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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
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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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Fractionating a Design
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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Fractionating a Design
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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Fractionating a Design
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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Fractionating a Design Example
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Fractionating a Design Example
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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Catalog of Defining Relations
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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Catalog of Defining Relations
4 Run Fractional Factorial Designs
Design Defining Relations Generators132 !
III 321 xxxI = 213 xxx =
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Catalog of Defining Relations
8 Run Fractional Factorial Designs
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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Catalog of Defining Relations
16 Run Fractional Factorial Designs
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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Catalog of Defining Relations
16 Run Fractional Factorial Designs
Design Defining Relations Generators592 !
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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Catalog of Defining Relations
16 Run Fractional Factorial Designs
Design Defining Relations Generators7112 !
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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Catalog of Defining Relations
16 Run Fractional Factorial Designs
Design Defining Relations Generators9132 !
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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Catalog of Defining Relations
16 Run Fractional Factorial Designs
Design Defining Relations Generators11152 !
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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Interactions Example
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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Interactions Example
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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Interactions Example
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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Interactions Example
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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Interactions Example
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.
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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.
Experimental Designs for Screening
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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Saturated Designs Example
From the four basic generatorsX1X2X4,X1X3X5,X2X3X6and
X1X2X3X7arising from the choice of aliases, the following defining
relation for this design can be formed,
Experimental Designs for Screening
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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Saturated Designs Example
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.
Experimental Designs for Screening
( )( )( )
( )( )( )( )
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.
Experimental Designs for Screening
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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.
Experimental Designs for Screening
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Example Saturated Design
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.
Experimental Designs for Screening
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Example Saturated Design
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 !!!!!!!! +++++++=
Experimental Designs for Screening
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Example Saturated Design
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
=
=
=
=
====
Experimental Designs for Screening
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Example Saturated Design
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
!!!!
!!!!
!!!!
!!!!
!!!!
!!!!
!!!!
Experimental Designs for Screening
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Example Saturated Design
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.
Experimental Designs for Screening
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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 +!!+!!!=
Experimental Designs for Screening
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Example Saturated Design
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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Example Saturated Design
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
Experimental Designs for Screening
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Example Saturated Design
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.
Experimental Designs for Screening
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Example Saturated Design
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.
Experimental Designs for Screening
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Example Saturated Design
( )( )
( )
( )( )
( )( )
( ) 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
ofestimateanofestimatean
ofestimatean
ofestimatean
Experimental Designs for Screening
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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
Experimental Designs for Screening
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Minitab output
Experimental Designs for Screening
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
Experimental Designs for Screening
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
Experimental Designs for Screening
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.
Experimental Designs for Screening
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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
!!!!!!!
!!!
!!!
!!!
!!!
!!!
!!!
!!!
Experimental Designs for Screening
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Example Foldover Design
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
=
!=
!=
!
=
=!=!=!=
Experimental Designs for Screening
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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
Experimental Designs for Screening
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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
!=!!!
=!!!
!=!!!
=!!!
=!!!
!=!!!
!=!!!
=
""""#""""#
""""#
""""#
""""#
""""#
""""#
"#
ofestimateanofestimatean
ofestimatean
ofestimatean
ofestimatean
ofestimatean
ofestimatean
ofestimatean
Experimental Designs for Screening
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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
ofestimateanofestimatean
ofestimatean
ofestimatean
ofestimatean
ofestimatean
ofestimatean
Experimental Designs for Screening
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Example Foldover Design
( ) 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
Experimental Designs for Screening
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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.
Experimental Designs for Screening
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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.
Experimental Designs for Screening
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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 !!!=
Experimental Designs for Screening
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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
Experimental Designs for Screening
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Minitab Exercise Foldover ExperimentEnter the Y response data in the Minitab worksheet
Experimental Designs for Screening
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
Experimental Designs for Screening
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Minitab Exercise Foldover ExperimentCoefficients table
Experimental Designs for Screening
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)
Experimental Designs for Screening
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
Experimental Designs for Screening
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