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N.J. Burn & Associates Inc. 581

Two-Level Fractional Factorial Designs

Motivation

Fractionating a Design

The Defining Relation

Confounding Pattern

Design Resolution

Catalog of Defining Relations

Interactions

Saturated DesignFoldover Design

Power and Sample Size

Experimental Designs for Screening

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Two-Level Fractional Factorial Designs Motivation

It is apparent that as k, the number of experimental factors that are

varied in a two-level full factorial experiment, becomes larger than

five, the number of runs required, 2k, becomes prohibitive.

Fractional factorial designs make more efficient use of full factorialdesigns by confounding potentially unimportant pieces of

information with important pieces of information.


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As the number of experimental factors increases in a two-level full

factorial experiment, so do the number and order of interaction

terms that are estimable in the linear model.

For example, in a five factor experiment, there are:

one constant

five main effects

ten second-order interactions

ten third-order interactions

five fourth-order interactions

one fifth-order interactions

that can be estimated. This adds up to a total of 32 effects that are

estimable from a 32-run experiment.


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A question that needs to be posed is this, is it very likely that the

interaction effect between all five factors is significant, especially in

comparison to the main effects?For that matter, are the

interactions between three and four factors likely to be thatimportant, or even likely to arise at all given the physical aspects of

the process or product under investigation?

If the answers to these questions is no,then we are assuming that

we are really only interested in estimating the main effects, and

perhaps the second-order interaction effects between pairs offactors. In this example then, we are executing 32 experimental run

conditions to estimate only 15 effects at most, plus the constant. Is

this the most efficient design?


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If we are willing to sacrifice information by mixing presumed

unimportant effects with important ones, then we can gain efficiency

by reducing the number of runs necessary to estimate the same

number of effects.In our example, we have a total of 16 model parameters to estimate

(potentially, as we are not compelled to include all or any interaction

terms). Can we select an appropriate subset or fraction of the

original 32 runs that will still give us this information with the best

mixture of unimportant and important information?The answer is yesbut which 16 runs do we choose from the

original 32? The solution is held in the defining relation for fractional

factorial designs.


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Fractionating a Design Example

Let us look at a simpler experiment that demonstrates how mixtures

of pieces of information arise in fractional factorial designs.

Suppose there are four operating variables of interest in a screening

study and restricted resources permit only eight tests to be carriedout. A full 24design requires 16 runs and it is decided to select the

eight runs for our experiment from this design.

There are 12870 different subsets of eight tests that can be selected

from sixteen, each subset yielding different mixtures of information.

Which subset should we then select?The particular subset chosen for this demonstration is that for which

the four factor interaction termx1x2x3x4has the value 1.


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The eight selected runs can be re-ordered into the more familiar

format:

x x x x1 2 3 4

! ! ! !

! !

! !

! !

! !

! !

! !

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

1 1 1 1

The pattern of -1s and +1s follows the same full

factorial pattern as a 23two-level full factorialdesign for the first three factors,x1,x2andx3.

What about the settings forx4? Is there an easier

way to determine its values other than writing outthe corresponding 24full factorial design and

selecting an appropriate number of runs that meetsome criteria such asx1x2x3x4= 1?

The pattern can be generated from the defining

relation of the fractional factorial experiment.


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The Defining Relation

The defining relation for a fractional factorial design can be used for

two purposes:

Generating the pattern of -1s and +1s for additional factors includedin the experiment beyond the k factors in a 2kfull factorial design.

Generating the confounding patternof the design. This will be explained

in more detail later.

The defining relation for the previous example is I =x1x2x3x4where Iis used to denote a column of 1s.


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The Defining Relation Operating Rules

1. Multiplication of any factor by I leaves the column of values for

that factor unchanged. For example,

x1 I =x1 x2x3I =x2x3

2.

Multiplication of any factor by itself produces a column of 1s or I.For example,

x3 x3= (x3)2= I (x1x3x4)

2= I

3. Any operation, such as multiplication, that is performed on one

side of the defining relation equality, must be performed on the

other side.4. There can be more than one defining relation.

5. Multiplication of two or more defining relations results in anotherdefining relation.


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The Defining Relation Example

We can use the defining relation from our example, I =x1x2x3x4, to

generate thex4column.

So, columnx4is the result of the product of columnsx1,x2andx3.


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Confounded Effects

From the previous example with the defining relation we were left

withx4=x1x2x3.

This means that thex4column is identical to that for thex1x2x3

column.Thus, we cannot independently estimate the effects of bothx4and

x1x2x3with this experimental design.

If we include columns in the calculation matrix for both of these

terms, the resulting matrix will be singular.

Singular matrices are a problem when software is used for DOEanalysis. The software algorithm will blow up.


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Confounded EffectsSince we cannot independently estimate these two effects, we saythat they are confoundedor aliasedwith each other. Theimplications of this confounding is that if, in the analysis of theexperimental results, it is determined that thex4/x1x2x3column has

a significant effect, we cannot be sure whether the effect is duesolely tox4orx1x2x3or a mixture of the two.

This is the loss of information (the price you have to pay) forfractionating a design. A smaller number of runs is required, but alleffects cannot be independently estimated in the final analysis.

However, if third and higher order interaction effects which are notlikely to be important are confounded with main effects and second-order interactions, we can then assume that any effect observed isdue mostly to the main or second-order effect. If we assume that thethird-order interactionx1x2x3is zero, then we can attribute anysignificance solely tox4.


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Confounding Pattern

The fact thatx4is confounded withx1x2x3in our example is not the

only instance of confounding. When a design is fractionated, all

effects which can be estimated from the corresponding full factorial

design are mixed up with each other and the pattern of confoundingcan be derived from the defining relation. For example, if I =

x1x2x3x4, we have the following confounding pattern:


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Confounding Pattern

Thus each main effect is confounded with a third-order interaction

(which is probably okay) and the second-order interactions are

confounded with each other (which may or may not be okay).

The degree to which main effects and second-order interactions areconfounded with each other or with higher order interactions is

defined by the resolutionof the design which will be presented

shortly.


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Confounding and the Model

In the above example, the linear model that can be fit to response

data can be written as follows:

Each parameter in the model, !i, i= 1, 2,!, 8 is really estimating

the mixed effects:

411431132112443322110

~~~~~~~~xxxxxxxxxxY

!!!!!!!! +++++++=

~

~

~

~

! ! !

! ! !

! ! !

! ! !

1 1 234

2 2 134

3 3 124

4 4 123

= +

= +

= +

= +

~

~

~

~

! ! !

! ! !

! ! !

! ! !

0 0 1234

12 12 34

13 13 24

14 14 23

= +

= +

= +

= +

~


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Nomenclature

The number of runs required in two-level full factorial designs is a

power of two.

The degree to which a full factorial design is reduced by

fractionating it is also a power of two. For example, half fractions,quarter fractions, eighth fractions and so on are taken off of full

factorial designs.

The order of reduction can be denoted by qwhere 1/2qrepresents

the degree of fractionation.

For example, 24-1denotes a half fraction of a 24full factorial design.Algebraically, 24-1= 23= 8 experimental runs, but a 24-1fractionalfactorial design is not the sameas a 23full factorial design which

also has 8 experimental runs.


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Degree of FractionationA full factorial design can only be fractionated to the extent that

For example, a fractional factorial design involving sevenexperimental factors can be carried out in eight runs as a sixteenthfraction of a 27full factorial design since 27-4= 23= 8 !7 + 1.

However, the effects of seven experimental factors cannot beestimated in four experimental runs as a 1/32 fraction since27-5= 22= 4 < 7 + 1.

There are simply not enough degrees of freedom to estimate somany effects with so few runs.

2 1k q k! " +


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Design Resolution

The resolution of a fractional factorial design describes the extent to

which main effects and second-order interactions are confounded

with each other and with third and higher order interactions.

As the term "resolution" suggests, the higher the resolution thebetter in the sense that potentially important effects are not

confounded with each other.

The important effects can be resolved or estimated without worrying

about the effects they are confounded with.

Three common classes of resolution for 2k-qfractional factorialdesigns are defined.


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Design Resolution

Resolution III designs are 2k-qdesigns for which

(i) no individual operating variable, such asx1, is confounded with

any other individual operating variable, such asx2and

(ii) at least one individual operating variable is confounded with atwo variable interaction.

An example of a resolution III design is the 23-1design with definingrelation I =x1x2x3. This is often denoted as:

23 1

III

!


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Design Resolution

Resolution IV designs are 2k-qdesigns for which

(i) no individual operating variable is confounded with any other

individual operating variable or with any two variable interaction and

(ii) at least one two variable interaction is confounded with anothertwo variable interaction.

An example of a resolution IV design is the 24-1design with definingrelation I =x1x2x3x4. This is often denoted as:

24 1

IV

!


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Design Resolution

Resolution V designs are 2k-qdesigns for which

(i) no individual operating variable is confounded with any other

individual operating variable or with any two variable interaction and

(ii) no two variable interaction is confounded with another twovariable interaction and

(iii) at least one two variable interaction is confounded with a three

variable interaction.

An example is the 25-1design with defining relation I =x1x2x3x4x5.

This is often denoted as:

25 1

V

!


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Most fractional factorial experiments are 4, 8 or 16 run designs.

Work has been done by people like George Box to identify the best

confounding patterns for a variety of fractional factorial designs that

achieve the highest resolution.The following series of slides provide the defining relations that can

be used to generate 4, 8 and 16 run fractional factorial designs.


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4 Run Fractional Factorial Designs

Design Defining Relations Generators132 !

III 321 xxxI = 213 xxx =


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Design Defining Relations Generators14

2 !IV

4321 xxxxI = 3214 xxxx =

252 !III

531

421

xxx

xxxI

=

=

315

214

xxx

xxx

=

=

362 !III

632

531

421

xxx

xxx

xxxI

=

=

=

326

315

214

xxx

xxx

xxx

=

=

=

472 !III

7321

632

531

421

xxxx

xxx

xxx

xxxI

=

=

=

=

3217

326

315

214

xxxx

xxx

xxx

xxx

=

=

=

=


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Design Defining Relations Generators15

2 !V

54321 xxxxxI = 43215 xxxxx =

262 !IV

6432

5321

xxxx

xxxxI

=

=

4326

3215

xxxx

xxxx

=

=

372 !IV

7431

6432

5321

xxxx

xxxx

xxxxI

=

=

=

4317

4326

3215

xxxx

xxxx

xxxx

=

=

=

482 !

IV

8421

7431

6432

5321

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

4218

4317

4326

3215

xxxx

xxxx

xxxx

xxxx

=

=

=

=


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III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

6102 !III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

1021 xxxI =

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

2110 xxx =

9 52

IV

!


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III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

1131

1021

xxx

xxxI

=

=

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

3111

2110

xxx

xxx

=

=

8122 !III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

1241

1131

1021

xxx

xxx

xxxI

=

=

=

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

4112

3111

2110

xxx

xxx

xxx

=

=

=


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III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

1332

1241

1131

1021

xxx

xxx

xxx

xxxI

=

=

=

=

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

3213

4112

3111

2110

xxx

xxx

xxx

xxx

=

=

=

=

10142 !III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

1442

1332

1241

1131

1021

xxx

xxx

xxx

xxx

xxxI

=

=

=

=

=

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

4214

3213

4112

3111

2110

xxx

xxx

xxx

xxx

xxx

=

=

=

=

=


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III

94321

8421

7431

6432

5321

xxxxx

xxxx

xxxx

xxxx

xxxxI

=

=

=

=

=

1543

1442

1332

1241

1131

1021

xxx

xxx

xxx

xxx

xxx

xxxI

=

=

=

=

=

=

43219

4218

4317

4326

3215

xxxxx

xxxx

xxxx

xxxx

xxxx

=

=

=

=

=

4315

4214

3213

4112

3111

2110

xxx

xxx

xxx

xxx

xxx

xxx

=

=

=

=

=

=


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Minitab Exercise Fractional Factorial Designs

Click on Stat!DOE!Factorial!Create Factorial Design

Click on the Display Available Designs button


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Defining Relations and Confounding Patterns

Rule 5 governing defining relations states that a new defining

relation can be determined by multiplying defining relations together.

For example, a resolution III 25-2design has defining relations

I =x1x2x4=x1x3x5. A third defining relation can be found bymultiplying these two together.

( )

( )

5432

5432

5432

2

1

531421

xxxx

xxxxI

xxxxx

xxxxxxI

=

=

=

= There can be many defining relationsthat determine the complete

confounding pattern for highlyfractionated designs (e.g. 215-11).


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Defining Relations and Confounding PatternsFrom the previous 25-2example where we haveI =x1x2x4=x1x3x5=x2x3x4x5, we have the following confounding (oraliasing) pattern:

x1withx2x4 andx3x5x2withx1x4x3withx1x5x4withx1x2x5withx1x3x2x3withx4x5

x2x5withx3x4

We are usually only interested in the confounding pattern up tosecond-order interactions.

This confounding pattern accountsfor the relationships between all

five main effects and the tensecond-order interactions.


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Defining Relations and Confounding Patterns

If we fit the following proposed model for a resolution III 25-2design:

Each parameter in the model, !i, i= 1, 2,!

, 8 is really estimatingthe mixed effects:

5225322355443322110

~~~~~~~~xxxxxxxxxY !!!!!!!! +++++++=

1355

1244

1533

1422

352411

~

~

~

~

~

!!!

!!!

!!!

!!!

!!!!

+=

+=

+=

+=

++=

342525

452323

234513512400

~

~

~

!!!

!!!

!!!!!

+=

+=

+++=


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Interactions

For resolution V designs, all of the main effects and second-order

interactions can be estimated independentlyin that they are

confounded with third and higher-order interactions that are

assumed to be insignificant.However, it is possible in some situations to obtain independent

estimates of second-order interactions in lower resolution designs.

It depends on the extent of prior knowledge the experimenter has with

the system/process/product/design.


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Interactions Example

Lets use the previous 25-2example where we have

I =x1x2x4=x1x3x5=x2x3x4x5. Ignoring second-order interactions, the

main effects model that can be fit is:

The alias pattern for the main effects is:

55443322110~~~~~~ xxxxxY !!!!!! +++++=

1355

1244

1533

1422

352411

~

~

~

~

~

!!!

!!!

!!!

!!!

!!!!

+=

+=

+=

+=

++=


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Note that we still have the freedom to include two more terms in the

proposed model corresponding to the confounded pairs of second-

order interactions (!23, !45) and (!25, !34).

Suppose that the five factors in the experiment are nitrogen contentin lawn fertilizer, amount of lawn watering, lawn aeration, type of

grass in the lawn and grade of top soil.

The response of interest is the total weight of grass clippings after a

season of mowing. More grass clippings is suggestive of a healthier,

thicker lawn.As an experienced gardener, you strongly suspect there to be a

significant interaction between nitrogen content and amount of

watering.


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The design initially proposed to you, based on the defining relations,

is as follows:

11111

11111

11111

11111

11111

11111

11111

11111

315214321

!!!

!!

!!!

!!

!!!

!!!!

!!!

== xxxxxxxxx

x1= nitrogen content

x2= aeration

x3= grade of top soil

x4= amount of water

x5= type of grass


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You see right away that the second-order interaction of interest,

between nitrogen content and amount of watering, corresponds to

thex1x4interaction the way the design is presently defined.

From examining the confounding pattern, you notice that thex1x4interaction is confounded with the main effect forx2.

Hence, as currently defined, the proposed design will not meet your

modeling objective.

However, by judicially reassigning the experimental factors, you can

make use of one of the two pairs of second-order interaction pairsthat are not in the model yet.


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You propose the following changes:

11111

11111

11111

11111

11111

11111

11111

11111

315214321

!!!

!!

!!!

!!

!!!

!!!!

!!!

== xxxxxxxxx

x1= aerationx2= nitrogen content

x3= amount of water

x4= grade of top soil

x5= type of grass


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Now the interaction of interest corresponds tox2x3which can be

added to the model without having it confounded with any of the five

main effects:

Depending on your prior knowledge about the process under

investigation, it may still be possible to obtain "independent"

estimates of main effects and some second-order interactions, even

with low resolution designs, if care is taken in the design stage toobtain a desirable confounding pattern.

The experimental design and subsequent confounding relationshipscan always be evaluated before committing to run the experiment.

322355443322110~~~~~~~ xxxxxxxY !!!!!!! ++++++=


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Saturated Designs

In the previous example, it was possible to use extra degrees of

freedom to include one or more interaction terms in the model.

There are a group of resolution III designs known as saturated

designsthat do not afford this freedom.With a design from this group, koperating variables can be

investigated simultaneously in k+1 tests where k+1 is a power of 2.

Examples of saturated two-level fractional factorial designs are:

In these designs, 3, 7, 15 and 31 factors are investigated in 4, 8, 16

and 32 runs respectively.


3 1 7 4 15 11 31 262 , 2 , 2 , 2

III III III III

! ! ! !

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Saturated Designs Example

Construction of a resolution III 27-4design is accomplished by first

writing a 23design in three of the seven operating variables,X1,X2

andX3.

Each of the remaining operating variables,X4,X5,X6andX7isconfounded with an interaction amongX1,X2andX3.

If the aliasesX4=X1X2,X5=X1X3,X6=X2X3andX7=X1X2X3are

used, then the resulting design is that shown below.


x x x x x x x1 2 3 4 5 6 7

! ! ! !

! ! ! !

! ! ! !

! ! ! !

! ! ! !

! ! ! !

! ! ! !

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

1 1 1 1 1 1 1

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From the four basic generatorsX1X2X4,X1X3X5,X2X3X6and

X1X2X3X7arising from the choice of aliases, the following defining

relation for this design can be formed,


I x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x

= = = =

= = = = = =

= = = =

1 2 4 1 3 5 2 3 6 1 2 3 7

2 3 4 5 1 3 4 6 3 4 7 1 2 5 6 2 5 7 1 6 7

4 5 6 1 4 5 7 2 4 6 7 3 5 6 7

(taking basic generators

one at a time)

(products of two basic

generators)

(products of three basic

generators)

(products of four basic

generators)

= x x x x x x x1 2 3 4 5 6 7

Note: These are also referred to as the words of a defining relation.

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Notice that the smallest order interaction in this defining relation is

three, verifying that the resolution of the design is indeed III.

Again ignoring interactions involving more than two operating

variables, the following eight estimates can be obtained from thisdesign.


( )( )( )

( )( )( )( )

l

l

l

l

l

l

l

l

1 24 35 67

2 14 36 57

3 15 26 47

4 12 56 37

5 13 46 27

6 23 45 17

7 34 25 16

0 0

,

,

,

,

,

,

,

,

which estimates

which estimates

which estimates

which estimates

which estimates

which estimates

which estimates

which estimates

1

2

3

4

5

6

7

! ! ! !

! ! ! !

! ! ! !

! ! ! !! ! ! !

! ! ! !

! ! ! !

!

+ + +

+ + +

+ + +

+ + +

+ + +

+ + +

+ + +

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Saturated Designs Interpretation of Results

Interpretation of results from a saturated design may be ambiguous

because each operating variable is confounded with a number of

two variable interactions.

As will be shown later, ambiguities can be partially resolved bycarrying out another saturated design from the same "family", that is,

a design for which the signs of all values of one or more of the

operating variables are reversed.

Thus, saturated designs are more useful for the first step in a

screening study of several operating variables.


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Example Saturated Design

This example has been slightly modified from the Course Notes:

Referring to Table 23.5 in the notes, the following changes have been

made:

x4

= recycle

x5= rate of addition of NaOH

x6= type of filter cloth

x7= holdup time

The standard generators ofx4=x1x2,x5=x1x3,x6=x2x3andx7=x1x2x3

are used instead of the ones in Equation 23.3.

The columns in the design matrix in Table 23.6 have been accordinglyadjusted.

"s are used in stead of !s for the model coefficients.

None of these changes affects the resulting analysis and interpretation.


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During the startup of a new process in a chemical plant, trouble was

encountered in a filtration operation. Filtration was requiring about

70 minutes per batch instead of 40 minutes, the required time for a

similar operation at other plant sites. An investigation was

undertaken to identify the operating variables that affected filtration

time and to determine how these variables might be altered in order

to reduce the filtration time. The operating variables selected for the

initial study are shown in the following table. The low levels

represent the operating conditions prior to this screening study. The

high levels are changes in operating conditions chosen to identifywhich, if any, of these seven operating variables affected the

filtration time. It will be noted that four of the operating variables,x1,

x2,x4andx6are qualitative variables.


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Model to be fit:

Operating Variable Level-1 1

x1 , water supply municipal reservoir well

x2 , raw material made on site made at another site

x3 , filtration temperature low high

x4 , recycle included omitted

x5 , rate of addition of NaOH fast slow

x6 , type of filter cloth new old

x7 , holdup time short long

( ) 776655443322110 xxxxxxxYE !!!!!!!! +++++++=


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As a first step in the study, a 27-4resolution III design was employed

because of its economy in tests and its facility for use as a building

block for further tests that might be required.

Basic generators chosen for the design were:

3217

326

315

214

7321632531421

xxxx

xxx

xxx

xxx

xxxxxxxxxxxxxI

=

=

=

=

====


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The design and the measured steady state filtration time for each

test are:

(min.)7654321 yxxxxxxx

7.38

7.68

2.41

6.78

0.81

4.66

7.77

4.68

1111111

1111111

1111111

1111111

1111111

1111111

1111111

1111111

!!!!

!!!!

!!!!

!!!!

!!!!

!!!!

!!!!


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When these results were examined, there may well have been a

temptation to conclude that either the sixth test or the eighth test

from the experiment had resolved the problem since both tests

produced filtration times in the order of 40 minutes, the target figure.

As will be shown shortly, a conclusion that changes in x1, x3and x5

produced this favourable result is only one of several possible

interpretations of these data.

In any case, before making a change in such an important operating

variable as water supply (x1), other interpretations would have to beassessed.


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Example Minitab Output

Fitted coefficients are:

The fitted model is then:

Estimated Effects and Coefficients for Filt. (coded units)

Term Effect Coef

Constant 65.09

Water Su -10.87 -5.44

Raw Mate -2.77 -1.39

Filt. Te -16.58 -8.29

Recycle 3.17 1.59

NaOH Add -22.83 -11.41

Type Fil -3.42 -1.71

Holdup 0.53 0.26

7654321 26.071.141.1159.129.839.144.509.65 xxxxxxxy +!!+!!!=


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From the defining relation for this design it can be confirmed that the

eight coefficient estimates are confounded with a number of two-

factor interactions. Interactions involving more than two operating

variables have been ignored.

x1+ x

2x4+ x

3x5+ x

6x7

x2+ x

1x4+ x

3x6+ x

5x7

x3+ x

1x5+ x

2x6+ x

4x7

x4+ x

1x2+ x

3x7+ x

5x6

x5+ x1x3+ x2x7+ x4x6x6+ x

1x7+ x

2x3+ x

4x5

x7+ x

1x6+ x

2x5+ x

3x4


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Because further tests were carried out in this study, a second

subscript has been added to these estimates to denote that they

arise from the first set of tests.

( )( )( )

( )

( )

( )( )( ) 26.0,

71.1,

41.11,

59.1,

29.8,

39.1,

44.5,

09.65,

342516771

452317661

462713551

563712441

472615331

573614221

673524111

001

=+++

!=+++

!=+++

=+++

!=+++

!=+++

!=+++

=

""""#

""""#

""""#

""""#

""""#

""""#

""""#

"#

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean


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Because no estimate of the pure error variance is available, one

interpretation of these estimates can be made on the basis of their

relative magnitudes. Among the coefficients of operating variables,

the estimates "5

, "3

and "1

are much larger in magnitude than the

other estimates. A number of alternative interpretations are possible.


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A simple explanation of these three large estimates might be that

only the terms !1x1, !3x3and !5x5are important, all two variable

interactions being of negligible size.

Another explanation might be that only operating variablesx1andx3are affecting the filtration time, their influence being explained by

terms !1x1, !3x3and !13x1x3.

A third possibility is that only operating variablesx1andx5are

important, their effect being accounted for by terms !1x1, !5x5and

!15x1x5.Another alternative is that only operating variablesx3andx5are

causing the response to change via terms !3x3, !5x5and !35x3x5.


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( )( )

( )

( )( )

( )( )

( ) 26.0,71.1,

41.11,

59.1,

29.8,

39.1,

44.5,

09.65,

342516771

452317661

462713551

563712441

472615331

573614221

673524111

001

=+++

!=+++

!=+++

=+++

!=+++

!=+++

!=+++

=

""""#

""""#

""""#

""""#

""""#

""""#

""""#

"#

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean ( )( )

( )( )

( )

( )( )

( ) 26.0,71.1,

41.11,

59.1,

29.8,

39.1,

44.5,

09.65,

342516771

452317661

462713551

563712441

472615331

573614221

673524111

001

=+++

!=+++

!=+++

=+++

!=+++

!=+++

!=+++

=

""""#

""""#

""""#

""""#

""""#

""""#

""""#

"#

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

( )

( )

( )( )( )

( )( )

( ) 26.0,71.1,

41.11,

59.1,

29.8,39.1,

44.5,

09.65,

342516771

452317661

462713551

563712441

472615331

573614221

673524111

001

=+++

!=+++

!=+++

=+++

!=+++!

=+++

!=+++

=

""""#

""""#

""""#

""""#

""""#""""#

""""#

"#

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimateanofestimatean

ofestimatean

ofestimatean ( )

( )

( )( )( )

( )

( )( ) 26.0,

71.1,

41.11,

59.1,

29.8,39.1,

44.5,

09.65,

342516771

452317661

462713551

563712441

472615331

573614221

673524111

001

=+++

!=+++

!=+++

=+++

!=+++

!

=+++

!=+++

=

""""#

""""#

""""#

""""#

""""#""""#

""""#

"#

ofestimatean

ofestimatean

ofestimatean

ofestimatean


ofestimatean

ofestimatean


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Minitab Exercise Saturated Experiment

Open the file Topic06SatExp.MTW

Select Stat!DOE!Factorial!Analyze Factorial Design

Select C12 Y as the Response

Click on the Graphs buttonClick on the Four in one radio button

Click on OK

Click on OK


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Minitab output


Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Model 7 1887.53 269.65 * *

Linear 7 1887.53 269.65 * *

X1 1 236.53 236.53 * *

X2 1 15.40 15.40 * *X3 1 549.46 549.46 * *

X4 1 20.16 20.16 * *

X5 1 1041.96 1041.96 * *

X6 1 23.46 23.46 * *

X7 1 0.55 0.55 * *

Error 0 * *

Total 7 1887.53

Model Summary

S R-sq R-sq(adj) R-sq(pred)

* 100.00% * *

Note that test statistics

and p-values cannot be

calculated because this

is an exact fit as

exhibited by an R2

of100%.

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Minitab output


Coded Coefficients

SE

Term Effect Coef Coef T-Value P-Value VIF

Constant 65.09 * * *

X1 -10.875 -5.437 * * * 1.00

X2 -2.775 -1.387 * * * 1.00

X3 -16.575 -8.288 * * * 1.00

X4 3.175 1.587 * * * 1.00

X5 -22.82 -11.41 * * * 1.00

X6 -3.425 -1.712 * * * 1.00

X7 0.5250 0.2625 * * * 1.00

Regression Equation in Uncoded Units

Y = 65.09 - 5.437 X1 - 1.387 X2 - 8.288 X3 + 1.587 X4 - 11.41 X5 - 1.712 X6

+ 0.2625 X7

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Minitab output


Aliases

I + ABD + ACE + AFG + BCF + BEG + CDG + DEF

A + BD + CE + FG + BCG + BEF + CDF + DEG

B + AD + CF + EG + ACG + AEF + CDE + DFG

C + AE + BF + DG + ABG + ADF + BDE + EFG

D + AB + CG + EF + ACF + AEG + BCE + BFG

E + AC + BG + DF + ABF + ADG + BCD + CFG

F + AG + BC + DE + ABE + ACD + BDG + CEG

G + AF + BE + CD + ABC + ADE + BDF + CEF

* NOTE * Could not graph the specified residual type because MSE = 0 or the

degrees of freedom for error = 0.

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Example Foldover DesignIt is well known that low resolution fractional factorial designs (III andIV) confound main effects with second-order interactions (III) andsecond-order interactions with each other (IV).

This sometimes makes the interpretation of a DOE analysis difficult.

Foldover designs can be used to increase the resolution of a lowresolution design and help to resolve lingering questions from theinitial design.

Foldover designs are a nice sequential strategy to employ whenresources are limited.

Because of the ambiguity in interpreting the results of these tests, asecond foldoverset of eight tests was conducted using a 27-4resolution III design formed from the first design by reversing thesigns of all values for all seven operating variables.

This second design is shown on the next slide along with themeasured filtration times obtained for these additional tests.


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Example Foldover Design

(min.)7654321 yxxxxxxx

6.67

6.42

0.59

8.47

9.61

4.860.65

7.66

1111111

1111111

1111111

1111111

1111111

11111111111111

1111111

!!!!!!!

!!!

!!!

!!!

!!!

!!!

!!!

!!!


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Switching the signs of all values for one operating variablexiin a

27-4resolution III design is equivalent to replacingxiwith -xi.

Because of the manner in which this second design has been

constructed from the first design, its defining relation can be

obtained by replacing every operating variablexiin the defining

relation for the first design, by -xi. The resulting defining relation is

then:

3217

326

315

214

7321632531421

xxxx

xxx

xxx

xxx

xxxxxxxxxxxxxI

=

!=

!=

!

=

=!=!=!=


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Example Minitab OutputMinitab was used to obtain the following results for the second

design on its own:

Estimated Effects and Coefficients for Filt. (coded units)

Term Effect Coef

Constant 62.125

Water Su -2.500 -1.250

Raw Mate -5.000 -2.500

Filt. Te 15.750 7.875

Recycle 2.250 1.125

NaOH Add -15.600 -7.800

Type Fil 3.300 1.650Holdup -9.150 -4.575


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Example Foldover DesignFrom the defining relation for the foldover design it can be confirmed

that the eight coefficient estimates are confounded in the following

matter. Interactions involving more than two operating variables

have been ignored.

x1- x

2x4- x

3x5- x

6x7

x2- x

1x4- x

3x6- x

5x7

x3- x

1x5- x

2x6- x

4x7

x4- x

1x2- x

3x7- x

5x6

x5- x1x3- x2x7- x4x6x6- x

1x7- x

2x3- x

4x5

x7- x

1x6- x

2x5- x

3x4


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Example Foldover DesignSo, we have:

which creates even more possibilities.

( )( )( )( )( )( )

( )( ) 575.4,65.1,

80.7,

125.1,

875.7,

50.2,

25.1,

125.62,

342516772

452317662

462713552

563712442

472615332

573614222

673524112

002

!=!!!

=!!!

!=!!!

=!!!

=!!!

!=!!!

!=!!!

=

""""#""""#

""""#

""""#

""""#

""""#

""""#

"#


ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean


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Example Foldover DesignHowever, we can combine the results from the original design with

those from the foldover design in the following manner:

( )( )( )

( )

( )( )

( ) 16.2,203.0,261.9,2

36.1,2

21.0,2

94.1,2

34.3,2

6.63,2

77271

66261

55251

44241

33231

22221

11211

00201

!=+

!=+

!

=+

=+

!=+

!=+

!=+

=+

"##

"##"##

"##

"##

"##

"##

"##

ofestimatean


ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean


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( ) 5.1,,20201

=! effectblocktheofestimatean""

( ) ( )( ) ( )( ) ( )

( ) ( )( ) ( )( ) ( )( ) ( ) 09.2,2

68.1,2

42.2,2

08.8,2

56.0,2

81.1,2

231.0,2

6735241211

4523176261

3425167271

4726153231

5736142221

4627135251

5637124241

!=++!

!=++!

=++!

!=++

!

=++!

!=++!

=++!

"""##

"""##

"""##

"""##

"""##

"""##

"""##

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean

ofestimatean


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Example Foldover DesignThe block effectis the difference in average response values

between the two sets of eight runs.

Had it been large, it would have indicated the presence of some

other variables, beyond the seven being studied, whose change

between the two sets of tests strongly affected the filtration time.

By combining the original with the foldover experiment, we have

effectively created a 27-3resolution IV experimental design.


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Example Foldover DesignAmong the above sixteen estimates, the largest is -9.61, an estimateof !5and 8.08, an estimate of the linear combination of two variableinteractions !15+ !26+ !47. The next largest estimate is -3.34, anestimate of !1.

The investigators concluded that operating variables !1and !5alone,the water supply and the rate of addition of caustic soda, affected thefiltration time, and the estimate -8.08 occurred primarily because of theinteraction !1!5.

Even at this stage, of course, other interpretations are possible. Forexample, the estimate -8.08 might have been due to the interactionx2x6and/or the interactionx4x7. It is noted, however, that the estimatesof !2, !6, !4, and !7are all relatively small and, although this does notnecessarily mean that interactions among these variables must alsobe small, this is often the case.


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Example Foldover DesignThe investigatorsinterpretation can be summarized conveniently

by the following chart which shows the average filtration time

obtained at each of the four sets of operating conditions of water

supply and rate of addition of NaOH. Evidence of the large negative

interaction between the two variables is very strong.

reservoir well

slow

fast

water supply

rate of addition

of NaOH

68.5 78.0

65.4 42.6


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Example Foldover DesignThe corrective action implied by these results was to change the

water supply from the municipal reservoir to the well and reduce the

rate of addition of caustic soda. These changes were made and

satisfactory filtration times close to 40 minutes were obtained in

subsequent plant operation.


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Exercise Foldover DesignThe reduced fitted model from this example is:

What is the predicted filtration time at the new operating conditionsusing well water and a slow rate of NaOH addition?

5151 08.861.934.36.63 xxxxy !!!=


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Minitab Exercise Foldover ExperimentClick on Stat!DOE!Factorial!Create Factorial Design

Use the Number of factors drop down menu to select 7

Click on the Designs button

Select the first row for a 2^7-4 fractional factorial design

Click OK

Click on the Factors button

Change the Factor Names from A through G to X1 through X7

Click OK

Click on the Options button

Click on the Fold on all factors radio button

Uncheck the Randomize runs box

Click OK

Click OK


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Minitab Exercise Foldover ExperimentEnter the Y response data in the Minitab worksheet


X1 X2 X3 X4 X5 X6 X7 Y

-1 -1 -1 1 1 1 -1 68.4

1 -1 -1 -1 -1 1 1 77.7

-1 1 -1 -1 1 -1 1 66.4

1 1 -1 1 -1 -1 -1 81.0

-1 -1 1 1 -1 -1 1 78.6

1 -1 1 -1 1 -1 -1 41.2

-1 1 1 -1 -1 1 -1 68.7

1 1 1 1 1 1 1 38.7

1 1 1 -1 -1 -1 1 66.7

-1 1 1 1 1 -1 -1 65.01 -1 1 1 -1 1 -1 86.4

-1 -1 1 -1 1 1 1 61.9

1 1 -1 -1 1 1 -1 47.8

-1 1 -1 1 -1 1 1 59.0

1 -1 -1 1 1 -1 1 42.6

-1 -1 -1 -1 -1 -1 -1 67.6

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Minitab Exercise Foldover ExperimentSelect Stat!DOE!Factorial!Analyze Factorial Design

Select C12 Y as the Response

Click on the Graphs button

Click on the Four in one radio button

Click on OK

Click on OK


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Minitab Exercise Foldover ExperimentCoefficients table


Coded Coefficients

SE

Term Effect Coef Coef T-Value P-Value VIF

Constant 63.61 * * *X1 -6.687 -3.344 * * * 1.00

X2 -3.888 -1.944 * * * 1.00

X3 -0.4125 -0.2062 * * * 1.00

X4 2.712 1.356 * * * 1.00

X5 -19.213 -9.606 * * * 1.00

X6 -0.06250 -0.03125 * * * 1.00

X7 -4.313 -2.156 * * * 1.00

X1*X2 0.4625 0.2312 * * * 1.00

X1*X3 -3.613 -1.806 * * * 1.00

X1*X4 1.1125 0.5563 * * * 1.00

X1*X5 -16.163 -8.081 * * * 1.00

X1*X6 4.838 2.419 * * * 1.00

X1*X7 -3.362 -1.681 * * * 1.00

X2*X4 -4.188 -2.094 * * * 1.00

X1*X2*X4 2.963 1.481 * * * 1.00

f S

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Minitab Exercise Foldover ExperimentReduced model

Start removing higher order terms with the smallest magnitude

coefficient value (highest p-value)


Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Model 3 2700.3 900.09 23.13 0.000

Linear 2 1655.4 827.69 21.27 0.000

X1 1 178.9 178.89 4.60 0.053

X5 1 1476.5 1476.48 37.94 0.000

2-Way Interactions 1 1044.9 1044.91 26.85 0.000

X1*X5 1 1044.9 1044.91 26.85 0.000

Error 12 467.1 38.92Total 15 3167.3

E i l D i f S i

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Minitab Exercise Foldover Experiment


Model Summary

S R-sq R-sq(adj) R-sq(pred)

6.23867 85.25% 81.57% 73.78%

Coded Coefficients

Term Effect Coef SE Coef T-Value P-Value VIF

Constant 63.61 1.56 40.78 0.000

X1 -6.69 -3.34 1.56 -2.14 0.053 1.00

X5 -19.21 -9.61 1.56 -6.16 0.000 1.00

X1*X5 -16.16 -8.08 1.56 -5.18 0.000 1.00

Regression Equation in Uncoded Units

Y = 63.61 - 3.34 X1 - 9.61 X5 - 8.08 X1*X5

Documents

Handout_10 SV(1)