introduction factorial designs

8/10/2019 Introduction Factorial Designs

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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 30

212 2

30 52 20 4011

2 2

52 20 30 401

2 2

A A

B B

A y y

B y y

AB


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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


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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

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 SurfaceSuppose that we add an

interaction term to the

model:

1 2

1 2

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8

y x x

x x

Interactionis actuallya 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 thesignificance of main effects.


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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 effectsA+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

alevels for factor A, blevels for factor B and n

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

temperatures (3 levels), and n = 4: 32factorial 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: alevels of factor

A, blevels of factor B, nreplicates


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Statistical (effects) model:

is an overall mean, iis the effect of theith level

of the row factor A, jis the effect of thejthcolumn of column factor B and ()ijis the

interaction between iand j.

Testing hypotheses:

1,2,...,

( ) 1, 2,...,

1,2,...,

ijk i j ij ijk

i a

y j b

k n

0)(oneleastat:v.s.,0)(:

0oneleastat:v.s.0:

0oneleastat: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

k

ijk

ij

ij

n

k

ijkij

ja

i

j

n

k

ijkj

ib

j

i

n

k

ijki

abn

yyyy

n

yyyy

an

yyyy

bnyyyy

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 j

i j k i ja b a b n

ij i j ijk ij

i 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 E SS 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

bn

aSSEMSE

EE

a

i

b

j

ij

ABAB

b

j

j

BB

a

i

i

AA


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The ANOVA table:


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Response: Life

ANOVA for Selected Factorial Model

Analysis of variance table [Partial sum of squares]

Sum of Mean F

Source Squares DF Square Value Prob > F

Model 59416.22 8 7427.03 11.00 < 0.0001

A 10683.72 2 5341.86 7.91 0.0020

B 39118.72 2 19559.36 28.97 < 0.0001

AB 9613.78 4 2403.44 3.56 0.0186

Pure E 18230.75 27 675.21

C Total 77646.97 35

Std. Dev. 25.98 R-Squared 0.7652

Mean 105.53 Adj R-Squared 0.6956

C.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: Temperature

Y = A: Material

A1 A1

A2 A2

A3 A3

A: Material

Interaction 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 factorB at a specific level and apply Turkeys test to

the means of factor A at this level.

See Page 174

Compare all abcells means to determine which

one differ significantly


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5.3.3 Model Adequacy Checking

Residual analysis: ijijkijkijkijk yyyye

DESIGN-EXPERT PlotLife

Residual

Normal%p

robab

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 Plot

Life

Predicted

Residuals

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 Plot

Life

Material

Residuals

Residuals vs. Material

-60.75

-34.25

-7.75

18.75

45.25

1 2 3

DESIGN-EXPERT Pl ot

Life

Temperature

Res

iduals

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

i

ijj

a

i

ij

i

b

j

ij

b

j

jii

a

i

b

j

ij

b

j

j

a

i

i

ynnnn

ynannan

ynnbnbn

ynanbnabn

)(:)(

)(:

)(:

)(:

11

11

1 111

0,0,01111

b

j

ij

a

i

ij

b

j

j

a

i

i


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Estimations:

The fitted value:

Choice of sample size: Use OC curves to choose

the proper sample size.

yyyy

yyyy

y

jiijij

jj

ii

ijijjiijk yy


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Consider a two-factor model without interaction:

Table 5.8

The fitted values: yyyy jiijk


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One observation per cell:

The error variance is not estimable because the

two-factor interaction and the error can not beseparated.

Assume no interaction. (Table 5.9)

Tukey (1949): assume ()ij= r

i

j(Page 183)

Example 5.2

27


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Degree of freedom:

Main effect: # of levels1

Interaction: the product of the # of degrees offreedom 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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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 lifeHierarchy principle


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Involve both quantitativeand qualitativefactors

This can be accounted for in the analysis to produce

regression modelsfor the quantitative factors at each

level (or combination of levels) of the qualitativefactors

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A= Material type

B= Linear effect of Temperature

B2= Quadratic effect of

Temperature

AB= Material typeTempLinear

AB2= Material type - TempQuad

B3= Cubic effect of

Temperature (Aliased)


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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

40


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Two randomization restrictions: Latin square

design

An example in Page 200.

Model:

Tables 5.23 and 5.24

ijklljkkjiijkly )(

introduction factorial designs

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