.1262 32.1762 23.8948 51.0252 6.73078 25.9632 0.301865

11.4565 20.3366 15.1024 32.2499 4.25411 16.4097 0.19079

B, D, and AC + BD are the largest three effects. Now because the main effects of B and D are large, the large effect estimate for the AC + BD alias chain probably indicates that the BD interaction is important. It is also possible that the AB interaction is actually the CD interaction. This is not an easy decision. Additional experimental runs may be required to de-alias these two interactions.

Design Expert Output Response: Crack Lengthin mm x 10^-2 ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean Source Squares DF Square Model 157.92 6 26.32 A 18.13 1 18.13 B 32.18 1 32.18 C 23.89 1 23.89 D 51.03 1 51.03 AB 6.73 1 6.73 BD 25.96 1 25.96 Residual 0.30 1 0.30 Cor Total 158.22 7

F Value 87.19 60.05 106.59 79.16 169.03 22.30 86.01

Prob > F 0.0818 0.0817 0.0615 0.0713 0.0489 0.1329 0.0684

The Model F-value of 87.19 implies there is a 8.18% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. Mean

0.55 12.04

R-Squared Adj R-Squared

0.9981 0.9866

not significant

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY

Coefficient Factor Estimate Intercept 12.04 A-Pour Temp 1.51 B-Titanium Content 2.01 C-Heat Treat Method -1.73 D-Grain Refiner 2.53 AB 0.92 BD -1.80

DF 1 1 1 1 1 1 1

Standard Error 0.19 0.19 0.19 0.19 0.19 0.19 0.19

95% CI Low 9.57 -0.96 -0.46 -4.20 0.057 -1.55 -4.27

95% CI High 14.50 3.97 4.47 0.74 4.99 3.39 0.67

VIF 1.00 1.00 1.00 1.00 1.00 1.00

Final Equation in Terms of Coded Factors: Crack Length +12.04 +1.51 +2.01 -1.73 +2.53 +0.92 -1.80

= *A *B *C *D *A*B *B*D

Final Equation in Terms of Actual Factors: Crack Length +12.03500 +1.50525 +2.00550 -1.72825 +2.52550 +0.91725 -1.80150

= * Pour Temp * Titanium Content * Heat Treat Method * Grain Refiner * Pour Temp * Titanium Content * Titanium Content * Grain Refiner

8.3. Consider the plasma etch experiment described in Example 6.1. Suppose that only a one-half fraction of the design could be run. Set up the design and analyze the data. Because Example 6.1 is a replicated 2 factorial experiment, a half fraction of this design is a 2 four runs. The experiment is replicated to assure an adequate estimate of the MSE.

A – – + + – – + +

B – – – – + + + +

C=AB + + – – – – + +

Etch Rate (A/min) 1037 1052 669 650 633 601 729 860

A (Gap, cm) B (C 2F6 flow, SCCM) C (Power, W)

Factor Low (-) 0.80 125 275

with

Levels High (+) 1.20 200 325

The analysis shown below identifies all three main effects as significant. Because this is a resolution III design, the main effects are aliased with two factor interactions. The original analysis from Example 6.1 identifies factors A, C, and the AC interaction as significant. In our replicated half fraction experiment, factor B is aliased with the AC interaction. This problem points out the concerns of running small resolution III designs.

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Design Expert Output Response: Etch Rate ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Model 2.225E+005 3 74169.79 31.61 A 21528.13 1 21528.13 9.18 B 42778.13 1 42778.13 18.23 C 1.582E+005 1 1.582E+005 67.42 Pure Error 9385.50 4 2346.37 Cor Total 2.319E+005 7

Prob > F 0.0030 0.0388 0.0130 0.0012

significant

The Model F-value of 31.61 implies the model is significant. There is only a 0.30% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. Mean C.V. PRESS

48.44 778.88 6.22 37542.00

Factor Intercept A-Gap B-C2F6 Flow C-Power

Coefficient Estimate 778.88 -51.88 -73.13 140.63

R-Squared Adj R-Squared Pred R-Squared Adeq Precision

DF 1 1 1 1

Standard Error 17.13 17.13 17.13 17.13

0.9595 0.9292 0.8381 12.481 95% CI Low 731.33 -99.42 -120.67 93.08

95% CI High 826.42 -4.33 -25.58 188.17

VIF 1.00 1.00 1.00

Final Equation in Terms of Coded Factors: Etch Rate +778.88 -51.88 -73.13 +140.63

= *A *B *C

Final Equation in Terms of Actual Factors: Etch Rate -332.37500 -259.37500 -1.95000 +5.62500

= * Gap * C2F6 Flow * Power

8.4. Problem 6.30 describes a process improvement study in the manufacturing process of an integrated circuit. Suppose that only eight runs could be made in this process. Set up an appropriate 25-2 design and find the alias structure. Use the appropriate observations from Problem 6.30 as the observations in this design and estimate the factor effects. What conclusions can you draw? I = ABD = ACE = BCDE A B C D E BC BE

(ABD) (ABD) (ABD) (ABD) (ABD) (ABD) (ABD)

=BD =AD =ABCD =AB =ABDE =ACD =ADE

A B C D E BC BE

(ACE) (ACE) (ACE) (ACE) (ACE) (ACE) (ACE)

=CE =ABCE =AE =ACDE =AC =ABE =ABC

A B C D E BC BE

(BCDE) (BCDE) (BCDE) (BCDE) (BCDE) (BCDE) (BCDE)

=ABCDE =CDE =BDE =BCE =BCD =DE =CD

A=BD=CE=ABCDE B=AD=ABCE=CDE C=ABCD=AE=BDE D=AB=ACDE=BCE E=ABDE=AC=BCD BC=ACD=ABE=DE BE=ADE=ABC=CD

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY A – + – + – + – + Design Expert Output Term Model Intercept Model A Model B Model C Model D Error E Error BC Error BE Lenth's ME Lenth's SME

B – – + + – – + +

C – – – – + + + +

D=AB + – – + + – – +

E=AC + – + – – + – +

de a be abd cd ace bc abcde

Effect

SumSqr

% Contribtn

11.25 33.25 10.75 7.75 2.25 -1.75 1.75 28.232 67.5646

253.125 2211.13 231.125 120.125 10.125 6.125 6.125

8.91953 77.9148 8.1443 4.23292 0.356781 0.215831 0.215831

6 9 35 50 18 22 40 63

The main A, B, C, and D are large. However, recall that we are really estimating A+BD+CE, B+AD, C+DE and D+AD. There are other possible interpretations of the experiment because of the aliasing. Design Expert Output Response: Yield ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean Source Squares DF Square Model 2815.50 4 703.88 A 253.13 1 253.13 B 2211.12 1 2211.12 C 231.13 1 231.13 D 120.13 1 120.13 Residual 22.38 3 7.46 Cor Total 2837 88 7

F Value 94.37 33.94 296.46 30.99 16.11

Prob > F 0.0017 0.0101 0.0004 0.0114 0.0278

significant

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY a 0.17% chance that a "Model F-Value" this large could occur due to noise. Std. Dev. Mean C.V. PRESS

2.73 30.38 8.99 159.11

R-Squared Adj R-Squared Pred R-Squared Adeq Precision

Coefficient Factor Estimate Intercept 30.38 A-Aperture 5.63 B-Exposure Time 16.63 C-Develop Time 5.37 D-Mask Dimension3.87

DF 1 1 1 1 1

Standard Error 0.97 0.97 0.97 0.97 0.97

0.9921 0.9816 0.9439 25.590 95% CI Low 27.30 2.55 13.55 2.30 0.80

95% CI High 33.45 8.70 19.70 8.45 6.95

VIF 1.00 1.00 1.00 1.00

Final Equation in Terms of Coded Factors: Yield +30.38 +5.63 +16.63 +5.37 +3.87

= *A *B *C *D

Final Equation in Terms of Actual Factors: Aperture Mask Dimension Yield -6.00000 +0.83125 +0.71667

small Small =

Aperture Mask Dimension Yield +5.25000 +0.83125 +0.71667

large Small =

Aperture Mask Dimension Yield +1.75000 +0.83125 +0.71667

small Large =

Aperture Mask Dimension Yield +13.00000 +0.83125 +0.71667

large Large =

* Exposure Time * Develop Time

* Exposure Time * Develop Time

* Exposure Time * Develop Time

* Exposure Time * Develop Time

8.5. Continuation of Problem 8.4. Suppose you have made the eight runs in the 25-2 design in Problem 8.4. What additional runs would be required to identify the factor effects that are of interest? What are the alias relationships in the combined design? We could fold over the original design by changing the signs on the generators D = AB and E = AC to produce the following new experiment.

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY

A B C D E BC BE

(–ABD) (–ABD) (–ABD) (–ABD) (–ABD) (–ABD) (–ABD)

=–BD =–AD =–ABCD =–AB =–ABDE =–ACD =–ADE

A – + – + – + – +

B – – + + – – + +

C – – – – + + + +

A B C D E BC BE

(–ACE) (–ACE) (–ACE) (–ACE) (–ACE) (–ACE) (–ACE)

D=-AB – + + – – + + – =–CE =–ABCE =–AE =–ACDE =–AC =–ABE =–ABC

E=-AC – + – + + – + – A B C D E BC BE

(1) ade bd abe ce acd bcde abc

(BCDE) (BCDE) (BCDE) (BCDE) (BCDE) (BCDE) (BCDE)

7 10 32 52 15 21 41 60 =ABCDE =CDE =BDE =BCE =BCD =DE =CD

A=–BD=–CE=ABCDE B=–AD=–ABCE=CDE C=–ABCD=–AE=BDE D=–AB=–ACDE=BCE E=–ABDE=–AC=BCD BC=–ACD=–ABE=DE BE=–ADE=–ABC=CD

Assuming all three factor and higher interactions to be negligible, all main effects can be separated from their two-factor interaction aliases in the combined design. 8.6. In the Example 6.6, a 24 factorial design was used to improve the response rate to a credit card marketing offer. Suppose that the researchers had used the 24-1 fraction factorial design with I=ABCD instead. Set up the design and select the responses for the runs from the full factorial data in Example 6.6. Analyze the data and draw conclusions. Compare your findings with those from the full factorial in Example 6.6. Based on the Pseudo p-Value, the effects appear to be much less significant than those found in Example 6.6. The estimates for the Long-term Interest Rate, Account Opening Fee, and Annual Fee * Long-term Interest Rate are similar to the estimates found in Example 6.6; however, the other estimates are not as similar. JMP Output Response Response rate Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts)

1 . . 2.3375 8

Sorted Parameter Estimates Term

Estimate

Long-term interest rate[Low] Account-opening fee[No] Initial interest[Current] Annual fee[Current] Annual fee[Current]*Long-term interest rate[Low] Annual fee[Current]*Account-opening fee[No] Annual fee[Current]*Initial interest[Current]

0.275 0.255 -0.17 -0.15 -0.0775 -0.0725 0.0525

Relative Std Error 0.353553 0.353553 0.353553 0.353553 0.353553 0.353553 0.353553

Pseudo t- Pseudo t-Ratio Ratio 1.22 1.13 -0.76 -0.67 -0.34 -0.32 0.23

Pseudo p-Value 0.3307 0.3600 0.5188 0.5649 0.7591 0.7739 0.8344

No error degrees of freedom, so ordinary tests uncomputable. Relative Std Error corresponds to residual standard error of 1. Pseudo t-Ratio and p-Value calculated using Lenth PSE = 0.225 and DFE=2.3333

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Prediction Profiler

8.7. Continuation of Problem 8.6. In Example 6.6, we found that all four main effects and the twofactor AB interaction were significant. Show that if the alternate fraction (I=-ABCD) is added to the 24-1 design in Problem 8.6 that the analysis from the combined design produced results identical to those found in Example 6.6 The estimates for the effects from the alternate fraction are shown below. By combining the estimates from Problem 8.6 with the estimates below, the same estimates as those shown in Example 6.6 are found. For example:

1 1 A ] + [A′ ]) = ( − 0.15 + ( − 0.2575)) = − 0.20375 → A ( [ 2 2 1 1 AB ] + [AB ′] ) = ( −0.0725 + ( −0.23) ) = −0.15125 → AB ( [ 2 2 1 1 AB ] − [AB ′ ]) = ( −0.0725 − (−0.23 ) ) = 0.07875 → CD ( [ 2 2 The analysis can also be performed as full factorial in two blocks with the block effect confounded with the ABCD interaction. This JMP analysis for the full factorial in two blocks is also shown below. The effect estimates are the same as those shown in Example 6.6. JMP Output Response Response rate Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts)

1 . . 2.39 8

Sorted Parameter Estimates Term

Estimate

Account-opening fee[No] Annual fee[Current] Annual fee[Current]*Account-opening fee[No] Long-term interest rate[Low] Initial interest[Current] Annual fee[Current]*Initial interest[Current] Annual fee[Current]*Long-term interest rate[Low]

0.2625 -0.2575 -0.23 0.2225 -0.0825 -0.05 -0.03

Relative Std Error 0.353553 0.353553 0.353553 0.353553 0.353553 0.353553 0.353553

Pseudo Pseudo t-Ratio t-Ratio 0.79 -0.77 -0.69 0.67 -0.25 -0.15 -0.09

Pseudo p-Value 0.5035 0.5109 0.5529 0.5649 0.8248 0.8929 0.9355

No error degrees of freedom so ordinary tests uncomputable Relative Std Error corresponds to residual standard error of 1 Pseudo t Ratio and

Solutions from Montgomery, D. C. (2017) Design and Analysis of Experiments, Wiley, NY Prediction Profiler

JMP Output Response Response rate Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts)

1 . . 2.36375 16

Sorted Parameter Estimates Term

Estimate

Account-opening fee[No] Long-term interest rate[Low] Annual fee[Current] Annual fee[Current]*Account-opening fee[No] Initial interest rate[Current] Initial interest rate[Current]*Long-term interest rate[Low] Annual fee[Current]*Long-term interest rate[Low] Account-opening fee[No]*Initial interest rate[Current]*Long-term interest rate[Low] Account-opening fee[No]*Long-term interest rate[Low] Annual fee[Current]*Account-opening fee[No]*Long-term interest rate[Low] Block[1] Annual fee[Current]*Account-opening fee[No]*Initial interest rate[Current] Account-opening fee[No]*Initial interest rate[Current] Annual fee[Current]*Initial interest rate[Current]*Long-term interest rate[Low] Annual fee[Current]*Initial interest rate[Current]

0.25875 0.24875 -0.20375 -0.15125 -0.12625 0.07875 -0.05375 0.05375

Relative Pseudo Std Error t-Ratio 0.25 3.63 0.25 3.49 0.25 -2.86 0.25 -2.12 0.25 -1.77 0.25 1.11 0.25 -0.75 0.25 0.75

0.05125 -0.04375 -0.02625 0.02625 -0.02375 -0.00375 0.00125

0.25 0.25 0.25 0.25 0.25 0.25 0.25

0.72 -0.61 -0.37 0.37 -0.33 -0.05 0.02

Pseudo t-Ratio

Pseudo p-Value 0.0150* 0.0174* 0.0354* 0.0872 0.1366 0.3194 0.4846 0.4846 0.5042 0.5661 0.7276 0.7276 0.7524 0.9601 0.9867

No error degrees of freedom, so ordinary tests uncomputable. Relativ...