two-level fractional factorial...

1 Andy Guo

Two-Level FractionalFactorial Design

Reference• DeVor, Statistical Quality Design and Control, Ch. 19, 20

2 Andy Guo

Types of Experimental Design

• One-factor design (2 levels)– Hypothesis testing, confidence interval

(randomized design)– Paired comparison (block design)

• One-factor design (k levels)– Completely randomized design– Randomized complete block design– Two block design (Latin square)

• Two-factor design• Full factorial design (2 level)• Fractional factorial design (2 level)• Robust design• Nested design• Split-plot design

• Response surface method design– Central composite design– Box-Behnken design– Computer-aided design

(D, G optimal design)• EVOP• Steepest ascent

Parallel-type approach Sequential-type approach

3 Andy Guo

Redundancy in Two-Level Factorials• A 10-variable experiment require 210 = 1024 tests. Such a test plan is simply

prohibitive in size.• In theory, the following effects are estimated from the test plan:

1 Mean response 10 Main effects 45 Two-factor interaction effects 120 Three-factor interaction effects 210 Four-factor interaction effects 252 Five-factor interaction effects 210 Six-factor interaction effects 120 Seven-factor interaction effects 45 Eight-factor interaction effects 10 Nine-factor interaction effects 1 Ten-factor interaction effects---------------------------------------------------1024 Variable effects.

• In reality, interaction effects involving three factors or more are small and can besimply ignored. This fact provides the opportunity for fractional factorial designs.

4 Andy Guo

Why do Fractional Factorial Designs Work?

• The sparsity of effects principle– There may be lots of factors, but few are important– System is dominated by main effects, low-order interactions

• The projection property– Every fractional factorial contains full factorials in fewer factors

• Sequential experimentation– Can add runs to a fractional factorial to resolve difficulties (or

ambiguities) in interpretation

5 Andy Guo

A 23-1 Fractional Factorial DesignTest 1 2 3=12

1 - - +2 + - -3 - + -4 + + +

1. 3 variables are studied2. in 23-1 = 4 tests3. p = 1 of the variables is introduced into a 22 full factorial4. by assigning it to the interaction 12 (i.e., let 3 = 12)

(+,-,-)

(-,-,+)

(+,+,+)

(-,+,-)

3

1

2

6 Andy Guo

A 24-1 Fractional Factorial Design

Test 1 2 3 4=1231 - - - -2 + - - +3 - + - +4 + + - -5 - - + +6 + - + -7 - + + -8 + + + +

1. 4 variables are studied2. in 24-1 = 8 tests3. p = 1 of the variables is introduced into a 23 full factorial4. by assigning it to the interaction 123 (i.e., let 4 = 123)

7 Andy Guo

A 24-1 Fractional Factorial Design

(+,+,-)

(-,+,+)

(+,-,+)

(-,-,-) (+,-,-)

(-,-,+)

(+,+,+)

(-,+,-)

3

1

2

4- +

8 Andy Guo

Calculation Matrix and Confounding Patterns

The following pairs of variables are confounded1 and 234 12 and 342 and 134 13 and 243 and 124 23 and 144 and 123 average(I) and 1234

Test I 1 2 3 4 12 13 14 23 24 34 123 124 134 234 12341 + - - - - + + + + + + - - - - +2 + + - - + - - + + - - + - - + +3 + - + - + - + - - + - + - + - +4 + + + - - + - - - - + - - + + +5 + - - + + + - - - - + + + - - +6 + + - + - - + - - + - - + - + +7 + - + + - - - + + - - - + + - +8 + + + + + + + + + + + + + + + +

9 Andy Guo

Effect Estimation

If three-factor and four-factor interactions can be neglected, we have:

l0 estimates mean + (1/2)(1234)l1 estimates 1 + 234l2 estimates 2 + 134l3 estimates 3 + 124

l12 estimates 12 + 34l13 estimates 13 + 24l23 estimates 23 + 14l123 estimates 4 + 123

l0 estimates mean l1 estimates 1l2 estimates 2l3 estimates 3

l12 estimates 12 + 34l13 estimates 13 + 24l23 estimates 23 + 14l123 estimates 4

10 Andy Guo

Procedure for Fractional Factorial Designs

Step 1: Define the base design

Example: 6 variables, only 8 tests are allowed.

Base design: 23 full factorial design

Test I 1 2 3 12 13 23 123 y1 + - - - + + + - y1

2 + + - - - - + + y2

3 + - + - - + - + y3

4 + + + - + - - - y4

5 + - - + + - - + y5

6 + + - + - + - - y6

7 + - + + - - + - y7

8 + + + + + + + + y8

Divisor 8 4 4 4 4 4 4 4

11 Andy Guo

Step 2: Introduction of Additional Variables

4 = 125 = 136 = 23

Design matrix: 26-3 fractional factorial design

Test 1 2 3 4 5 61 - - - + + +2 + - - - - +3 - + - - + -4 + + - + - -5 - - + + - -6 + - + - + -7 - + + - - +8 + + + + + +

12 Andy Guo

Step 3: Obtain the Defining Relation• 4 = 12

4 x 4 = 12 x 4,but since 4 x 4 = I, a column of (+) signs, we have I = 124.

• The defining relation is given byI = 124, I = 135, I = 236 (the generators)plus two-at-a-time products:(124)(135) = 2345(124)(236) = 1346(135)(236) = 1256plus the three-at-a-time products:(124)(135)(236) = 456.

• The complete defining relation I is thereforeI = 124 = 135 = 236 = 2345 = 1346 = 1256 = 456.

13 Andy Guo

Step 4: Reveal the Complete Confounding Structure

• With (1)I = (1)124 = (1)135 = (1)236 = (1)2345 = (1)1346 = (1)1256 = (1)456We have 1 = 24 = 35 = 1236 = 12345 = 346 = 256 = 1456.

• Assuming that third-and higher-order interactions can be neglected,l1 estimates 1 + 24 + 35.

• Summary of the confounding structure:l0 estimates mean l12 estimates 12 + 4 + 56l1 estimates 1 + 24 + 35 l13 estimates 13 + 5 + 46l2 estimates 2 + 14 + 36 l23 estimates 23 + 6 + 45l3 estimates 3 + 15 + 26 l123 estimates 34 + 25 + 16

14 Andy Guo

Concept of Resolution

• Resolution = the number of letters (numbers) in the shortest length word (term) inthe defining relation, excluding I.I = 124 = 135 = 2345 => Resolution IIII = 1235 = 2346 = 1456 => Resolution IV

• Resolution III => some main effects are confounded with two-factor interactions.

• Resolution IV => some main effects are confounded with three-factor interactions,and some two-factor interactions are confounded with othertwo-factor interactions.

• Resolution V => some main effects are confounded with four-factor interactionsand some two-factor interactions are confounded with three-factor interactions.

15 Andy Guo

Concept of Design Resolution

• Higher-resolution designs seem more desirable since they provide the opportunityfor low-order effect estimates to be determined in an un-confounded state,assuming that higher-order interaction effects can be neglected.

• There is a limit to the number of variables that can be considered in a fixed numberof tests while maintaining a pre-specified resolution requirement.

• No more than (n-1) variables can be examined in n tests (n is a power of 2) tomaintain a design resolution of at least III. Such designs are commonly referred toas saturated designs.

– Examples are 23-1, 27-4, 215-11, 231-26

– For saturated designs all interactions in the base design variables are used to introduceadditional variables.

16 Andy Guo

Two-Level Fractional Factorial Designswith Maximum Resolution

Number ofFactors Fraction Number of

RunsDefining Relation

(omitting generalized interactions)

3 4 I=123

4 8 I=1234

5 16 I=12345

8 I=124=135

6 32 I=123456

16 I=1235=2346

8 I=124=135=236

13III2 −

14IV2 −

15V2 −

25III2 −

16VI2 −

26IV2 −

36III2 −

17 Andy Guo

Two-Level Fractional Factorial Designswith Maximum Resolution

Number ofFactors Fraction Number of

RunsDefining Relation

(omitting generalized interactions)

7 64 I=1234567

32 I=12346=12457

16 I=1235=2346=1347

8 I=124=135=236=1237

8 64 I=12347=12568

32 I=1236=1247=23458

16 I=2345=1346=1237=1248

9 128 I=134678=235679

64 I=12347=13568=34569

32 I=23456=13457=12458=12359

16 I=1235=2346=1347=1248=12349

17VII2 −

27IV2 −

28V2 −

38IV2 −

48IV2 −

29VI2 −

39IV2 −

47III2 −

37IV2 −

49IV2 −

59III2 −

18 Andy Guo

Orthogonal Arrays and Two-Level Fractional Factorial Designs

L8 (27) Orthogonal Array

2III7-4 Fractional Factorial Design

FactorA B C D E F G

Test 1 2 3 4 5 6 7 Result1 1 1 1 1 1 1 1 y1

2 1 1 1 2 2 2 2 y23 1 2 2 1 1 2 2 y3

4 1 2 2 2 2 1 1 y45 2 1 2 1 2 1 2 y56 2 1 2 2 1 2 1 y6

7 2 2 1 1 2 2 1 y78 2 2 1 2 1 1 2 y8

-12 -13 -23 +123Test 1 2 3 4 5 6 7

1 - - - - - - -2 + - - + + - +3 - + - + - + +4 + + - - + + -5 - - + - + + +6 + - + + - + -7 - + + + + - -8 + + + - - - +

D B A F E C G(4) (2) (1) (6) (5) (3) (7)

19 Andy Guo

Orthogonal Arrays and Two-Level Fractional Factorial Designs

L16 (215) Orthogonal Array

2III8-4 Fractional Factorial Design

234 -14 123 -1234 Test 1 2 3 4 5 6 7 8

1 - - - - - - - - 2 + - - - - + + + 3 - + - - + - + + 4 + + - - + + - - 5 - - + - + - + + 6 + - + - + + - - 7 - + + - - - - - 8 + + + - - + + + 9 - - - + + + - + 10 + - - + + - + - 11 - + - + - + + - 12 + + - + - - - + 13 - - + + - + + - 14 + - + + - - - + 15 - + + + + + - + 16 + + + + + - + - C B A F E H G D

A B Ax x x

F A e B e B E C H e D e D G DTest 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 12 1 1 1 1 1 1 1 2 2 2 2 2 2 2 23 1 1 1 2 2 2 2 1 1 1 1 2 2 2 24 1 1 1 2 2 2 2 2 2 2 2 1 1 1 15 1 2 2 1 1 2 2 1 1 2 2 1 1 2 26 1 2 2 1 1 2 2 2 2 1 1 2 2 1 17 1 2 2 2 2 1 1 1 1 2 2 2 2 1 18 1 2 2 2 2 1 1 2 2 1 1 1 1 2 29 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

10 2 1 2 1 2 1 2 2 1 2 1 2 1 2 111 2 1 2 2 1 2 1 1 2 1 2 2 1 2 112 2 1 2 2 1 2 1 2 1 2 1 1 2 1 213 2 2 1 1 2 2 1 1 2 2 1 1 2 2 114 2 2 1 1 2 2 1 2 1 1 2 2 1 1 215 2 2 1 2 1 1 2 1 2 2 1 2 1 1 216 2 2 1 2 1 1 2 2 1 1 2 1 2 2 1

20 Andy Guo

Sequential Experimentation

A 27-4 fractional factorial designResolution = III

I = 124 = 135 = 236 = 1237

Test 1 2 3 4 5 6 7 Overrun(%)

1 - - - + + + - 1152 + - - - - + + 813 - + - - + - + 1104 + + - + - - - 695 - - + + - - + 1746 + - + - + - - 997 - + + - - + - 808 + + + + + + + 63

21 Andy Guo

Estimation of Effects and Results

• The main effects of variables 1, 2, 6 alone are important.• The main effects of variables 1and 6, as well as the interactions 14

and/or 36 are important.• We might conclude that l23 is large because interactions 17 and/or

23 are important instead of variable 6.• We might conclude that l1 is large because interactions 24 and/or

67 are important instead of variable 1.

-60 -40 -20 0 20 40 60

0.52.05.010.020.0

50.0

80.090.095.098.099.5

23+6+17+452+14+36+57

1+24+35+67

Estimate of linear combination of effects

Cum

ulat

ive

prob

abili

ty (%

)

22 Andy Guo

Mirror Image Design

Switch the signs for all of the columns in the original designA 27-4 fractional factorial design

Resolution = IIII = -124 = -135 = -236 = 1237

Test 1 2 3 4 5 6 7 Overrun(%)

9 + + + - - - + 8410 - + + + + - - 6911 + - + + - + - 5612 - - + - + + + 16113 + + - - + + - 5614 - + - + - + + 4015 + - - + + - + 9216 - - - - - - - 208

23 Andy Guo

Comparison of Original and Mirror Image Tests

Mirror Image Design Original Eight Tests Additional Eight Tests

l0 = 98.875 and estimates mean l'0 = 95.750 and estimates mean l1 = -41.750 and estimates 1+24+35+67 l'1 = -47.500 and estimates 1-24-36-57 l2 = -36.750 and estimates 2+14+36+57 l'2 = -67.000 and estimates 2-14-36-57 l3 = 10.250 and estimates 3+15+26+47 l'3 = -6.500 and estimates 3-15-26-47 l12 = 12.750 and estimates 12+4+37+56 l'12 = 63.000 and estimates 12-4+37+56 l13 = -4.250 and estimates 13+5+27+46 l'13 = 2.500 and estimates 13-5+27+46 l23 = -28.250 and estimates 23+6+17+45 l'23 = 35.000 and estimates 23-6+17+45 l123 = 16.250 and estimates 34+25+16+7 l'123 = -3.000 and estimates -34-25-16+7

24 Andy Guo

Unconfounding the Main Effects

( ) ( )[ ] ( ) ( )[ ]

7. estimate 6.6252

'

174523 estimates 375.32

'

462713 estimates 875.02

'

563712 estimates 875.372

'

3 estimate 875.12

'

2 estimate 875.512

'

Similarly,.1 estimate 625.44

6735241673524121 estimates 750.47750.41

21

2'

2323

2323

1313

1212

33

22

11

=+

++=+

++−=+

++=+

=+

−=+

−=

−−−++++��

�−+−��

��

�=+

ll

ll

ll

ll

ll

ll

ll

25 Andy Guo


( ) ( )[ ] ( ) ( )[ ]

16.2534 estimates 625.92

'

6 estimates 625.312

'

5 estimates 375.32

'

4 estimates 125.252

'

476215 estimates 375.82

'

573614 estimates 125.152

'673524 estimates 2.875

6735241673524121 estimates 750.47750.41

21

2'

123123

2323

1313

1212

33

22

11

++=−

−=−

−=−

−=−

++=−

++=−++=

−−−−+++��

�−−−��

��

�=−

ll

ll

ll

ll

ll

ll

ll

26 Andy Guo


1234567456167257347236135124 estimates 125.3'

:obtained isresult following the,' and between difference theBy taking

mean. estimates 313.972

'

00

00

00

+++++++=−

=+

llll

ll

27 Andy Guo

Results of the Combined DesignsThe combined designs are the same as a 27-3 design.

Resolution = IVI = 1237 = 2345 = 1346

Estimate of 1 = -44.325Estimate of 2 = -51.875Estimate of 3 = 1.875Estimate of 12+37+56 = 37.875Estimate of 13+27+46 = -0.875Estimate of 23+45+17 = 3.375Estimate of 7 = 6.625Estimate of error = 3.125Estimate of 24+35+67 = 2.875Estimate of 14+36+57 = 15.125Estimate of 15+26+47 = 8.375Estimate of 4 = -25.125Estimate of 5 = -3.375Estimate of 6 = -31.675Estimate of 34+25+16 = 9.625

28 Andy Guo

Results Based on Tests 1 to 8 and 9 to 16

• The main effects of variables 1, 2, 4, 6 are important.• The interaction 12 is important.

-60 -40 -20 0 20 40 60

0.52.05.010.020.0

50.0

80.090.095.098.099.5

12+37+56

1


Cum

ulat

ive

prob

abili

ty (%

)

2

64

29 Andy Guo

Alternative Experimental Strategies

Principal fraction• Original 27-4 design: I = 124 = 135 = 236 = 1237

Alternate fractions• Mirror image: I = -124 = -135 = -236 = 1237• Family of fractional factorials I = ±124 = ± 135 = ± 236 = ± 1237

Depending on the interpretation of the results of the principal fraction, wemay choose any one of several other alternate fractions to achieve aparticular result when two fractions are combined.

30 Andy Guo

One Alternative Design ExampleSuppose that the larger linear combinations from the first experiment,l1 estimates 1 + 24 + 67 = - 41.75l2 estimates 2 + 14 + 36 = - 36.75l3 estimates 23 + 6 + 17 = - 28.25

Suppose that our knowledge suspects that variable 1 was important.=> folding only the first column

Test 1 2 3 4 5 6 7 Overrun(%)

17 + - - + + + - 6618 - - - - - + + 17119 + + - - + - + 14720 - + - + - - - 12221 + - + + - - + 5122 - - + - + - - 14823 + + + - - + - 4924 - + + + + + + 14

Generators：I = -124, I = -135, I = 236, I = -1237

31 Andy Guo

Comparison of Test Results

Principal Fraction Design Tests 1-8

Alternative Fraction Design Tests 17-24

l0 = 98.875 and estimates mean l''0 = 96.000 and estimates mean l1 = -41.750 and estimates 1+24+35+67 l''1 = -35.500 and estimates 1-24-35-67 l2 = -36.750 and estimates 2+14+36+57 l''2 = -26.000 and estimates 2-14+36+57 l3 = 10.250 and estimates 3+15+26+47 l''3 = -61.000 and estimates 3-15+26+47 l12 = 12.750 and estimates 12+4+37+56 l''12 = -65.500 and estimates 12-4-37-56 l13 = -4.250 and estimates 13+5+27+46 l''13 = 4.500 and estimates 13-5-27-46 l23 = -28.250 and estimates 23+6+17+45 l''23 = -42.000 and estimates 23+6-17+45 l123 = 16.250 and estimates 34+25+16+7 l''123 = 0.500 and estimates -34-25+16-7

32 Andy Guo

Results Based on Tests 1 to 8 and 17 to 24• Variable 1 and the interactions 12 and 15 are important.• By folding a single column, we can estimate the main effect

of that variable and its two-factor interactions better.

-60 -40 -20 0 20 40 60

0.52.05.010.020.0

50.0

80.090.095.098.099.5

23+6+451


Cum

ulat

ive

prob

abili

ty (%

)

2+36+574+37+56

3+26+47

12

15

33 Andy Guo

27-4III Family of Fractional Factorials

Fraction Generators Combined with Principal Fraction Gives Estimates of：*

Principal I=+124 I=+135 I=+236 I=+1237 -- A1 I=-124 I=+135 I=+236 I=+1237 4,14,24,34,45,46,47 A2 I=+124 I=-135 I=+236 I=+1237 5,15,25,35,45,56,57 A3 I=-124 I=-135 I=+236 I=+1237 A4 I=+124 I=+135 I=-236 I=+1237 6,16,26,36,46,56,67 A5 I=-124 I=+135 I=-236 I=+1237 A6 I=+124 I=-135 I=-236 I=+1237 A7 I=-124 I=-135 I=-236 I=+1237 All main effects A8 I=+124 I=+135 I=+236 I=-1237 7,17,27,37,47,57,67 A9 I=-124 I=+135 I=+236 I=-1237 A10 I=+124 I=-135 I=+236 I=-1237 A11 I=-124 I=-135 I=+236 I=-1237 1,12,13,14,15,16,17 A12 I=+124 I=+135 I=-236 I=-1237 A13 I=-124 I=+135 I=-236 I=-1237 2,12,23,24,25,26,27 A14 I=+124 I=-135 I=-236 I=-1237 3,13,23,34,46,56,67 A15 I=-124 I=-135 I=-236 I=-1237

* Assuming that third- and higher-order interactions are negligible.

34 Andy Guo

Summary of Results

Principal Fraction + Mirror Image Design (A7)

Principal Fraction + Alternative Design (A11)

1 estimated as -44.625 1 estimated as -38.625 2 estimated as -51.875 2+36+57 estimated as -31.375 12+37+56 estimated as 37.875 12 estimated as 39.125 4 estimated as -25.125 4+37+56 estimated as -26.375 6 estimated as -31.625 23+6+45 estimated as -35.125 3+26+47 estimated as -25.375 15 estimated as 35.625

35 Andy Guo

Interpretation• Both combined designs produce an unconfounded estimates of the

main effect of variable 1.• Both combined designs show that interaction 12 is important.• Both combined design show that main effect 4 is important.• Both combined design show that main effect 6 is important.• Based on (principal + A11), interaction 15 is important.• Both combined design show that main effect 2 is important.• The interactions (36+57) and (26+47) are important.

4523 estimates 500.32

''2

'


''2

'


''2

'

23232323

13131313

12121212

+−=+−+

+−=+−+

+−=+−+

llll

llll

llll

25.34 estimates 250.12

''2

'


''2

'


''2

'

123123123123

3333

2222

+=−−−

+−=−−−

+=−−−

llll

llll

llll

36 Andy Guo

Possible Strategies for Sequential Experimentation

Temperature TimePr

essu

re

(a) Move to new locationto explore an apparent

trend in response

(f) Augment to modelapparent curvature

(e) Replicate to improve estimatesof effects or because some runs

were incorrectly made

(d) Drop and add factorsbecause the original factor

catalyst feed rate is negligible

(c) Rescale some factors becausethey may have been variedover inappropriate ranges

(b) Add another fractionto resolve ambiguities

from the original fractionInitial design

Temperature

Pres

sure

Catalys

tfee

d rate

37 Andy Guo

Case Study: Epitaxy Process

• The objective is to study how the control factorsaffect the thickness of the the epitaxial layer.

• Eight factors were studied:– A: rotation method– B: the code of the wafers– C: deposition temperature– D: deposition time– E: arsenic gas flow rate– F: HCl etch temperature– G: HCl flow rate– H: nozzle position

38 Andy Guo

Experimental Design and Results• 28-4 fractional factorial design• A full 24 factorial in A, B, C, E• D=-ABC, F=ABE, G=ACE, H=CBE

A B C D E F G H log(s2)- - - + - - - - 14.821 -0.4425- - - + + + + + 14.888 -1.1989- - + - - - + + 14.037 -1.4307- - + - + + - - 13.880 -0.6505- + - - - + - + 14.165 -1.4230- + - - + - + - 13.860 -0.4969- + + + - + + - 17.757 -0.3267- + + + + - - + 14.921 -0.6270+ - - - - + + - 13.972 -0.3467+ - - - + - - + 14.032 -0.8563+ - + + - + - + 14.843 -0.4369+ - + + + - + - 14.415 -0.3131+ + - + - - + + 14.878 -0.6154+ + - + + + - - 14.932 -0.2292+ + + - - - - - 13.907 -0.1190+ + + - + + + + 13.914 -0.8625

y

39 Andy Guo

Analysis Results of the Main Effects

• Mean thicknessA = -0.055, B = 0.056, C = -0.109, D = 0.836E = -0.067, F = 0.060, G = -0.098, H = 0.142

• Variability of thicknessA = 0.352, B = 0.122, C = 0.105, D = 0.249E = -0.012, F = -0.072, G = -0.101, H = -0.566

• Factor D has the largest impact on the mean level.• Factors A and H affect the variability.

40 Andy Guo

Case Study: Plasma Etching

• A nitride etch process on a single-wafer plasma etcher• Output: etch rate• Inputs: gap, pressure, C2F6 flow rate, power

Design FactorGap Pressure C2F6 Flow Power A B C D

Level (cm) (m Torr) (SCCM) (W)Low (-) 0.80 450 125 275High (+) 1.20 550 200 325

41 Andy Guo

24-1 Fractional Factorial Design

Run A B C D=ABC Etch Rate1 - - - - 5502 + - - + 7493 - + - + 10524 + + - - 6505 - - + + 10756 + - + - 6427 - + + - 6018 + + + + 729

42 Andy Guo

Estimated Effects

lA = A + BCD = -127.00lB = B + ACD = 4.00lC = C + ABD = 11.50lD = D + ABC = 290.51lAB = AB + CD = -10.00lAC = AC + BD = -25.50lAD = AD + BD = 197.50

A, D, AD are significant

two-level fractional factorial...

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