1 prof. indrajit mukherjee, school of management, iit bombay + - + - high (2 pounds ) low (1pound)...

1Prof. Indrajit Mukherjee, School of Management, IIT Bombay

+-

+

-

High

(2 pounds)

Low (1pound)

B=60(18+19+23)

Ab=90(31+30+29)

(1)=80(28+25+27)

A=100(36+32+32)

Am

ount

of

cata

lyst

, B

Reactant concentration A

“-” and “+” denote the low and high levels of a factor, respectively

• Low and high are arbitrary terms

• Geometrically, the four runs form the corners of a square

• Factors can be quantitative or qualitative, although their treatment in the final model will be different


Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

FactorTreatment

combination ReplicateA B 1 2 3 Total- - Alow,Blow 28 25 27 80+ - Ahigh Blow 36 32 32 100- + Alow Bhigh 18 19 23 60+ + Ahigh,Bhigh 31 30 29 90


Statistical Testing - ANOVA

The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?

Analysis of variance for the experiment

Source of variation

Sum of squares

Degrees of freedom

Mean square F0P-value

A208.33

1 208.33 53.15 0.0001

B75.00

1 75.00 19.13 0.0024

AB8.33

1 8.33 2.13 0.1826

Error24.84

8 3.92

Total323.00

11


Effects in The 23 Factorial Design

(a)Geometric View

High

Low

-

+

Factor A

Fact

or B

Fact

or

C

(1) a

b

c

ab

ac

bcabc

-

+

High

Low

+HighLow

-

The 23 Factorial Design

RunFactor

A B C1 - - -2 + - -3 - + -4 + + -5 - - +6 + - +7 - + +8 + + +

(b) Design Matrix


etc, etc, ...

A A

B B

C C

A y y

B y y

C y y

Analysis done via computer

Geometric presentation of contrasts correspondingTo the main effects And Interactions in the 23 design

(a) Main Effects

(b)Two-Factor interaction

(c)Three-Factor interaction

Effects in The 23 Factorial Design


An Example of a 23 Factorial Design

A = gap, B = Flow, C = Power, y = Etch Rate

The Plasma Etch Experiment

Coded Factors Etch Rate Factor Levels

Run A B C Replicate 1 Replicate 2 Total Low(-1) High(+!)

1 -1 -1 -1 550 604 (1)=1154 A(Gap, cm) 0.80 1.2

2 1 -1 -1 669 650 a=1319 B (C2F6 flow, SCCM) 125 200

3 -1 1 -1 633 601 b=1234 C(Power, W) 275 325

4 1 1 -1 642 635 ab=1277

5 -1 -1 1 1037 1052 c=2089

6 1 -1 1 749 868 ac=1617

7 -1 1 1 1075 1063 bc=2138

8 1 1 1 729 860 abc=1589


Estimation of Factor Effects

Effect Estimate Summary

Factor Effect estimate Sum of squaresPercent

Contribution

A -101.625 41,310.5625 7.7736

B 7.375 217.5625 0.0406

C 306.125 374,850.0625 70.5373

AB -24.875 2475.0625 0.4657

AC -153.625 94,402.5625 17.7642

BC -2.125 18.0625 0.0034

ABC 5.625 126.5625 0.0238


ANOVA Summary – Full ModelAnalysis of variance for the Plasma Etching Experiment

Source of variation

Sum of squares Degrees of freedom


Gap (A) 41,310.5625 1 41,310.5625 18.34 0.0027

Gas Flow (B) 217.5625

1217.5625

0.10 0.7639

Power (C) 374,850.0625 1 374,850.0625 166.41 0.0001

AB 2475.0625 1 2475.0625 1.10 0.3252

AC 94,402.5625 1 94,402.5625 41.91 0.0002

BC 18.0625 1 18.0625 0.01 0.9308

ABC 126.5625 1 126.5625 0.06 0.8186

Error 18,020.5000 8 2252.5625Total 531,420.9375 15


Model Coefficients – Full Model

FactorCoefficient Estimated DF

Standard Error

95% CI Low

95% CI High VIF

Inter 776.06 1 11.87 748.7 803.42A -50.81 1 11.87 -78.17 -23.45 1B 3.69 1 11.87 -23.67 31.05 1C 153.06 1 11.87 125.7 180.42 1

AB -12.44 1 11.87 -39.8 14.92 1AC -76.81 1 11.87 -104.17 -49.45 1BC -1.06 1 11.87 -28.42 26.3 1

ABC 2.81 1 11.87 -24.55 30.17 1


Model Coefficients – Reduced Model

FactorCoefficient Estimated DF

Standard Error

95% CI Low

95% CI High VIF

Inter 776.06 1 10.42 753.35 798.77A Gap -50.81 1 10.42 -73.52 28.10 1

C Power 153.06 1 10.42 130.35 175.77 1AC -76.81 1 10.42 -99.52 -54.10 1


Model Interpretation

Cube plots are often useful visual displays of experimental results

200sccm

275w

Gap A

C 2F 6

Flow

Pow

er

C

C=2089

ab

ac=1617

bc=2138 abc=1589

325w

125sccm

1.20cm0.80cm

a=1319

b=1234 ab=1277

(1)=1154

+

- +-

+

-

The 23 Design for the Plasma etch experiment


200sccm

225w

Gap A

C 2F 6

Flow

Pow

er

C

R=15

ab

R=119

R=12 R=131

325w

125sccm

1.20cm0.80cm

R=19

R=32R=7

R=34

+

- +-

+

-

Ranges of etch rates

What do the large ranges

when gap and power are at the

high level tell you?

Cube Plot of Ranges


Spacing of Factor Levels in the Unreplicated 2k Factorial Designs

If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data

More aggressive spacing is usually best

+- +-


The Resin Plant Experiment

Pilot Plant Filtration Value Experiment

Run Number

Factors Run label

Filtration Rate(gal/h)A B C D

1 - - - - (1) 452 + - - - a 713 - + - - b 484 + + - - ab 655 - - + - c 686 + - + - ac 607 - + + - bc 808 + + + - abc 659 - - - + d 43

10 + - - + ad 10011 - + - + bd 4512 + + - + abd 10413 - - + + cd 7514 + - + + acd 8615 - + + + bcd 7016 + + + + abcd 96


The Resin Plant Experiment

BC

D - +

6548

68 60

65

7145

80

10445

75 86

96

10043

70

Data from the Pilot Plant Filtration rate Experiment


A B AB C AC BC ABC D AD BD ABD CD ACD BCD ABCD1 - - + - + + - - + + - + - - +a + - - - - + + - - + + + + - -b - + - - + - + - + - + + - + -ab + + + - - - - - - - - + + + +c - - + + - - + - + + - - + + -ac + - - + + - - - - + + - - + +bc - + - + - + - - + - + - + - +abc + + + + + + + - - - - - - - -d - - + - + + - + + - + - + + -ad + - - - - + + + - - - - - + +bd - + - - + - + + + + - - + - +abd + + + - - - - + - + + - - - -cd - - + + - - + + + - + + - - +acd + - - + + - - + - - - + + - -bcd - + - + - + - + + + - + - + -abcd + + + + + + = + - + + + +r + +

Contrast constant for the 24 design


Factor Effect Estimates and sums of Squares for the 24 Factorial

Model TermEffect

EstimateSum of Squares

Percent Contribution

A 21.625 1870.563 32.6397

B 3.125 39.0625 0.681608

C 9.875 390.0625 6.80626

D 14.625 855.5625 14.9288

AB 0.125 0.0625 0.00109057

AC -18.125 1314.063 22.9293

AD 16.625 1105.563 19.2911

BC 2.375 22.5625 0.393696

BD -0.375 0.5625 0.00981515

CD -1.125 5.0625 0.0883363

ABC 1.875 14.0625 0.45379

ABD 4.125 68.0625 1.18763

ACD -1.625 10.5625 0.184307

BCD -2.625 27.5625 0.480942

(ABCD) 1.375 7.5625 0.131959


Design Projection: ANOVA Summary for the Model as a 23 in Factors A, C, and D

Analysis of variance for the Pilot Plant filtration Rate Experiment in A, C, and D

Source of variation

Sum of squares

Degrees of freedom


A1870.5625

1 1870.5625 83.36 <0.0001

C390.0625

1 390.0625 17.38 <0.0001

D855.5625

1 855.5625 38.13 <0.0001

AC1314.0625

1 1314.0625 58.56 <0.0001

AD1105.5625

1 1105.5625 49.27 <0.0001

CD5.06

1 5.06 <1

ACD10.56

1 10.56 <1


Model Interpretation – Main Effects and Interactions

90807060

+

Resp

onse

50 -

90807060

+

Resp

onse

50 -

90807060

+

Resp

onse

50 -

1009080706050

+

Resp

onse

40

1009080706050

+

Resp

onse

40- -


The Drilling Experiment Example 6.3

A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill

BC

D - +

5.704.98

3.24 3.44

9.07

1.981.68

9.97

9.437.77

4.09 4.53

16.03

2.442.07

11.75

Data from the Drilling Experiment


The Box-Cox MethodA log transformation is recommended

The procedure provides a confidence interval on the transformation parameter lambda

If unity is included in the confidence interval, no transformation would be needed


Effect Estimates Following the Log Transformation

Three main effects are large

No indication of large interaction effects

What happened to the interactions?


ANOVA Following the Log Transformation

Analysis of variance for the Log Transformation

Source of variation

Sum of squares

Degrees of freedom


B5.345

1 5.345 381.79 <0.0001

C1.339

1 1.339 95.64 <0.0001

D0.431

1 0.431 30.79 <0.0001

Error0.173

12 0.014

Total7.288

15


Adjusted multipliers for Lenth’s method

Suggested because the original method makes too many type I errors, especially for small designs (few contrasts)

Simulation was used to find these adjusted multipliers

Lenth’s method is a nice supplement to the normal probability plot of effects

JMP has an excellent implementation of Lenth’s method in the screening platform

Number of Contrasts 7 15 31

Original ME 3.764 2.571 2.218Adjusted ME 2.295 2.14 2.082Original SME 9.008 5.219 4.218Adjusted SME 4.891 4.163 4.03


The least squares estimate of β is

1

0

14

2

12

ˆ

4 0 0 0 (1)

0 4 0 0 (1)

0 0 4 0 (1)

0 0 0 4 (1)

(1)

4ˆ (1) (ˆ (1)1ˆ (1)4

(1)ˆ

a b ab

a ab b

b ab a

a b ab

a b ab

a b ab a ab ba ab b

b ab a

a b ab

-1β = (X X) X y

I

1)

4(1)

4(1)

4

b ab a

a b ab

The matrix is diagonal – consequences of an orthogonal design

The regression coefficient estimates are exactly half of the ‘usual” effect estimates

The “usual” contrasts


0.00 0.25 0.50 0.75 1.00

0.00

0.25

0.50

0.75

1.00

Fraction of Design Space

Std

Err

Meal

Min StdErr Mean 0.500Max StdErr Mean 1.000CubicalRadius = 1 Points = 10000


no "curvature"F Cy y The hypotheses are:

01

11

: 0

: 0

k

iii

k

iii

H

H

2

Pure Quad

( )F C F C

F C

n n y ySS

n n

This sum of squares has a single degree of freedom


ANOVA for Example 6.6 (A Portion of Table 6.22)

Analysis for the Reduce Model

Source of variation

Sum of squares

Degrees of freedom


Model5535.81

5 1107.16 59.02 <0.000

A1870.5625

1 1870.5625 99.71 <0.0001

C390.0625

1 390.0625 20.79 0.0005

D855.5625

1 855.5625 45.61 <0.000

AC1314.0625

1 1314.0625 70.05 <0.000

AD1105.5625

1 1105.5625 58.93 <0.000


If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a second-order response

surface model

x2

(a) Two Factors (b)Three Factors

x1x1

Central Composite Designs

x3x2


Catalyst Type

Time

Tem

pera

ture

A 23 factorial design with one qualitativefactor and center points

Center Points and Qualitative Factors

1 prof. indrajit mukherjee, school of management, iit bombay + - + - high (2 pounds ) low (1pound)...

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b c2f6 flow