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Chapter 5 Introduction to Factorial Designs

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5.1 Basic Definitions and Principles

• Study the effects of two or more factors.• Factorial designs• Crossed: factors are arranged in a factorial design• Main effect: the change in response produced by a

change in the level of the factor

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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

4

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 2

ˆ 35.5 10.5 5.5

8

y x x

x x

Interaction is actually a form of curvature

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• When an interaction is large, the corresponding main effects have little practical meaning.

• A significant interaction will often mask the significance of main effects.

5.2 The Advantage of Factorials

• One-factor-at-a-time desgin • Compute the main effects of factors

A: A+B- - A-B-

B: A-B- - A-B+

Total number of experiments: 6• Interaction effects

A+B-, A-B+ > A-B- => A+B+ is

better???8

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5.3 The Two-Factor Factorial Design

5.3.1 An Example• a levels for factor A, b levels for factor B and n

replicates• Design a battery: the plate materials (3 levels) v.s.

temperatures (3 levels), and n = 4: 32 factorial design• Two questions:

– What effects do material type and temperature have on the life of the battery?

– Is there a choice of material that would give uniformly long life regardless of temperature?

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• The data for the Battery Design:

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• Completely randomized design: a levels of factor A, b levels of factor B, n replicates

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• Statistical (effects) model:

is an overall mean, i is the effect of the ith level

of the row factor A, j is the effect of the jth

column of column factor B and ( )ij is the

interaction between i and j .

• Testing hypotheses:

1,2,...,

( ) 1,2,...,

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

i a

y j b

k n

0)( oneleast at : v.s., 0)(:

0 oneleast at : v.s.0:

0 oneleast at : v.s.0:

10

110

110

ijij

jb

ia

HjiH

HH

HH

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• 5.3.2 Statistical Analysis of the Fixed Effects Model

a

i

b

j

n

kijk

ijij

n

kijkij

ja

ij

n

kijkj

ib

ji

n

kijki

abn

yyyy

n

yyyy

an

yyyy

bn

yyyy

1 1

......

1...

..

1.

..

1..

1..

..

1..

1..

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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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• Mean squares

2

1 1

2

2

1

2

2

1

2

2

))1(

()(

)1)(1(

)(

))1)(1(

()(

1))1/(()(

1))1/(()(

nab

SSEMSE

ba

n

ba

SSEMSE

b

an

bSSEMSE

a

bnaSSEMSE

EE

a

i

b

jij

ABAB

b

jj

BB

a

ii

AA

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• The ANOVA table:

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Response: Life ANOVA for Selected Factorial 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 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

Example 5.1

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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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• Multiple Comparisons:– Use the methods in Chapter 3.– Since the interaction is significant, fix the

factor B at a specific level and apply Turkey’s test to the means of factor A at this level.

– See Page 174– Compare all ab cells means to determine which

one differ significantly

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5.3.3 Model Adequacy Checking• Residual analysis: ijijkijkijkijk yyyye ˆ

DESIGN-EXPERT PlotLife

Residual

No

rma

l % p

rob

ab

ility

Normal plot of residuals

-60.75 -34.25 -7.75 18.75 45.25

1

5

10

20

30

50

70

80

90

95

99

DESIGN-EXPERT PlotLife

Predicted

Re

sid

ua

ls

Residuals vs. Predicted

-60.75

-34.25

-7.75

18.75

45.25

49.50 76.06 102.62 129.19 155.75

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DESIGN-EXPERT PlotLife

Run Number

Re

sid

ua

ls

Residuals vs. Run

-60.75

-34.25

-7.75

18.75

45.25

1 6 11 16 21 26 31 36

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DESIGN-EXPERT PlotLife

Material

Re

sid

ua

lsResiduals vs. Material

-60.75

-34.25

-7.75

18.75

45.25

1 2 3

DESIGN-EXPERT PlotLife

Temperature

Re

sid

ua

ls

Residuals vs. Temperature

-60.75

-34.25

-7.75

18.75

45.25

1 2 3

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5.3.4 Estimating the Model Parameters• The model is

• The normal equations:

• Constraints:

ijkijjiijky )(

ijijjiij

j

a

iijj

a

iij

i

b

jij

b

jjii

a

i

b

jij

b

jj

a

ii

ynnnn

ynannan

ynnbnbn

ynanbnabn

)(:)(

)(:

)(:

)(:

11

11

1 111

0,0,01111

b

jij

a

iij

b

jj

a

ii

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• Estimations:

• The fitted value:

• Choice of sample size: Use OC curves to choose the proper sample size.

yyyy

yy

yy

y

jiijij

jj

ii

ˆ

ˆ

ˆ

ijijjiijk yy ˆˆˆˆ

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• Consider a two-factor model without interaction:– Table 5.8– The fitted values: yyyy jiijkˆ

• One observation per cell: – The error variance is not estimable because the

two-factor interaction and the error can not be separated.

– Assume no interaction. (Table 5.9)

– Tukey (1949): assume ()ij = rij (Page 183)

– Example 5.2

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5.4 The General Factorial Design

• More than two factors: a levels of factor A, b levels of factor B, c levels of factor C, …, and n replicates.

• Total abc … n observations.• For a fixed effects model, test statistics for each

main effect and interaction may be constructed by dividing the corresponding mean square for effect or interaction by the mean square error.

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• Degree of freedom:– Main effect: # of levels – 1 – Interaction: the product of the # of degrees of

freedom associated with the individual components of the interaction.

• The three factor analysis of variance model:–

– The ANOVA table (see Table 5.12)– Computing formulas for the sums of squares

(see Page 186)– Example 5.3

ijklijkjkik

ijkjiijkly

)()()(

)(

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• Example 5.3: Three factors: the percent carbonation (A), the operating pressure (B); the line speed (C)

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32

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5.5 Fitting Response Curves and Surfaces• An equation relates the response (y) to the factor

(x).• Useful for interpolation.• Linear regression methods• Example 5.4

– Study how temperatures affects the battery life– Hierarchy principle

– Involve both quantitative and qualitative factors

– 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

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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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37

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5.6 Blocking in a Factorial Design

• A nuisance factor: blocking• A single replicate of a complete factorial

experiment is run within each block.• Model:

– No interaction between blocks and treatments• ANOVA table (Table 5.20)

ijkkijjiijky )(

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• Example 5.6: – Two factors: ground clutter and filter type– Nuisance factor: operator

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• Two randomization restrictions: Latin square design

• An example in Page 200.• Model:

• Tables 5.23 and 5.24

ijklljkkjiijkly )(