Design of Engineering Experiments Chapter 6 – The 2k...

Montgomery_Chap6 1

Design of Engineering ExperimentsChapter 6 – The 2k Factorial Design

• Text reference, Chapter 6• Special case of the general factorial design; k factors,

all at two levels• The two levels are usually called low and high (they

could be either quantitative or qualitative)• Very widely used in industrial experimentation• Form a basic “building block” for other very useful

experimental designs (DNA)• Special (short-cut) methods for analysis• We will make use of Design-Expert

Montgomery_Chap6 2

The Simplest Case: The 22

“-” and “+” denote the low and high levels of a factor, respectively

Low and high are arbitrary terms

Geometrically, the four runs form the corners of a square

Factors can be quantitative or qualitative, although their treatment in the final model will be different

Montgomery_Chap6 3

Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

Montgomery_Chap6 4

Analysis Procedure for a Factorial Design

• Estimate factor effects• Formulate model

– With replication, use full model– With an unreplicated design, use normal probability

plots• Statistical testing (ANOVA)• Refine the model• Analyze residuals (graphical)• Interpret results

Montgomery_Chap6 5

Estimation of Factor Effects

1

1

1

(1)2 2[ (1)]

(1)2 2[ (1)]

(1)2 2

[ (1) ]

A A

n

B B

n

n

A y y

ab a bn n

ab a bB y y

ab b an n

ab b aab a bAB

n nab a b

+ −

+ −

= −

+ += −

= + − −

= −

+ += −

= + − −

+ += −

= + − −

See textbook, pg. 221 For manual calculations

The effect estimates are: A = 8.33, B = -5.00, AB = 1.67

Practical interpretation?

Design-Expert analysis

Montgomery_Chap6 6

Estimation of Factor EffectsForm Tentative Model

Term Effect SumSqr % ContributionModel InterceptModel A 8.33333 208.333 64.4995Model B -5 75 23.2198Model AB 1.66667 8.33333 2.57998Error Lack Of Fit 0 0Error P Error 31.3333 9.70072

Lenth's ME 6.15809Lenth's SME 7.95671

Montgomery_Chap6 7

Statistical Testing - ANOVAResponse:Conversion

ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 291.67 3 97.22 24.82 0.0002A 208.33 1 208.33 53.19 < 0.0001B 75.00 1 75.00 19.15 0.0024AB 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11

Std. Dev. 1.98 R-Squared 0.9030Mean 27.50 Adj R-Squared 0.8666C.V. 7.20 Pred R-Squared 0.7817

PRESS 70.50 Adeq Precision 11.669

The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?

Montgomery_Chap6 8

Statistical Testing - ANOVA

Coefficient Standard 95% CI 95% CI

Factor Estimate DF Error Low High VIFIntercept 27.50 1 0.57 26.18 28.82A-Concent 4.17 1 0.57 2.85 5.48 1.00B-Catalyst -2.50 1 0.57 -3.82 -1.18 1.00AB 0.83 1 0.57 -0.48 2.15 1.00

General formulas for the standard errors of the model coefficients and the confidence intervals are available. They will be given later.

Montgomery_Chap6 9

Refine ModelResponse:Conversion


Sum of Mean FSource Squares DF Square Value Prob > FModel 283.33 2 141.67 32.14 < 0.0001A 208.33 1 208.33 47.27 < 0.0001B 75.00 1 75.00 17.02 0.0026Residual 39.67 9 4.41Lack of Fit 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11



There is now a residual sum of squares, partitioned into a “lack of fit” component (the AB interaction) and a “pure error” component

Montgomery_Chap6 10

Regression Model for the ProcessCoefficient Standard 95% CI 95% CI

Factor Estimate DF Error Low High VIFIntercept 27.5 1 0.60604 26.12904 28.87096A-Concent 4.166667 1 0.60604 2.79571 5.537623 1B-Catalyst -2.5 1 0.60604 -3.87096 -1.12904 1

Final Equation in Terms of Coded Factors:

Conversion =27.5

4.166667 * A-2.5 * B

Final Equation in Terms of Actual Factors:

Conversion =18.333330.833333 * Concentration

-5 * Catalyst

Montgomery_Chap6 11

Residuals and Diagnostic CheckingDESIGN-EXPERT Plo tConversion

Res idual

Nor

mal

% p

roba

bilit

yNormal plot of residuals

-2.83333 -1.58333 -0.333333 0.916667 2.16667

1

5

10

20

30

50

7080

90

95

99

DESIGN-EXPERT P lo tConversion

22

Predicted

Res

idua

ls

Residuals vs. Predicted

-2.83333

-1.58333

-0.333333

0.916667

2.16667

20.83 24.17 27.50 30.83 34.17

Montgomery_Chap6 12

The Response Surfacet Conversion

A: Concentration

B: C

atal

yst

15.00 17.50 20.00 22.50 25.00

1.00

1.25

1.50

1.75

2.00

23.0556

25.2778

27.529.7222

31.9444

3 3

3 3 n

20.8333

24.1667

27.5

30.8333

4.1667

C

on

vers

ion

15.00

17.50

20.00

22.50

25.00

1 .00

1.25

1.50

1.75

2.00

A : Concen tra tion B : Ca ta lyst

3

Montgomery_Chap6 13

The 23 Factorial Design

Montgomery_Chap6 14

Effects in The 23 Factorial Design

etc, etc, ...

A A

B B

C C

A y y

B y y

C y y

+ −

+ −

+ −

= −

= −

= −

Analysis done via computer

Montgomery_Chap6 15

An Example of a 23 Factorial Design

A = carbonation, B = pressure, C = speed, y = fill deviation

Montgomery_Chap6 16

Table of – and + Signs for the 23 Factorial Design (pg. 231)

Factorial Effect

TreatmentCombination

I A B AB C AC BC ABC

(1) = -4 + - - + - + + -

a = 1 + + - - - - + +

b = -1 + - + - - + - +

ab = 5 + + + + - - - -

c = -1 + - - + + - - +

ac = 3 + + - - + + - -

bc = 2 + - + - + - + -

abc = 11 + + + + + + + +

Contrast 24 18 6 14 2 4 4

Effect 3.00 2.25 0.75 1.75 0.25 0.50 0.50

Montgomery_Chap6 17

Properties of the Table • Except for column I, every column has an equal number of + and –

signs• The sum of the product of signs in any two columns is zero• Multiplying any column by I leaves that column unchanged (identity

element)• The product of any two columns yields a column in the table:

• Orthogonal design• Orthogonality is an important property shared by all factorial designs

2

A B ABAB BC AB C AC× =

× = =

Montgomery_Chap6 18

Estimation of Factor Effects

Term Effect SumSqr % ContributionModel InterceptError A 3 36 46.1538Error B 2.25 20.25 25.9615Error C 1.75 12.25 15.7051Error AB 0.75 2.25 2.88462Error AC 0.25 0.25 0.320513Error BC 0.5 1 1.28205Error ABC 0.5 1 1.28205Error LOF 0Error P Error 5 6.41026


Montgomery_Chap6 19

ANOVA Summary – Full ModelResponse:Fill-deviation


Sum of Mean FSource Squares DF Square Value Prob > FModel 73.00 7 10.43 16.69 0.0003A 36.00 1 36.00 57.60 < 0.0001B 20.25 1 20.25 32.40 0.0005C 12.25 1 12.25 19.60 0.0022AB 2.25 1 2.25 3.60 0.0943AC 0.25 1 0.25 0.40 0.5447BC 1.00 1 1.00 1.60 0.2415ABC 1.00 1 1.00 1.60 0.2415Pure Error 5.00 8 0.63Cor Total 78.00 15



Montgomery_Chap6 20

Model Coefficients – Full Model

CoefficientStandard 95% CI 95% CI

Factor Estimate DF Error Low High VIF

Intercept 1.00 1 0.20 0.54 1.46

A-Carbonation 1.50 1 0.20 1.04 1.96 1.00B-Pressure 1.13 1 0.20 0.67 1.58 1.00C-Speed 0.88 1 0.20 0.42 1.33 1.00AB 0.38 1 0.20 -0.081 0.83 1.00AC 0.13 1 0.20 -0.33 0.58 1.00BC 0.25 1 0.20 -0.21 0.71 1.00ABC 0.25 1 0.20 -0.21 0.71 1.00

Montgomery_Chap6 21

Refine Model – Remove Nonsignificant FactorsResponse: Fill-deviation


Sum of Mean FSource Squares DF Square Value Prob > FModel 70.75 4 17.69 26.84 < 0.0001A 36.00 1 36.00 54.62 < 0.0001B 20.25 1 20.25 30.72 0.0002C 12.25 1 12.25 18.59 0.0012AB 2.25 1 2.25 3.41 0.0917Residual 7.25 11 0.66LOF 2.25 3 0.75 1.20 0.3700Pure E 5.00 8 0.63C Total 78.00 15



Montgomery_Chap6 22

Model Coefficients – Reduced Model

Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low HighIntercept 1.00 1 0.20 0.55 1.45A-Carbonation 1.50 1 0.20 1.05 1.95B-Pressure 1.13 1 0.20 0.68 1.57C-Speed 0.88 1 0.20 0.43 1.32AB 0.38 1 0.20 -0.072 0.82

Montgomery_Chap6 23

Model Summary Statistics (pg. 239)• R2 and adjusted R2

• R2 for prediction (based on PRESS)

2

2

73.00 0.935978.00

/ 5.00 / 81 1 0.8798/ 78.00 /15

Model

T

E EAdj

T T

SSRSS

SS dfRSS df

= = =

= − = − =

2 20.00PRESSR = − = − =Pred 1 1 0.743678.00TSS

Montgomery_Chap6 24

Model Summary Statistics (pg. 239)

• Standard error of model coefficients

• Confidence interval on model coefficients

2 0.625ˆ ˆ( ) ( ) 0.202 2 2(8)

Ek k

MSse Vn nσ

β β= = = = =

/ 2, / 2,ˆ ˆ ˆ ˆ( ) ( )

E Edf dft se t seα αβ β β β β− ≤ ≤ +

Montgomery_Chap6 25

The Regression ModelFinal Equation in Terms of Coded Factors:

Fill-deviation =+1.00+1.50 * A+1.13 * B+0.88 * C+0.38 * A * B

Final Equation in Terms of Actual Factors:

Fill-deviation =+9.62500-2.62500 * Carbonation-1.20000 * Pressure+0.035000 * Speed+0.15000 * Carbonation * Pressure

Montgomery_Chap6 26

Residual Plots are SatisfactoryDESIGN-EXPERT PlotFi l l -devia tion

Studentized Res iduals

Nor

mal

% p

roba

bilit

y

Normal plot of residuals

-1.67 -0.84 0.00 0.84 1.67

1

5

10

20

30

50

7080

90

95

99

Montgomery_Chap6 27

Model InterpretationDESIGN-EXPERT Plot

Fi l l -deviation

X = A: CarbonationY = B: Pressure

B- 25.000B+ 30.000

Actual FactorC: Speed = 225.00

B: Pres s ureInteraction Graph

Fill-

devi

atio

n

A: Carbonation

10.00 10.50 11.00 11.50 12.00

-3

-0.75

1.5

3.75

6

Moderate interaction between carbonation level and pressure

Montgomery_Chap6 28

Model InterpretationDESIGN-EXPERT P lo t

Fi l l -deviationX = A: CarbonationY = B: PressureZ = C: Speed

Cube GraphFill-deviation

A: Carbonation

B: P

ress

ure

C : Speed

A- A+B-

B+

C-

C+

-2.13

-0.37

-0.63

1.13

0.12

1.88

3.13

4.88 Cube plots are often useful visual displays of experimental results

Montgomery_Chap6 29

Contour & Response Surface Plots –Speed at the High Level

DESIGN-EXPERT P lo t

Fi l l -deviationX = A: CarbonationY = B: Pressure

Design Poin ts

Actua l FactorC: Speed = 250.00

Fill-deviation

A: Carbonation

B: P

ress

ure

10.00 10.50 11.00 11.50 12.00

25.00

26.25

27.50

28.75

30.00

0.5

1.375

2.25

3.125

2 2

2 2

-0.375

0.9375

2.25

3.5625

4.875

Fill

-dev

iatio

n

10.00

10.50

11.00

11.50

12.00

2 5.00

26.25

27.50

28.75

30.00

A: Carbonation B: Pressure

Montgomery_Chap6 30

The General 2kFactorial Design• Section 6-4, pg. 242, Table 6-9, pg. 243• There will be k main effects, and

two-factor interactions2

three-factor interactions3

1 factor interaction

k

k

k

−M

Montgomery_Chap6 31

Unreplicated 2kFactorial Designs

• These are 2k factorial designs with oneobservation at each corner of the “cube”

• An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k

• These designs are very widely used• Risks…if there is only one observation at each

corner, is there a chance of unusual response observations spoiling the results?

• Modeling “noise”?

Montgomery_Chap6 32

Spacing of Factor Levels in the Unreplicated 2k

Factorial Designs

If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data

More aggressive spacing is usually best

Montgomery_Chap6 33

Unreplicated 2kFactorial Designs

• Lack of replication causes potential problems in statistical testing– Replication admits an estimate of “pure error” (a better

phrase is an internal estimate of error)– With no replication, fitting the full model results in zero

degrees of freedom for error• Potential solutions to this problem

– Pooling high-order interactions to estimate error– Normal probability plotting of effects (Daniels, 1959)– Other methods…see text, pp. 246

Montgomery_Chap6 34

Example of an Unreplicated 2k Design

• A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin

• The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate

• Experiment was performed in a pilot plant

Montgomery_Chap6 35

The Resin Plant Experiment

Montgomery_Chap6 36

The Resin Plant Experiment

Montgomery_Chap6 37

Estimates of the EffectsTerm Effect SumSqr % Contribution

Model InterceptError A 21.625 1870.56 32.6397Error B 3.125 39.0625 0.681608Error C 9.875 390.062 6.80626Error D 14.625 855.563 14.9288Error AB 0.125 0.0625 0.00109057Error AC -18.125 1314.06 22.9293Error AD 16.625 1105.56 19.2911Error BC 2.375 22.5625 0.393696Error BD -0.375 0.5625 0.00981515Error CD -1.125 5.0625 0.0883363Error ABC 1.875 14.0625 0.245379Error ABD 4.125 68.0625 1.18763Error ACD -1.625 10.5625 0.184307Error BCD -2.625 27.5625 0.480942Error ABCD 1.375 7.5625 0.131959


Montgomery_Chap6 38

The Normal Probability Plot of EffectsDESIGN-EXPERT PlotFi l tra tion Rate

A: T em peratureB: PressureC: ConcentrationD: Sti rring Rate

Normal plot

Nor

mal

% p

roba

bilit

y

Effect

-18.12 -8.19 1.75 11.69 21.62

1

5

10

20

30

50

70

80

90

95

99

A

CD

AC

AD

Montgomery_Chap6 39

The Half-Normal Probability Plot DESIGN-EXPERT PlotFi l tra tion Rate

A: T em peratureB: PressureC: ConcentrationD: Sti rring Rate

Half Normal plot

Hal

f Nor

mal

% p

roba

bilit

y

|Effect|

0.00 5.41 10.81 16.22 21.63

0

20

40

60

70

80

85

90

95

97

99

A

CD

AC

AD

Montgomery_Chap6 40

ANOVA Summary for the ModelResponse:Filtration Rate


Sum of Mean FSource Squares DF Square Value Prob >FModel 5535.81 5 1107.16 56.74 < 0.0001A 1870.56 1 1870.56 95.86 < 0.0001C 390.06 1 390.06 19.99 0.0012D 855.56 1 855.56 43.85 < 0.0001AC 1314.06 1 1314.06 67.34 < 0.0001AD 1105.56 1 1105.56 56.66 < 0.0001Residual 195.12 10 19.51Cor Total 5730.94 15



Montgomery_Chap6 41

The Regression Model


Filtration Rate =+70.06250+10.81250 * Temperature+4.93750 * Concentration+7.31250 * Stirring Rate-9.06250 * Temperature * Concentration+8.31250 * Temperature * Stirring Rate

Montgomery_Chap6 42

Model Residuals are SatisfactoryDESIGN-EXPERT PlotFi l tration Rate

Studentized Res iduals

Nor

mal

% p

roba

bilit

y


-1.83 -0.96 -0.09 0.78 1.65

1

5

10

20

30

50

7080

90

95

99

Montgomery_Chap6 43

Model Interpretation – InteractionsDESIGN-EXPERT P lo t

Fi l tra tion Rate

X = A: T em peratureY = C: Concentra tion

C- -1 .000C+ 1.000

Actua l FactorsB: Pressure = 0 .00D: Sti rring Rate = 0 .00

C: ConcentrationInteraction Graph

Filtr

atio

n R

ate

A: Tem perature

-1.00 -0.50 0.00 0.50 1.00

41.7702

57.3277

72.8851

88.4426

104

DESIGN-EXPERT Plo t

Fi l tra tion Rate

X = A: T em pera tureY = D: S ti rring Rate

D- -1 .000D+ 1.000

Actua l FactorsB: Pressure = 0 .00C: Concentration = 0.00

D: Stirring RateInteraction Graph

Filtr

atio

n R

ate

A: Tem perature

-1.00 -0.50 0.00 0.50 1.00

43

58.25

73.5

88.75

104

Montgomery_Chap6 44

Model Interpretation – Cube PlotIf one factor is dropped, the unreplicated 24

design will projectinto two replicates of a 23

Design projection is an extremely useful property, carrying over into fractional factorials

DESIGN-EXPERT Plo t

Fi l tra tion RateX = A: T em peratureY = C: ConcentrationZ = D: Sti rring Rate

Actua l FactorB: Pressure = 0.00

Cube GraphFiltration Rate

A: Tem perature

C: C

once

ntra

tion

D : Stirring Rate

A- A+C-

C+

D-

D+

46.25

44.25

74.25

72.25

69.38

100.63

61.13

92.38

Montgomery_Chap6 45

Model Interpretation – Response Surface Plots

DESIGN-EXPERT P lo t

Fi l tra tion RateX = A: T em peratureY = D: S ti rring Rate

Actua l FactorsB: Pressure = 0 .00C: Concentra tion = -1.00

44.25

58.3438

72.4375

86.5313

100.625

Filt

ratio

n R

ate

-1.0 0

-0.50

0.00

0.50

1.00

-1 .00

-0.50

0.00

0.50

1.00

A : T em perature D: Sti rring Rate

DESIGN-EXPERT Plo t

Fi l tra tion RateX = A: T em peratureY = D: Sti rring Rate

Actua l FactorsB: Pressure = 0 .00C: Concentra tion = -1 .00

Filtration Rate

A: Tem perature

D: S

tirrin

g R

ate

-1.00 -0.50 0.00 0.50 1.00

-1.00

-0.50

0.00

0.50

1.00

64.62571

77.375

83.75

90.125

56.93551.9395

With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates

Montgomery_Chap6 46

The Drilling Experiment Example 6-3, pg. 257

A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill

Montgomery_Chap6 47

Effect Estimates - The Drilling Experiment

Term Effect SumSqr % ContributionModel InterceptError A 0.9175 3.36722 1.28072Error B 6.4375 165.766 63.0489Error C 3.2925 43.3622 16.4928Error D 2.29 20.9764 7.97837Error AB 0.59 1.3924 0.529599Error AC 0.155 0.0961 0.0365516Error AD 0.8375 2.80563 1.06712Error BC 1.51 9.1204 3.46894Error BD 1.5925 10.1442 3.85835Error CD 0.4475 0.801025 0.30467Error ABC 0.1625 0.105625 0.0401744Error ABD 0.76 2.3104 0.87876Error ACD 0.585 1.3689 0.520661Error BCD 0.175 0.1225 0.0465928Error ABCD 0.5425 1.17722 0.447757


Montgomery_Chap6 48

Half-Normal Probability Plot of EffectsDESIGN-EXPERT Plotadv._rate

A: loadB: flowC: speedD: m ud

Half Normal plot

Hal

f Nor

mal

% p

roba

bilit

y

|Effect|

0.00 1.61 3.22 4.83 6.44

0

20

40

60

70

80

85

90

95

97

99

B

C

D

BCBD

Montgomery_Chap6 49

Residual Plotsot

Res idual

Nor

mal

% p

roba

bilit

y


-1.96375 -0.82625 0.31125 1.44875 2.58625

1

5

10

20

30

50

7080

90

95

99

t

Predicted

Res

idua

ls


-1.96375

-0.82625

0.31125

1.44875

2.58625

1.69 4.70 7.70 10.71 13.71

DE SIG N- EXP ERT Pl o ta d v._ r a te

P redi cted

Re

sid

uals

Residuals vs. P redicted

-1.96375

-0.82625

0.31125

1.44875

2.58625

1.69 4.70 7.70 10.71 13.71

Montgomery_Chap6 50

Residual Plots• The residual plots indicate that there are problems

with the equality of variance assumption• The usual approach to this problem is to employ a

transformation on the response• Power family transformations are widely used

• Transformations are typically performed to – Stabilize variance– Induce normality– Simplify the model

*y yλ=

Montgomery_Chap6 51

Selecting a Transformation

• Empirical selection of lambda• Prior (theoretical) knowledge or experience can

often suggest the form of a transformation• Analytical selection of lambda…the Box-Cox

(1964) method (simultaneously estimates the model parameters and the transformation parameter lambda)

• Box-Cox method implemented in Design-Expert

Montgomery_Chap6 52

The Box-Cox MethodDESIGN-EXPERT Plotadv._rate

Lam bdaCurrent = 1Best = -0 .23Low C.I. = -0 .79High C.I. = 0 .32

Recom m end transform :Log (Lam bda = 0)

Lam bda

Ln(R

esid

ualS

S)

Box-Cox Plot for Power Transforms

1.05

2.50

3.95

5.40

6.85

-3 -2 -1 0 1 2 3

A log transformation is recommended

The procedure provides a confidence interval on the transformation parameter lambda

If unity is included in the confidence interval, no transformation would be needed

Montgomery_Chap6 53

Effect Estimates Following the Log Transformation

DESIGN-EXPERT PlotLn(adv._rate)

A: loadB: flowC: speedD: m ud

Half Normal plotH

alf N

orm

al %

pro

babi

lity

|Effect|

0.00 0.29 0.58 0.87 1.16

0

20

40

60

70

80

85

90

95

97

99

B

C

D

Three main effects are large

No indication of large interaction effects

What happened to the interactions?

Montgomery_Chap6 54

ANOVA Following the Log TransformationResponse: adv._rate Transform: Natural logConstant: 0.000

ANOVA for Selected Factorial Model

Analysis of variance table [Partial sum of squares]Sum of Mean F

Source Squares DF Square Value Prob > FModel 7.11 3 2.37 164.82 < 0.0001B 5.35 1 5.35 371.49 < 0.0001C 1.34 1 1.34 93.05 < 0.0001D 0.43 1 0.43 29.92 0.0001Residual 0.17 12 0.014Cor Total 7.29 15



Montgomery_Chap6 55

Following the Log Transformation


Ln(adv._rate) =+1.60+0.58 * B+0.29 * C+0.16 * D

Montgomery_Chap6 56

Following the Log TransformationDESIGN-EXPERT Plo tLn(adv._rate)

Res idual

Nor

mal

% p

roba

bilit

yNormal plot of residuals

-0.166184 -0.0760939 0.0139965 0.104087 0.194177

1

5

10

20

30

50

7080

90

95

99

DESILn(adv

GN-EXPERT Plo t._rate)

PredictedR

esid

uals


-0.166184

-0.0760939

0.0139965

0.104087

0.194177

0.57 1.08 1.60 2.11 2.63

Montgomery_Chap6 57

The Log Advance Rate Model

• Is the log model “better”?• We would generally prefer a simpler model

in a transformed scale to a more complicated model in the original metric

• What happened to the interactions?• Sometimes transformations provide insight

into the underlying mechanism

Montgomery_Chap6 58

Other Examples of Unreplicated 2k Designs

• The sidewall panel experiment (Example 6-4, pg. 260)– Two factors affect the mean number of defects– A third factor affects variability– Residual plots were useful in identifying the dispersion

effect

• The oxidation furnace experiment (Example 6-5, pg. 265)– Replicates versus repeat (or duplicate) observations?– Modeling within-run variability

Montgomery_Chap6 59

Other Analysis Methods for Unreplicated 2k Designs

• Lenth’s method (see text, pg. 254)– Analytical method for testing effects, uses an estimate

of error formed by pooling small contrasts– Some adjustment to the critical values in the original

method can be helpful– Probably most useful as a supplement to the normal

probability plot

• Conditional inference charts (pg. 255 & 256)

Montgomery_Chap6 60

Addition of Center Points to a 2k Designs

• Based on the idea of replicating some of the runs in a factorial design

• Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models:

01 1

20

1 1 1

First-order model (interaction)

Second-order model

k k k

i i ij i ji i j i

k k k k

i i ij i j ii ii i j i i

y x x x

y x x x x

β β β ε

β β β β ε

= = >

= = > =

= + + +

= + + + +

∑ ∑∑

∑ ∑∑ ∑

Montgomery_Chap6 61

no "curvature"F Cy y= ⇒

The hypotheses are:

01

11

: 0

: 0

k

iiik

iii

H

H

β

β

=

=

=

≠

∑

∑2

Pure Quad( )F C F C

F C

n n y ySSn n

−=

+

This sum of squares has a single degree of freedom

Montgomery_Chap6 62

Example 6-6, Pg. 273

5Cn =

Usually between 3 and 6 center points will work well

Design-Expert provides the analysis, including the F-test for pure quadratic curvature

Montgomery_Chap6 63

ANOVA for Example 6-6Response: yield


Sum of Mean FSource Squares DF Square Value Prob > FModel 2.83 3 0.94 21.92 0.0060A 2.40 1 2.40 55.87 0.0017B 0.42 1 0.42 9.83 0.0350AB 2.500E-003 1 2.500E-003 0.058 0.8213Curvature 2.722E-003 1 2.722E-003 0.063 0.8137Pure Error 0.17 4 0.043Cor Total 3.00 8

Std. Dev. 0.21 R-Squared 0.9427Mean 40.44 Adj R-Squared 0.8996

C.V. 0.51 Pred R-Squared N/A

PRESS N/A Adeq Precision 14.234

Montgomery_Chap6 64

If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective

design for fitting a second-order response surface model

Montgomery_Chap6 65

Practical Use of Center Points (pg. 275)

• Use current operating conditions as the center point

• Check for “abnormal” conditions during the time the experiment was conducted

• Check for time trends• Use center points as the first few runs when there

is little or no information available about the magnitude of error

• Center points and qualitative factors?

Montgomery_Chap6 66

Center Points and Qualitative Factors

Design of Engineering Experiments Chapter 6 – The 2k...

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