Present Goal

0% Knowledge 100%

Objective:

No. of Factors:

Model:

Information:

Designs:

Screening

6 - 20Continuous &/or Discrete

Linear

Identify important variables;Crude predictions of effects

Fractional-Factorial orPlackett-Burman

Constrained Optimization

3 - 8Continuous &/or Discrete

Linear + Cross-products (interactions)

Good predictions of effects and interactions

2-Level Factorial (+ Center Points)

Unconstrained Optimization

3 - 6Continuous only

Linear + Cross-products + Quadratics

Good predictions of effects, interactions, and curvature.

Central Composite orBox-Behnken

Extrapolationor Optimization

1 - 5

Mechanistic Model

Estimate parameters in theoretical model

Special (computer generated)

Constrained Optimization

3 - 8Continuous &/or Discrete

Linear + Cross-products (interactions)

Good predictions of effects and interactions

2-Level Factorial (+ Center Points)

Constrained Optimization

3-8Continuous &/or Discrete

Linear + Cross-products (interactions)

Good predictions of effects and interactions

2-Level Factorial (+ Center Points)

S

C

U

Relative Importance of Three Stages of Experimentation

U-Unconstrained Optimization

(Response Surfaces Chapter 10)

C-Constrained Optimization

(Main effects and interactions)

S-Screening Experiments

(What Factors are important)

A Poor Solution is to Use One-at-a-Time Experiments

Run A B C D E F G H 1 - - - - - - - - 2 + - - - - - - - 3 - + - - - - - - 4 - - + - - - - - 5 - - - + - - - - 6 - - - - + - - - 7 - - - - - + - - 8 - - - - - - + - 9 - - - - - - - +

Fractional Factorial Experiments

• Method for Strategically Picking a Subset of a 2k Design

• Used for Screening purposes

• Has much Higher Power for detecting Effects through hidden replication

• Can be used to estimate some interactions and limited optimization

A B C AB AC BC ABC-1 -1 -1 1 1 1 -11 -1 -1 -1 -1 1 1

-1 1 -1 -1 1 -1 11 1 -1 1 -1 -1 -1

-1 -1 1 1 -1 -1 11 -1 1 -1 1 -1 -1

-1 1 1 -1 -1 1 -11 1 1 1 1 1 1

I = C A = C I = ABC

Half-Fraction of 23

Paradigms That Justify Use of Fractional Factorials

•

•

Venus – Moon – Jupiter align

Jupiter Mars Venus Crescent Moon

Hierarchical Ordering Principle

▪Although its possible that three planets may align with the moon, its more often that two planets will align with moon than three

▪Likewise though three factor interactions and higher order interactions are possible, its more likely that large effects will be main effects or two factor interactions

142

Creating a half fraction design in SAS

I = ABCDE

Would these conclusions have been reached using one at a time experimentation?

powerfraction theis factors, ofnumber theis 222

1pkpkk

p

· In a one half fraction of a 2k experiment every effect that could be estimated was confounded with one other effect, thus one half the effects had to be assumed negligible in order to interpret or explain the results

· In a one quarter fraction of a 2k experiment every effect that can be estimated is confounded with three other effects, thus three quarters of the effects must be assumed negligible in order to interpret or explain the results

· In a one eighth fraction of a 2k experiment every effect that can be estimated is confounded with seven other effects, thus seven eights of the effects must be assumed negligible in order to interpret or explain the results, etc.

Creating a 2k-p Design

1. Create a full two-level factorial in k-p factors

2. Add each of the remaining p factors by assigning them to a column of signs for an interaction among the first k-p columns

These are the generators

the generalized interaction

the generators

the defining relation

Defining Relation

Confounding Pattern orAlias Structure

26-3 design base design in 6-3 = 3Factors A, B, C

The three factor generalized interaction is

The defining relation is

New Two-Level Design ► Define Variables… ► Add>Select Design…

Example ¼ Fraction of 26

One possible set of generators is:

Resulting in the following Alias Structure

Another possible set of generators is:

Resulting in the following Alias Structure

R, the resolution, is the length of the shortest word in the defining relation.

Resolution as a criteria for choosing generators

Resolution III – main effects confounded with two-factor interactions

Resolution IV – main effects confounded with three-factor interactions, and two factor interactions confounded with other two-factor interactions

Resolution V – main effects confounded with four-factor interactions, two-factor interactions confounded with three-factor interactions. In this case if you are willing to assume three factor interactions and higher are negligible, you can estimate all main effects and two factor interactions

Higher Resolution means main effects are confounded with higher order interactions

Minimum Aberration as a criteria for choosing generators

I = ABCDF = ABCEG = DEFG

272 IV

d2 F = ABC, G = ADE

d1 F = ABCD, G = ABCE

I = ABCF = ADEG = BCDEFG

Which is better?

d1 (0, 1, 2)

Word length pattern: length 3 length 4

length 5

d2 (0, 2, 0, 1)

Symbolically: (A3, A4, A5, …) Ar is number of words of length r

Number of clear Effects as a criteria for choosing generators

An effect is defined to be clear if none of its aliases are main effects or two factor interactions

See Example64.sas

Only 56% Eucalyptus used in Brazilian forests

Hemicellulose hydrolyzate

acid treatment

Paecilomyces variolii

Fermentation

Edible Biomassrich inessential amino acids

Generators for minimum aberration 482 IV

BH

EG

Maximum appears to be with Ammonium Sulfate and Sodium Phosphate both at 2 g/L

CH

BEG

Recap

8 Factors would require 28 = 256 for full factorial

16 + 8 = 24 resulted in plausible interpretation and identification of optimal results

Label Factor Optimal Setting

B Rice Bran 30.0 g/L

E Ammonium Sulfate 2.0 g/L

G Sodium Phosphate 0.0 g/L

BECDAF

DEBCF

DFACE

EFABD

BFAEC

CFADB

CEBDA

ABEFACDFBCDEDEFBCFACEABDI

Reverse signs of coded factor levels for Factor B

ABEFACDFBCDEDEFBCFACEABDI +

BCDABF

BCFABD

BDEBF

ABCBE

ABEBC

BEFBD

ADECEFCDAF

CFAD

AB

ACDDEF

DFACE

ACFEFD

ADFAEC

B

CDFCEA

BCDEACEABDI +

Example

Creating Design Augmented by Foldover in SAS Data Step

ADX

Augmenting a resolution IV by mirror image or foldoverdoes not break strings of confounded two factor interactions

AH

AG

AE

AD

AF

AC

AB

FGCEBD

FHDEBC

CHDGBF

BHEGCF

GHBECD

EHBGDF

DHCGEF

Augment by design with signsreversed on Factor A only

, H=ABD

Augment by design Reversing signs on A

High concentration of arsenic reported in ground water in countries such as Bangladesh, Chile, India, Poland, Nepal …causing people to be prone to various forms of cancer

Example:

Simple IOCS filters have been used in Bangladesh and Nepal to remove arsenic from ground water

Simple household filters are effective

iron oxide coated sand

raw water

pourousmembrane

purified water

Coating solution made of ferric nitrate and

sodium hydroxide with NAOH added to control pH.

IOCS

Ramakrishna et. al. (2006) conducted experiments to optimizeThe coating process.

Mix CoatingSolution

Age CoatingSolution

Pour overclean sand

Mix Dry FilterSpiked Water Samplere

peat

noyes

472 III

ABCDEFGCEFGBDFG

ADEGDEFAFGBEGABEFCDG

ACDFBCDEABCGBCFACEABDI

AFBECDG

DEAGBCF

DFBGACE

EFCGABD

DGBFAEC

EGCFADB

FGCEBDA

ABCDEFGCEFGBDFG

ADEGABEFACDFBCDEABCGI

AFBECD

DEAGBC

DFBGAC

EFCGAB

DGBFAE

EGCFAD

FGCEBD

G

F

E

D

C

B

A

AFBECD

DEAGBC

DFBGAC

EFCGAB

DGBFAE

EGCFAD

FGCEBD

G

F

E

D

C

B

A

What can be done to separateAD+CF

AD CF

- - - - + + + + - - - - + + + +

- - - - + + + + - - - - + + + +

- 0 - + 0 + 0 - -+ 0 - + 0 + 0 + -- 0 + + 0 + 0 - ++ 0 + + 0 + 0 + +

17181920

3333

No longer orthogonal

Fit modelY=A B F AD CFby regression

5.125.20

05)det( ,

11

5.1

01

5.1

11

XXX

Exchange Algorithm for maximizing det(X’X)

5.165.31

15)det( ,

11

5.1

11

5.1

11

XXX 0.1925.45.1

5.15)det( ,

11

5.1

11

11

11

XXX

0.2451

15)det( ,

11

11

11

11

11

XXX

Step 1 replace 0 with -1 Step 2 replace -.5 with -1

Step 3 replace .5 with 1

Candidate x’s-1, -.5, 0, .5, 1

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5Y

-1 -0.5 0 0.5 1

X

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

Y

-1 -0.5 0 0.5 1

X

Choose additional runs to maximize the || XX

●Plackett-Burman Designs are Resolution III, but there is no defining relation

●Main Effects are confounded with two-factor interactions, but rather than being completely confounded with a few two-factor interactions, they are partially confounded with many two-factor interactions

Alias Matrix shows the alias structure

Example

Implications of Partial Confounding

1. We can use Alias matrix to determine what two-factor interactions are confounded with large unassigned effects

2. Models involving main effects and some partially confounded can be fit by regression since X‘X matrix is not saingular

Creating a Plackett-Burman Design in SAS

Read in the data and merge it with the design created earlier

Fit the model and output the parameter estimates

Create interactions and do all subsets regression

Run 1 2 3 4 5

1 0 0 0 0 0

2 0 0 1 0 1

3 0 1 0 1 1

4 0 1 1 1 0

5 1 0 0 1 1

6 1 0 1 1 0

7 1 1 0 0 1

8 1 1 1 0 0

9 2 0 0 1 0

10 2 0 1 0 1

11 2 1 0 0 0

12 2 1 1 1 1

OA(12, 31, 24)

Data Similar to Experiment with Teaching Methods in Chapter 2

Dummy variablesrepresent effect ofchair style