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Design of Engineering ExperimentsPart 4 – Introduction to Factorials

• Text reference, Chapter 5• 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

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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 3021

2 230 52 20 40

112 2

52 20 30 401

2 2

A A

B B

A y y

B y y

AB

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The Case of Interaction:

50 12 20 401

2 240 12 20 50

92 2

12 20 40 5029

2 2

A A

B B

A y y

B y y

AB

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Regression Model & The Associated Response Surface

0 1 1 2 2 12 1 2

1 2 1 2 1 2

The least squares fit is

ˆ 35.5 10.5 5.5 0.5 35.5 10.5 5.5

y x x x x

y x x x x x x

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The Effect of Interaction on the Response Surface

Suppose that we add an interaction term to the model:

1 2 1 2ˆ 35.5 10.5 5.5 8y x x x x

Interaction is actually a form of curvature

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Example 5-1 The Battery Life ExperimentText reference pg. 165

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)?

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The General Two-Factor Factorial Experiment

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

This is a completely randomized design

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Statistical (effects) model:

1,2,...,

( ) 1,2,...,

1, 2,...,ijk i j ij ijk

i a

y j b

k n

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

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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 SS

df

abn a b a b ab n

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ANOVA Table – Fixed Effects Case

Design-Expert will perform the computations

Text gives details of manual computing (ugh!) – see pp. 169 & 170

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Design-Expert Output – Example 5-1

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Residual Analysis – Example 5-1

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Residual Analysis – Example 5-1

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Interaction Plot DESIGN-EXPERT Plot

Life

X = B: TemperatureY = A: Material

A1 A1A2 A2A3 A3

A: MaterialInteraction Graph

Life

B: Temperature

15 70 125

20

62

104

146

188

2

2

22

2

2

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Quantitative and Qualitative Factors

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

• Sometimes an experiment will involve both quantitative and qualitative 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

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Quantitative and Qualitative Factors

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Quantitative and Qualitative Factors

Candidate model terms from Design- Expert: Intercept

A B B2

AB B3

AB2

A = Material type

B = Linear effect of Temperature

B2 = Quadratic effect of Temperature

AB = Material type – TempLinear

AB2 = Material type - TempQuad

B3 = Cubic effect of Temperature (Aliased)

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Regression Model Summary of Results

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Regression Model Summary of Results

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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, Example 5-5

T A B AB AC

ABC AB K E

SS SS SS SS SS

SS SS SS