Design of Engineering Experiments Part 4 – Introduction to...

Montgomery_chap_5 Steve Brainerd 1

Design of Engineering ExperimentsChapter 5 – Introduction to Factorials

• Text reference, Chapter 5 page 170• General principles of factorial experiments• The two-factor factorial with fixed effects• The ANOVA for factorials• Extensions to more than two factors• Quantitative and qualitative factors –

response curves and surfaces


Factorial Designs • A factorial design is

one in which two or more factors are investigated at all possible combinations for their levels as:

• #Level#factors = Total Number of experiments

• We will discuss 2 level designs as : 22 or 24 or 2k In Chapter 6

FACTORS# Experments: 2 level full Factoria l= #level^#factor

# Experments: 3 level full Factorial

# Experments: 4 level full Factorial

2 4 9 16

3 8 27 64

4 16 81 256

5 32 243 1024

6 64 729 4096


Factorial Designs Definitions • If there are a levels of Factor A and b levels of Factor B, a Full

Factorial design is one in all ab combinations are tested.


Factorial Designs Definitions • If there are a levels of Factor A and b levels of Factor

B, a full factorial design is one in all ab combinations are tested. When factors are arranged in a factorial design, they are often called crossed.

• The effect of a factor is defined to be the change in the response Y for a change in the level of that factor. This is called a main effect, because it refers to the primary factors of interest in the experiment.


Some Basic Definitions

Definition of a factor effect: The change in the mean response when the factor is changed from low to high

40 52 20 30 212 2

30 52 20 40 112 2

52 20 30 40 12 2

A A

B B

A y y

B y y

AB

+ −

+ −

+ += − = − =

+ += − = − =

+ += − = −


The Case of Interaction:

50 12 20 40 12 2

40 12 20 50 92 2

12 20 40 50 292 2

A A

B B

A y y

B y y

AB

+ −

+ −

+ += − = − =

+ += − = − = −

+ += − = −


Regression Model & The Associated Response

Surface

0 1 1 2 2

12 1 2

1 2

1 2

1 2

The least squares fit isˆ 35.5 10.5 5.5

0.535.5 10.5 5.5

y x xx x

y x xx x

x x

β β ββ ε

= + ++ +

= + ++≅ + +


The Effect of Interaction on the Response Surface

Suppose that we add an interaction term to the model:

1 2

1 2

ˆ 35.5 10.5 5.58

y x xx x

= + ++

Interaction is actually a form of curvature


Example 5-1 The Battery Life ExperimentText reference pg. 175 3 2

A = Material type; B = Temperature (A quantitative variable)

1. What effects do material type & temperature have on life?

2. Is there a choice of material that would give long life regardless of temperature (a robust product)?


The General Two-Factor Factorial Experiment

a levels of factor A; b levels of factor B; n replicates

This is a completely randomized design


Statistical (effects) model:1,2,...,

( ) 1, 2,...,1, 2,...,

ijk i j ij ijk

i ay j b

k nµ τ β τβ ε

== + + + + = =

Other models (means model, regression models) can be useful


Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 177

2 2 2... .. ... . . ...

1 1 1 1 1

2 2. .. . . ... .

1 1 1 1 1

( ) ( ) ( )

( ) ( )

a b n a b

ijk i ji j k i j

a b a b n

ij i j ijk iji j i j k

y y bn y y an y y

n y y y y y y

= = = = =

= = = = =

− = − + −

+ − − + + −

∑∑∑ ∑ ∑

∑∑ ∑∑∑

breakdown:1 1 1 ( 1)( 1) ( 1)

T A B AB ESS SS SS SS SSdfabn a b a b ab n

= + + +

− = − + − + − − + −


ANOVA Table – Fixed Effects Case

Design-Expert will perform the computations

Text gives details of manual computing (ugh!) –see pp. 180 & 181


Design-Expert Output – Example 5-1

Response: LifeANOVA for Selected Factorial Model

Analysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 59416.22 8 7427.03 11.00 < 0.0001A 10683.72 2 5341.86 7.91 0.0020B 39118.72 2 19559.36 28.97 < 0.0001AB 9613.78 4 2403.44 3.56 0.0186Pure E 18230.75 27 675.21C Total 77646.97 35

Std. Dev. 25.98 R-Squared 0.7652Mean 105.53 Adj R-Squared 0.6956C.V. 24.62 Pred R-Squared 0.5826

PRESS 32410.22 Adeq Precision 8.178


Residual Analysis – Example 5-1DESIGN-EXPERT PlotL i fe

Res idual

Nor

mal

% p

roba

bilit

yNormal plot of residuals

-60.75 -34.25 -7.75 18.75 45.25

1

5

10

20

30

50

7080

90

95

99

DESIGN-EXPERT P lo tL i fe

Predicted

Res

idua

ls

Residuals vs. Predicted

-60.75

-34.25

-7.75

18.75

45.25

49.50 76.06 102.62 129.19 155.75


Residual Analysis – Example 5-1DESIGN-EXPERT Plo tL i fe

Run Num ber

Res

idua

ls

Residuals vs. Run

-60.75

-34.25

-7.75

18.75

45.25

1 6 11 16 21 26 31 36


Residual Analysis – Example 5-1DESIGN-EXPERT PlotL i fe

Material

Res

idua

lsResiduals vs. Material

-60.75

-34.25

-7.75

18.75

45.25

1 2 3

DESIGN-EXPERT P lo tL i fe

Tem perature

Res

idua

ls

Residuals vs. Temperature

-60.75

-34.25

-7.75

18.75

45.25

1 2 3


Interaction Plot DESIGN-EXPERT Plot

L i fe

X = B: T em peratureY = A: M ateria l

A1 A1A2 A2A3 A3

A: MaterialInteraction Graph

Life

B: Tem perature

15 70 125

20

62

104

146

188

2

2

22

2

2


Quantitative and Qualitative Factors

• The basic ANOVA procedure treats every factor as if it were qualitative

• Sometimes an experiment will involve both quantitative(temperature) and qualitative (Material type) factors, such as in Example 5-1

• This can be accounted for in the analysis to produce regression models for the quantitative factors at each level (or combination of levels) of the qualitative factors

• These response curves and/or response surfaces are often a considerable aid in practical interpretation of the results



Response:Life*** WARNING: The Cubic Model is Aliased! ***

Sequential Model Sum of SquaresSum of Mean F

Source Squares DF Square Value Prob > F

Mean 4.009E+005 1 4009E+005

Linear 49726.39 3 16575.46 19.00 < 0.0001 Suggested2FI 2315.08 2 1157.54 1.36 0.2730Quadratic 76.06 1 76.06 0.086 0.7709Cubic 7298.69 2 3649.35 5.40 0.0106 AliasedResidual 18230.75 27 675.21Total 4.785E+005 36 13292.97

"Sequential Model Sum of Squares": Select the highest order polynomial where theadditional terms are significant.


Quantitative and Qualitative FactorsA = Material type

B = Linear effect of Temperature

B2 = Quadratic effect of Temperature

AB = Material type – TempLinearAB2 = Material type - TempQuadB3 = Cubic effect of

Temperature (Aliased)

Candidate model terms from Design-Expert:Intercept

ABB2ABB3AB2



Lack of Fit Tests

Sum of Mean FSource Squares DF Square Value Prob > F

Linear 9689.83 5 1937.97 2.87 0.0333 Suggested

2FI 7374.75 3 2458.25 3.64 0.0252Quadratic 7298.69 2 3649.35 5.40 0.0106Cubic 0.00 0 AliasedPure Error 18230.75 27 675.21

"Lack of Fit Tests": Want the selected model to have insignificant lack-of-fit.



Model Summary Statistics

Std. Adjusted PredictedSource Dev. R-Squared R-Squared R-Squared PRESS

Linear 29.54 0.6404 0.6067 0.5432 35470.60 Suggested

2FI 29.22 0.6702 0.6153 0.5187 37371.08Quadratic 29.67 0.6712 0.6032 0.4900 39600.97Cubic 25.98 0.7652 0.6956 0.5826 32410.22 Aliased

"Model Summary Statistics": Focus on the model maximizing the "Adjusted R-Squared"and the "Predicted R-Squared".



Response: Life

ANOVA for Response Surface Reduced Cubic ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 59416.22 8 7427.03 11.00 < 0.0001A 10683.72 2 5341.86 7.91 0.0020B 39042.67 1 39042.67 57.82 < 0.0001B2 76.06 1 76.06 0.11 0.7398AB 2315.08 2 1157.54 1.71 0.1991AB2 7298.69 2 3649.35 5.40 0.0106Pure E 18230.75 27 675.21C Total 77646.97 35

Std. Dev. 25.98 R-Squared 0.7652Mean 105.53 Adj R-Squared 0.6956C.V. 24.62 Pred R-Squared 0.5826

PRESS 32410.22 Adeq Precision 8.178


Regression Model Summary of Results

Final Equation in Terms of Actual Factors:

Material A1Life =

+169.38017-2.50145 * Temperature+0.012851 * Temperature2

Material A2Life =

+159.62397-0.17335 * Temperature-5.66116E-003 * Temperature2

Material A3Life =

+132.76240+0.90289 * Temperature-0.010248 * Temperature2


Regression Model Summary of ResultsDESIGN-EXPERT Plot

L i fe

X = B: T em peratureY = A: M ateria l

A1 A1A2 A2A3 A3

A: MaterialInteraction Graph

Life

B: Tem perature

15.00 42.50 70.00 97.50 125.00

20

62

104

146

188

2

2

22

2

2


Factorials with More Than Two Factors

• Basic procedure is similar to the two-factor case; all abc…kn treatment combinations are run in random order

• ANOVA identity is also similar:

• Complete three-factor example in text, Section 5-4

T A B AB AC

ABC AB K E

SS SS SS SS SSSS SS SS

= + + + + ++ + + +L

L L

L

Design of Engineering ExperimentsChapter 5 – Introduction to FactorialsFactorial DesignsFactorial Designs DefinitionsFactorial Designs DefinitionsSome Basic DefinitionsThe Case of Interaction:Regression Model & The Associated Response SurfaceThe Effect of Interaction on the Response SurfaceExample 5-1 The Battery Life ExperimentText reference pg. 175 3 2The General Two-Factor Factorial ExperimentExtension of the ANOVA to Factorials (Fixed Effects Case) – pg. 177ANOVA Table – Fixed Effects CaseDesign-Expert Output – Example 5-1Residual Analysis – Example 5-1Residual Analysis – Example 5-1Residual Analysis – Example 5-1Interaction PlotQuantitative and Qualitative FactorsQuantitative and Qualitative FactorsQuantitative and Qualitative FactorsQuantitative and Qualitative FactorsQuantitative and Qualitative FactorsQuantitative and Qualitative FactorsRegression Model Summary of ResultsRegression Model Summary of ResultsFactorials with More Than Two Factors

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Design of Engineering Experiments Part 4 – Introduction to...

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