II.2 Four Factors in Eight Runs Demonstrating the Effects of Confounding* Response: Conductivity...

II.2 Four Factors in Eight RunsII.2 Four Factors in Eight RunsDemonstrating the Effects of Confounding*Demonstrating the Effects of Confounding*

Response: Conductivity (milliohm/cm), Corrected for Response: Conductivity (milliohm/cm), Corrected for Concentration (CC)Concentration (CC)

Factors (Lo, Hi)Factors (Lo, Hi)– A: Stirring Rate (Low, High)A: Stirring Rate (Low, High)– B: Gas Bubbling (Off, On)B: Gas Bubbling (Off, On)– C: Solution Temperature (25C: Solution Temperature (25ooC, 45C, 45ooC)C)– D: Solution Concentration (.076 M, .76 M NaOH)D: Solution Concentration (.076 M, .76 M NaOH)

* Based on DOE study done by Kamal Jha, USC Statistics 506 and Chemical Engineering * Based on DOE study done by Kamal Jha, USC Statistics 506 and Chemical Engineering StudentStudent

II.2 Four Factors in Eight Runs: A II.2 Four Factors in Eight Runs: A DemonstrationDemonstration

Full 2Full 244 Experiment Experiment Response: Conductivity Response: Conductivity

(milliohm/cm)(milliohm/cm) Factors (Lo, Hi)Factors (Lo, Hi)

– A: Stirring Rate (Low, High)A: Stirring Rate (Low, High)– B: Gas Bubbling (Off, On)B: Gas Bubbling (Off, On)– C: Solution TemperatureC: Solution Temperature

(25(25ooC, 45C, 45ooC)C)– D: Solution Concentration D: Solution Concentration

(.076 M, .76 M (.076 M, .76 M NaOH)NaOH)

Full 2Full 244 Experiment Experiment Response: Conductivity Response: Conductivity

(milliohm/cm)(milliohm/cm) Factors (Lo, Hi)Factors (Lo, Hi)

– A: Stirring Rate (Low, High)A: Stirring Rate (Low, High)– B: Gas Bubbling (Off, On)B: Gas Bubbling (Off, On)– C: Solution TemperatureC: Solution Temperature


(.076 M, .76 M (.076 M, .76 M NaOH)NaOH)

Effects100500

D

Main Effects Interaction EffectsActualOrder

Concy A B C D AB AC AD BC BD CD ABC AB

DACD BCD ABCD

4 16.5 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 112 16.6 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -12 15.0 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -15 2.0 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 110 20.0 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -18 20.5 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 19 17.7 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 115 0.2 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -13 134.3 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -16 134.3 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 113 123.0 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 17 106.0 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -111 152.0 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 116 154.0 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -11 150.0 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -114 136.0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SumDivisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 74.88 -7.36 -12.29 12.84 122.64 -8.01 0.112 0.113 1.638 -2.61 10.76 -0.49 -0.24 1.138 3.263 0.738


Response: Conductivity Response: Conductivity (milliohm/cm) , Corrected (milliohm/cm) , Corrected for Concentration (CC)for Concentration (CC)

Factors (Lo, Hi)Factors (Lo, Hi)– A: Stirring Rate (Low, High)A: Stirring Rate (Low, High)– B: Gas Bubbling (Off, On)B: Gas Bubbling (Off, On)– C: Solution TemperatureC: Solution Temperature


(.076 M, .76 M (.076 M, .76 M NaOH)NaOH)

Response: Conductivity Response: Conductivity (milliohm/cm) , Corrected (milliohm/cm) , Corrected for Concentration (CC)for Concentration (CC)

Factors (Lo, Hi)Factors (Lo, Hi)– A: Stirring Rate (Low, High)A: Stirring Rate (Low, High)– B: Gas Bubbling (Off, On)B: Gas Bubbling (Off, On)– C: Solution TemperatureC: Solution Temperature


(.076 M, .76 M (.076 M, .76 M NaOH)NaOH)

Effects100-10

B AB

A

CD

C


CCy A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

4 2.94 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 112 3.04 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -12 1.44 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -15 -11.56 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 110 6.44 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -18 6.94 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 19 4.14 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 115 -13.36 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -13 -1.90 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -16 -1.90 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 113 -13.20 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 17 -30.20 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -111 15.80 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 116 17.80 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -11 13.80 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -114 -0.20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SumDivisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 0 -7.36 -12.29 12.84 0 -8.01 0.112 0.113 1.638 -2.61 10.76 -0.49 -0.24 1.138 3.263 0.738


CCy A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

4 2.94 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 112 3.04 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -12 1.44 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -15 -11.56 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 110 6.44 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -18 6.94 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 19 4.14 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 115 -13.36 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -13 -1.90 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -16 -1.90 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 113 -13.20 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 17 -30.20 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -111 15.80 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 116 17.80 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -11 13.80 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -114 -0.20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

SumDivisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 0 -7.36 -12.29 12.84 0 -8.01 0.112 0.113 1.638 -2.61 10.76 -0.49 -0.24 1.138 3.263 0.738

Significant Effects

– A, B, AB– C, CD

Significant Effects

– A, B, AB– C, CD


Half-Fraction Signs TablesHalf-Fraction Signs Tables

ABCD = -ICCy

A-BCD

B-ACD

C-ABD

AB-CD

AC-BD

AD-BC

D-ABC

-1.90 -1 -1 -1 1 1 -1 13.04 1 -1 -1 -1 -1 -1 -11.44 -1 1 -1 -1 1 1 -1

-30.20 1 1 -1 1 -1 1 16.44 -1 -1 1 1 -1 1 -1

17.80 1 -1 1 -1 1 1 113.80 -1 1 1 -1 -1 -1 1

-13.36 1 1 1 1 1 -1 -1

8 4 4 4 4 4 4 4 -.3675 -10.63 -13.43 13.08 -18.78 2.73 -1.53 0.49

ABCD = -ICCy

A-BCD

B-ACD

C-ABD

AB-CD

AC-BD

AD-BC

D-ABC

-1.90 -1 -1 -1 1 1 -1 13.04 1 -1 -1 -1 -1 -1 -11.44 -1 1 -1 -1 1 1 -1

-30.20 1 1 -1 1 -1 1 16.44 -1 -1 1 1 -1 1 -1

17.80 1 -1 1 -1 1 1 113.80 -1 1 1 -1 -1 -1 1

-13.36 1 1 1 1 1 -1 -1

8 4 4 4 4 4 4 4 -.3675 -10.63 -13.43 13.08 -18.78 2.73 -1.53 0.49

24 DOE EstimatedSignificant Effects

– A: -7.36– B: -12.29– C: 12.84– AB: -8.01– CD: 10.76

24 DOE EstimatedSignificant Effects

– A: -7.36– B: -12.29– C: 12.84– AB: -8.01– CD: 10.76

ABCD = ICCy

A+BCD

B+ACD

C+ABD

AB+CD

AC+BD

AD+BC

D+ABC

2.94 -1 -1 -1 1 1 1 -1-1.90 1 -1 -1 -1 -1 1 1

-13.20 -1 1 -1 -1 1 -1 1-11.56 1 1 -1 1 -1 -1 -115.80 -1 -1 1 1 -1 -1 1

6.94 1 -1 1 -1 1 -1 -14.14 -1 1 1 -1 -1 1 -1

-0.20 1 1 1 1 1 1 1

8 4 4 4 4 4 4 4.3700 -4.1 -11.15 12.60 2.75 -2.50 1.75 -0.49

ABCD = ICCy

A+BCD

B+ACD

C+ABD

AB+CD

AC+BD

AD+BC

D+ABC

2.94 -1 -1 -1 1 1 1 -1-1.90 1 -1 -1 -1 -1 1 1

-13.20 -1 1 -1 -1 1 -1 1-11.56 1 1 -1 1 -1 -1 -115.80 -1 -1 1 1 -1 -1 1

6.94 1 -1 1 -1 1 -1 -14.14 -1 1 1 -1 -1 1 -1

-0.20 1 1 1 1 1 1 1

8 4 4 4 4 4 4 4.3700 -4.1 -11.15 12.60 2.75 -2.50 1.75 -0.49


Half-Fraction Probability PlotsHalf-Fraction Probability Plots

Effects

7 Effects Plot

1

2

3

4

5

6

7

ABCD = -I

100-10-20

AB-CD

B-ACD

A-BCD

C-ABD

Effects

7 Effects Plot

1

2

3

4

5

6

7

ABCD = -I

100-10-20

AB-CD

B-ACD

A-BCD

C-ABD

Effects

7 Effects Plot

1

2

3

4

5

6

7

100-10

B+ACD

C+ABD

ABCD = I

Effects

7 Effects Plot

1

2

3

4

5

6

7

100-10

B+ACD

C+ABD

ABCD = I

Effects

7 Effects Plot

1

2

3

4

5

6

7

ABCD = -I

100-10-20

AB-CD

B-ACD

A-BCD

C-ABD

Effects

7 Effects Plot

1

2

3

4

5

6

7

ABCD = -I

100-10-20

AB-CD

B-ACD

A-BCD

C-ABD


Sequential Design CommentsSequential Design Comments

Suppose the ABCD = -I Half-Fraction Suppose the ABCD = -I Half-Fraction Was Performed FirstWas Performed First– The estimate of the effect of D for the The estimate of the effect of D for the

concentration (not CC) would have been concentration (not CC) would have been 123.125.123.125.

– This with the ABCD = -I probability plot would This with the ABCD = -I probability plot would indicate that factors A, B, C, and D are indicate that factors A, B, C, and D are significant if you assume that 3-way significant if you assume that 3-way interactions are negligible.interactions are negligible.

– You would be unsure how to interpret the AB - You would be unsure how to interpret the AB - CD. This ambiguity would be resolved by CD. This ambiguity would be resolved by adding the half-fraction determined by ABCD = adding the half-fraction determined by ABCD = I.I.

– If the data were the same as given above for If the data were the same as given above for runs ABCD = Iruns ABCD = I

you would combine the data for the two you would combine the data for the two half-fractions (two blocks)half-fractions (two blocks)

the estimated effects would be obtained the estimated effects would be obtained from the 16-run signs tablefrom the 16-run signs table

ABCD is confounded with the block effectABCD is confounded with the block effect

Suppose the ABCD = -I Half-Fraction Suppose the ABCD = -I Half-Fraction Was Performed FirstWas Performed First– The estimate of the effect of D for the The estimate of the effect of D for the

concentration (not CC) would have been concentration (not CC) would have been 123.125.123.125.

– This with the ABCD = -I probability plot would This with the ABCD = -I probability plot would indicate that factors A, B, C, and D are indicate that factors A, B, C, and D are significant if you assume that 3-way significant if you assume that 3-way interactions are negligible.interactions are negligible.

– You would be unsure how to interpret the AB - You would be unsure how to interpret the AB - CD. This ambiguity would be resolved by CD. This ambiguity would be resolved by adding the half-fraction determined by ABCD = adding the half-fraction determined by ABCD = I.I.

– If the data were the same as given above for If the data were the same as given above for runs ABCD = Iruns ABCD = I

you would combine the data for the two you would combine the data for the two half-fractions (two blocks)half-fractions (two blocks)

the estimated effects would be obtained the estimated effects would be obtained from the 16-run signs tablefrom the 16-run signs table

ABCD is confounded with the block effectABCD is confounded with the block effect

Effects

7 Effects Plot

1

2

3

4

5

6

7

100-10

B+ACD

C+ABD

ABCD = I

Effects

7 Effects Plot

1

2

3

4

5

6

7

100-10

B+ACD

C+ABD

ABCD = I


Sequential Design CommentsSequential Design Comments

Suppose the ABCD = I Half-Fraction Were Suppose the ABCD = I Half-Fraction Were Performed FirstPerformed First– The estimate of the effect of D for the The estimate of the effect of D for the

concentration (not CC) would have been 122.15.concentration (not CC) would have been 122.15.– This with the ABCD = I probability plot would This with the ABCD = I probability plot would

indicate that factors B, C, and D are significant if indicate that factors B, C, and D are significant if you assume that 3-way interactions are you assume that 3-way interactions are negligible.negligible.

– Based on this half-fraction, you would not detect Based on this half-fraction, you would not detect the AB or CD interactions.the AB or CD interactions.

Suppose the ABCD = I Half-Fraction Were Suppose the ABCD = I Half-Fraction Were Performed FirstPerformed First– The estimate of the effect of D for the The estimate of the effect of D for the

concentration (not CC) would have been 122.15.concentration (not CC) would have been 122.15.– This with the ABCD = I probability plot would This with the ABCD = I probability plot would

indicate that factors B, C, and D are significant if indicate that factors B, C, and D are significant if you assume that 3-way interactions are you assume that 3-way interactions are negligible.negligible.

– Based on this half-fraction, you would not detect Based on this half-fraction, you would not detect the AB or CD interactions.the AB or CD interactions.

II.2 Four Factors in Eight Runs Demonstrating the Effects of Confounding* Response: Conductivity...

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