Design and Analysis of Experiments

Design and Analysis of Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Factorial Experiments

Dr. Tai-Yue Wang Department of Industrial and Information Management

National Cheng Kung UniversityTainan, TAIWAN, ROC

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Outline Basic Definition and Principles The Advantages of Factorials The Two Factors Factorial Design The General Factorial Design Fitting Response Curve and Surfaces Blocking in Factorial Design

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Basic Definitions and Principles Factorial Design—all of the possible

combinations of factors’ level are investigated When factors are arranged in factorial design,

they are said to be crossed Main effects – the effects of a factor is defined

to be changed Interaction Effect – The effect that the

difference in response between the levels of one factor is not the same at all levels of the other factors.

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Basic Definitions and Principles Factorial Design without interaction

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Basic Definitions and Principles Factorial Design with interaction

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Basic Definitions and Principles Average response – the average value at one

factor’s level Average response increase – the average value

change for a factor from low level to high level No Interaction:

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

8

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

Basic Definitions and Principles With Interaction:

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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

Basic Definitions and Principles Another way to look at interaction: When factors are quantitative

In the above fitted regression model, factors are coded in (-1, +1) for low and high levels

This is a least square estimates

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Basic Definitions and Principles Since the interaction is small, we can ignore it. Next figure shows the response surface plot

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Basic Definitions and Principles The case with interaction

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Advantages of Factorial design Efficiency Necessary if interaction effects are presented The effects of a factor can be estimated at several

levels of the other factors

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The Two-factor Factorial Design

Two factors a levels of factor A, b levels of factor B n replicates In total, nab combinations or experiments

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The Two-factor Factorial Design – An example

Two factors, each with three levels and four replicates

32 factorial design

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The Two-factor Factorial Design – An example

Questions to be answered: What effects do material type and temperature

have on the life the battery Is there a choice of material that would give

uniformly long life regardless of temperature?

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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

means model


nk

bj

,...,a,i

y ijkijijk

,...,2,1

,...,2,1

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Hypothesis Row effects:

Column effects:

Interaction:

0 oneleast at :

0: 210

ia

a

H

H

0 oneleast at :

0: 210

ia

a

H

H

0)( oneleast at :

0)(:0

ija

ij

H

H

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The Two-factor Factorial Design -- Statistical Analysis

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 square: A:

B:

Interaction:

11)( 1

2

2

a

bn

a

SSEMSE

a

ii

AA

11)( 1

2

2

b

an

b

SSEMSE

a

ii

BB

)1)(1(

)(

)1)(1()( 1 1

2

2

ba

n

ba

SSEMSE

a

i

b

jij

ABAB

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Mean square: Error:

2

)1()(

nab

SSEMSE E

A

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ANOVA table

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Example

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Example

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Example

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Example STATANOVA--GLMGeneral Linear Model: Life versus Material, Temp

Factor Type Levels ValuesMaterial fixed 3 1, 2, 3Temp fixed 3 15, 70, 125

Analysis of Variance for Life, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PMaterial 2 10683.7 10683.7 5341.9 7.91 0.002Temp 2 39118.7 39118.7 19559.4 28.97 0.000Material*Temp 4 9613.8 9613.8 2403.4 3.56 0.019Error 27 18230.7 18230.7 675.2Total 35 77647.0

S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56%

Unusual Observations for LifeObs Life Fit SE Fit Residual St Resid 2 74.000 134.750 12.992 -60.750 -2.70 R 8 180.000 134.750 12.992 45.250 2.01 R

R denotes an observation with a large standardized residual.

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Example STATANOVA--GLM

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Example STATANOVA--GLM

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Estimating the model parameters

1,2,...,

( ) 1, 2,...,

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

i a

y j b

k n

........

.....

.....

...

yyyy

yy

yy

y

jiijij

jj

ii

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Choice of sample size Row effects

Column effects

Interaction effects

D:difference, :standard deviation

2

22

2 a

nbD

2

22

2 b

naD

]1)1)(1[(2 2

22

ba

nD

30


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Appendix Chart V For n=4, giving D=40 on temperature, v1=2,

v2=27, Φ 2 =1.28n. β =0.06

n Φ2 Φ υ1 υ2 β

2 2.56 1.6 2 9 0.45

3 3.84 1.96 2 18 0.18

4 5.12 2.26 2 27 0.06

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The Two-factor Factorial Design -- Statistical Analysis – example with no interaction

Analysis of Variance for Life, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F PMaterial 2 10684 10684 5342 5.95 0.007Temp 2 39119 39119 19559 21.78 0.000Error 31 27845 27845 898Total 35 77647

S = 29.9702 R-Sq = 64.14% R-Sq(adj) = 59.51%

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The Two-factor Factorial Design – One observation per cell

Single replicate The effect model

bj

aiy ijijjiij ,...,2,1

,...,2,1 )(

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The Two-factor Factorial Design – One observation per cell

ANOVA table

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The Two-factor Factorial Design -- One observation per cell

The error variance is not estimable unless interaction effect is zero

Needs Tuckey’s method to test if the interaction exists.

Check page 183 for details.

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The General Factorial Design

In general, there will be abc…n total observations if there are n replicates of the complete experiment.

There are a levels for factor A, b levels of factor B, c levels of factor C,..so on.

We must have at least two replicate (n 2≧ ) to include all the possible interactions in model.

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The General Factorial Design

If all the factors are fixed, we may easily formulate and test hypotheses about the main effects and interaction effects using ANOVA.

For example, the three factor analysis of variance model:

nl

ckbj

ai

y ijklijkjkikijkjiijkl

,...,2,1

,...,2,1,...,2,1

,...,2,1

)()()()(

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The General Factorial Design ANOVA.

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The General Factorial Design where

a

i

b

j

c

k

n

lijklT abcn

yySS

1 1 1 1

2....2

a

iiA abcn

yy

bcnSS

1

2....2

...

1

b

jjB abcn

yy

acnSS

1

2....2

...

1

c

kkC abcn

yy

abnSS

1

2....2

...

1

BA

a

i

b

jijAB SSSS

abcn

yy

cnSS

1 1

2....2

..

1CA

a

i

c

kkiAC SSSS

abcn

yy

bnSS

1 1

2....2

..

1

CB

a

j

c

kjkBC SSSS

abcn

yy

anSS

1 1

2....2

..

1

ACBCABCBA

a

i

b

j

c

kijkABC SSSSSSSSSSSS

abcn

yy

nSS

1 1 1

2....2

.

1

a

i

b

j

c

kijkTE abcn

yy

nSSSS

1 1 1

2....2

.

1

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The General Factorial Design --example

Three factors: pressure, percent of carbonation, and line speed.

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The General Factorial Design --example

ANOVA

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Fitting Response Curve and Surfaces

When factors are quantitative, one can fit a response curve (surface) to the levels of the factor so the experimenter can relate the response to the factors.

These surface could be linear or quadratic. Linear regression model is generally used

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Fitting Response Curve and Surfaces -- example

Battery life data Factor temperature is quantitative

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Example STATANOVA—GLM Response life Model temp, material temp*temp,

material*temp, material*temp*temp Covariates temp


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General Linear Model: Life versus Material Factor Type Levels ValuesMaterial fixed 3 1, 2, 3

Analysis of Variance for Life, using Sequential SS for TestsSource DF Seq SS Adj SS Seq MS F PTemp 1 39042.7 1239.2 39042.7 57.82 0.000Material 2 10683.7 1147.9 5341.9 7.91 0.002Temp*Temp 1 76.1 76.1 76.1 0.11 0.740Material*Temp 2 2315.1 7170.7 1157.5 1.71 0.199Material*Temp*Temp 2 7298.7 7298.7 3649.3 5.40 0.011Error 27 18230.8 18230.8 675.2Total 35 77647.0S = 25.9849 R-Sq = 76.52% R-Sq(adj) = 69.56%

Term Coef SE Coef T PConstant 153.92 11.87 12.96 0.000Temp -0.5906 0.4360 -1.35 0.187Temp*Temp -0.001019 0.003037 -0.34 0.740Temp*Material 1 -1.9108 0.6166 -3.10 0.005 2 0.4173 0.6166 0.68 0.504Temp*Temp*Material 1 0.013871 0.004295 3.23 0.003 2 -0.004642 0.004295 -1.08 0.289

Two kinds of coding methods:1. 1, 0, -12. 0, 1, -1

coding method: -1, 0, +1


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Final regression equation:

]2[*004642.0]1[*013871.0]2[*4173.0

]1[*9108.1*001019.0]2[*7.5

]1[*46.15*5906.092.153

22

2

BABAAB

ABAB

BALife


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Tool life Factors: cutting speed, total angle Data are coded

Fitting Response Curve and Surfaces – example –32 factorial design

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Regression Analysis: Life versus Speed, Angle, ... The regression equation isLife = - 1068 + 14.5 Speed + 136 Angle - 4.08 Angle*Angle - 0.0496 Speed*Speed - 1.86 Angle*Speed + 0.00640 Angle*Speed*Speed + 0.0560 Angle*Angle*Speed - 0.000192 Angle*Angle*Speed*Speed

Predictor Coef SE Coef T PConstant -1068.0 702.2 -1.52 0.163Speed 14.480 9.503 1.52 0.162Angle 136.30 72.61 1.88 0.093Angle*Angle -4.080 1.810 -2.25 0.051Speed*Speed -0.04960 0.03164 -1.57 0.151Angle*Speed -1.8640 0.9827 -1.90 0.090Angle*Speed*Speed 0.006400 0.003272 1.96 0.082Angle*Angle*Speed 0.05600 0.02450 2.29 0.048Angle*Angle*Speed*Speed -0.00019200 0.00008158 - 2.35 0.043

S = 1.20185 R-Sq = 89.5% R-Sq(adj) = 80.2%

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Analysis of Variance

Source DF SS MS F PRegression 8 111.000 13.875 9.61 0.001Residual Error 9 13.000 1.444Total 17 124.000

Source DF Seq SSSpeed 1 21.333Angle 1 8.333Angle*Angle 1 16.000Speed*Speed 1 4.000Angle*Speed 1 8.000Angle*Speed*Speed 1 42.667Angle*Angle*Speed 1 2.667Angle*Angle*Speed*Speed 1 8.000

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We may have a nuisance factor presented in a factorial design

Original two factor factorial model:

Blocking in a Factorial Design

bj

aiy ijijjiij ,...,2,1

,...,2,1 )(

Two factor factorial design with a block factor model:

nk

bj

ai

y ijkkijjiijk

,...,2,1

,...,2,1

,...2,1

)(

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Blocking in a Factorial Design

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Blocking in a Factorial Design -- example

Response: intensity level Factors: Ground cutter and filter type Block factor: Operator

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General Linear Model: Intensity versus Clutter, Filter, Blocks

Factor Type Levels ValuesClutter fixed 3 High, Low, MediumFilter fixed 2 1, 2Blocks fixed 4 1, 2, 3, 4

Analysis of Variance for Intensity, using Sequential SS for Tests

Source DF Seq SS Adj SS Seq MS F PClutter 2 335.58 335.58 167.79 15.13 0.000Filter 1 1066.67 1066.67 1066.67 96.19 0.000Clutter*Filter 2 77.08 77.08 38.54 3.48 0.058Blocks 3 402.17 402.17 134.06 12.09 0.000Error 15 166.33 166.33 11.09Total 23 2047.83

S = 3.33000 R-Sq = 91.88% R-Sq(adj) = 87.55%

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General Linear Model: Intensity versus Clutter, Filter, Blocks

Term Coef SE Coef T PConstant 94.9167 0.6797 139.64 0.000Clutter High 4.3333 0.9613 4.51 0.000 Low -4.7917 0.9613 -4.98 0.000Filter 1 6.6667 0.6797 9.81 0.000Clutter*Filter High 1 2.0833 0.9613 2.17 0.047 Low 1 -2.2917 0.9613 -2.38 0.031Blocks 1 0.417 1.177 0.35 0.728 2 1.583 1.177 1.34 0.199 3 4.583 1.177 3.89 0.001

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Design and Analysis of Experiments