chapter 5 design & analysis of experiments 1 8e 2012

Chapter 5 Design & Analysis of Experiments

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Design of Engineering Experiments

– Introduction to Factorials

• Text reference, Chapter 5

• General principles of factorial experiments

• The two-factor factorial with fixed effects


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• 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

Some Basic Definitions


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Factor effect: The change in the mean response when the

factor is changed from low to high

40 52 20 3021

2 2

30 52 20 4011

2 2

52 20 30 401

2 2

A A

B B

A y y

B y y

AB

+ −

+ −

+ += − = − =

+ += − = − =

+ += − = −

The Case of Interaction:


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50 12 20 401

2 2

40 12 20 509

2 2

12 20 40 5029

2 2

A A

B B

A y y

B y y

AB

+ −

+ −

+ += − = − =

+ += − = − = −

+ += − = −

Regression Model & The Associated Response Surface


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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

β β β β ε= + + + +

= + + + ≅ + +

The Effect of Interaction on the Response Surface


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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

Example 5.1 The Battery Life Experiment

Text reference pg. 187


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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


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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

µ τ β τβ ε

=

= + + + + = =

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

Extension of the ANOVA to Factorials

(Fixed Effects Case) – pg. 189

2 2 2

... .. ... . . ...

1 1 1 1 1

2 2

. .. . . ... .

( ) ( ) ( )

( ) ( )

a b n a b

ijk i j

i j k i j

a b a b n

ij i j ijk ij

y y bn y y an y y

n y y y y y y

= = = = =

− = − + −

+ − − + + −

∑∑∑ ∑ ∑

∑∑ ∑∑∑


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. .. . . ... .

1 1 1 1 1

( ) ( )ij i j ijk ij

i j i j k

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

= + + +

− = − + − + − − + −

ANOVA Table – Fixed Effects Case


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JMP and Design-Expert will perform the computations

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

192


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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 P lot

L i fe

X = B: T em perature

Y = A: M ateria l

A1 A1

A2 A2

A3 A3

A: Materia l

Interaction Graph

Lif

e

104

146

188


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B: Tem perature

15 70 125

20

62

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

and qualitative factors, such as in Example 5.1

• This can be accounted for in the analysis to produce


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• 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


Candidate model

terms from Design-

Expert:

Intercept

A

A = Material type

B = Linear effect of Temperature

B2 = Quadratic effect of

Temperature


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A

B

B2

AB

B3

AB2

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:


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• ANOVA identity is also similar:

• Complete three-factor example in text, Example 5.3

T A B AB AC

ABC AB K E

SS SS SS SS SS

SS SS SS

= + + + + +

+ + + +L

L L

L

Homework

• Solve

– Problem 5.3

– Problem 5.18– Problem 5.18

• both manually and using DesignExpert


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chapter 5 design & analysis of experiments 1 8e 2012

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