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程序代写案例-STAT 453/558
时间：2021-04-04
STAT 453/558: The Design and Analysis of
Experiments
Michelle F. Miranda
University of Victoria
michellemiranda@uvic.ca Office: DTB 543
Chapter 8- Fractional Factorial Designs 2
Two-level Fractional Factorial Design
I For a 2k factorial designs, 2k increases very fast as k
increases
22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64 runs
.
I Sometimes there is not enough resources to run a complete
replicate for a 2k design.
I We can run a fraction of 2k runs, and this yields fractional
factorial designs.
I Major use is in screening experiments (when many factors
are considered with the goal of identifying those with large
effects).
Fractional Factorial Designs 3
One-half fraction of the 2k design
I Example: Consider the case when the experimenter cannot
run all 8 combinations of the 23 design, but can afford 4
runs.
I This is a one-half fraction of a 23 design
I This is called a 23−1 design
One-half fraction of the 2k design 4
I Suppose we select 4 treatments: a, b, c, and abc as our
one-half fraction
I Notice: the 23−1 design is formed by selecting the
combination with plus on ABC
I I = ABC is called the generator and this one-half fraction
is usually called the principal fraction . We can also refer
to ABC as a word or the defining relation for our design.
Effect Estimation 5
I From the table, we notice that the contrasts to estimate
the main effects are
[A] =
1
2
(a− b− c+ abc)
[B] =
1
2
(−a+ b− c+ abc)
[C] =
1
2
(−a− b+ c+ abc)
I Notation [A], [B], and [C] indicates the linear combinations
associated with the main effects
Aliases 6
I We can also verify that the linear combination of the
observations used to estimate the two-factor interactions
are
[BC] =
1
2
(a− b− c+ abc)
[AC] =
1
2
(−a+ b− c+ abc)
[AB] =
1
2
(−a− b+ c+ abc)
I Thus when we estimate [A] we are in fact estimating
[A] + [BC]; [A] and [BC] are aliases
I Notation to indicate aliases [A]→ A+BC, [B]→ B +AC,
[C]→ C +AB
One-half fraction of the 2k design 7
I Now suppose we select the last 4 treatments: ab, acb, bc,
and (1) as our one-half fraction
I In this case I = −ABC and this one-half fraction is called
the alternate or complementary fraction
One-half fraction of the 2k design 8
I When we estimate A,B, and C with this particular
fraction, we are estimating A−BC, B−AC, AND C −AB
I The alternate fraction gives us
[A′]→ A−BC
[B′]→ B −AC
[C ′]→ C −AB
One-half fraction of the 2k design 9
I When we estimate A,B, and C with this particular
fraction, we are estimating A−BC, B−AC, AND C −AB
I The alternate fraction gives us
[A′]→ A−BC
[B′]→ B −AC
[C ′]→ C −AB
I The two one-half fractions form a complete 23 design
One-half fraction of the 2k design 10
I If we run both halves of the design, we can obtain
de-aliased estimates of all the effects
1
2
([A] + [A′]) =
1
2
(A+BC +A−BC)→ A
1
2
([A]− [A′]) = 1
2
(A+BC −A+BC)→ BC
I By assembling the full 23 design with I = +ABC in the
first group of runs and I = −ABC in the second, the 23
confounds ABC with blocks.
Design resolution 11
I The previous 23−1 is called a resolution III design ( 23−1III )
I Definitions of designs with resolution III, IV, and V
1. Resolution III designs. No main effect is aliased with
any other main effect, but main effects are aliased with
two-factor interactions, and some two-factor interactions
may be aliased with each other.
2. Resolution IV designs. No main effect is aliased with
any other main effect or with any two-factor interactions,
but two-factor interactions are aliased with each other.
Example: A 24−1 design with I = ABCD is a resolution
IV design (24−1IV ).
3. Resolution V designs. No main effect or two-factor
interactions are aliased with any other main effect or
two-facvtor interactions, but two-factor interactions are
aliased with three-factor interactions.
Example: A 25−1 design with I = ABCDE is a
resolution V design (25−1V ).
Constructing 2k−1 with the highest resolution 12
I A one-half fraction of the 2k design of the highest
resolution can be constructed by writing down a basic
design consisting of the runs of a full 2k−1 factorial and
then adding the kth factor the following way:
I Identify the plus and minus signs of the kth factor with the
plus and minus signs of the highest order interaction
ABC . . . (K − 1)
I Example: The 23−1III fractional factorial is obtained by
writing down the full 22 factorial as the basic design and
then equating factor C to the AB interaction. The
alternate fraction is obtained by equating factor C to the
−AB interaction.
Constructing 2k−1 with the highest resolution 13
Constructing 2k−1 with the highest resolution 14
In general:
I The basic design always has the right number of runs
(rows) but it is missing one column.
I The missing column is K = ABC . . . (K − 1).
I Any interaction effect could be used to generate the
columns for the kth factor
I Using any effect other than K = ABC . . . (K − 1) will not
produce a design of the highest possible resolution.
I Another way of looking at it: to partition the runs into
two-blocks with the highest order interaction
K = ABC . . .K confounded. Each block is a 2k−1
fractional factorial design of the highest resolution.
Projection of fractions into factorials 15
I Any fractional design of resolution R contains complete
factorial designs in any subset of R− 1 factors.
I If several factors are of interest but only R− 1 have
important effects, then a fractional factorial design of
resolution R is the appropriate choice.
I The fractional factorial design of resolution R will project
into a full factorial in the R− 1 significant factors
Projection of fractions into factorials 16
I The maximum resolution of a one-half fraction of the 2k
design is R = k, every 2k−1 design will project into a full
factorial in any of the (k − 1) original k factors.
I A 2k−1 design may be projected into two replicates of a
full factorial in any subset of k − 2 factors.
I A 2k−1 design may be projected into four replicates of a
full factorial in any subset of k − 3 factors and so on.
Example 8.1 17
Consider the filtration rate experiment in Example 6.2
Example 8.1 18
What have happened it a half-fraction of the 24 design had
been run instead of the full factorial?
I To run the half-fraction of the 24 design we will use
I = ABCD as the generator.
I This choice of generator will produce the highest possible
resolution (IV) for the design.
I First step: to write down the basic design , which is a 23
design.
Example 8.1 19
I The basic design has the necessary number of runs (eight)
but only three columns, A,B, and C.
I To find the fourth factor, solve I = ABCD for D giving us
D = ABC
Aliases relations 20
I Using the defining relation I = ABCD we can find that
each main effect is aliased with a three-factor interaction.
Example:
AI = A(ABCD) = A2BCD = BCD
I Every two-factor interaction is aliased with another
two-factor interaction. Example:
ABI = ABABCD = CD
I The four main effects plus the three two-factor interaction
alias pair account for seven degrees of freedom.
Example 8.1 21
I The last two columns of Table 8.3 correspond to the runs
of the 24−1IV design
I The linear combination of observations associated with the
A effect is
[A] =
1
4
(−(1) + ad− bd+ ab− cd+ ac− bc+ abcd)
=
1
4
(−45 + 100− 45 + 65− 75 + 60− 80 + 96)
= 19→ A+BCD
I AB + CD alias chain has a small estimate, so either both
interactions are negligible, or both are high with opposite
signs.
I If A,C,and D are significant, then the significant
interactions are most likely AC and AD.
I Ockham’s principle: scientific principle that when one is
confronted with several different possible
interpretations of a phenomena, the simplest
interpretation is usually the correct one.
Final model 23
I Factor B is not significant so it can be dropped.
I Consequently, we can project this 24−1IV design into a single
replicate 23 design in factors A, C, and D.
I Finally, we can do further analysis such as regression
analysis.
I In this example, nothing is significant. We learned from the
full factorial model that this is not the case. Running the
fraction was not enough to find the important effects.
Sequences of Fractional Factorials 24
I Great economy if the runs can be made sequentially
I Suppose k = 4 (16 runs).
I It is preferable to run a 24−1IV fractional design (8 runs),
then decide on the best set of runs to perform next.
I If necessary, we can always run the alternate fraction and
complete the 24 design.
I Both one-half fractions represents blocks of the complete
design with the highest order interaction confounded with
blocks.
I Advantage: in many cases we learn enough from the first
fraction and can make adjustments if necessary.
The one-quarter fraction of the 2k design 25
I Useful if we have a large number of factors
I The design contains 2k−2 runs, called a 2k−2 fractional
design.
I Construct the design by writing down the basic design of a
full factorial in k − 2 factors.
I A one-quarter fraction of the 2k design has two generators.
I If P and Q are the chosen generators, then I = P and
I = Q are called the generating relations for the design.
I The signs of P and Q will determine which fraction is
produced.
I The four fractions P+Q+, P−Q+, P+Q−, and P−Q− are
members of the same family.
The one-quarter fraction of the 2k design 26
I P+Q+ is the principal fraction.
I The complete defining relation consists of all columns
that are equal to the identity column I: i.e. P , Q and PQ
(generalized interaction).
I P , Q and PQ in the defining relation are called words
I Each effect has three aliases (careful when choosing the
generator so important effects are not aliased with each
other)
Example: Consider the 26 design 27
I Suppose we choose I = ABCE and I = BCDF . The
generalized interaction of the generators is ADEF ,
therefore the complete defining relation for this design is:
I = ABCE = BCDF = ADEF
I To find the aliases of any effect, multiply that effect by
each word in the defining relation
A = BCE = ABCDF = DEF
Example - Alias Structure 28
I Every main effect is aliased by three- and five-factor
interactions
I Two-factor interactions are aliased with each other and
with higher order interactions
I This is a resolution IV design
Design construction 29
I Write down the basic design, which consists of 16 runs of
the 24 design in A,B,C, and D
I The two factors E and F are added by associating their
plus and minus levels with the plus and minus signs of the
interactions ABC and BCD, respectively
Example 30
Another way to construct the design 31
I Another way to construct this design is to derive the 4
blocks of the 26 design with ABCE and BCDF
confounded with blocks then choose the treatment
combinations that are positive on ABCE and BCDF .
I This would be a 26−2 fractional factorial with generating
relations I = ABCE and I = BCDF and because both
generators are positive, this is the principal fraction.
The alternate fraction 32
I There are three alternate fractions of this particular 26−2IV
coming from I = ABCE and I = −BCDF ; I = −ABCE
and I = BCDF ; I = −ABCE and I = −BCDF
I We can use similar strategy as Table 8.9. For example, if
we wish to find the fraction for I = ABCE and
I = −BCDF , we set E as before and F = −BCD
I The complete defining relation for this alternate fraction is
I = ABCE = −BCDF = −ADEF .
I The aliases for A are A = BCE = −DEF = −ABCDF
and this the linear combination of the observations [A]
actually estimates A+BCE −DEF −ABCDF
Projections 33
I The 26−2IV design will project into a single replicate of a 24
design in any subset of four factors that is not a word in
the defining relation
I It collapses to a replicated one-half fraction of a 24 in any
subset of four factors that is a word in the defining relation.
In this case, the design in Table 8.9 becomes two replicates
of a 24−1 in the factors ABCE, BCDF , and ADEF
I There are 12 other combinations of the 6 factors such as
ABCD,ABCF for which the design projects to a single
replicate of the 24.
I This designs also collapses to two replicates of a 23 in any
subset of three of the six factors or four replicates of a 22 in
any subset of two factors.
Example 8.4 34
I Parts manufactured in a molding process are showing
excessive shrinkage, causing problems in assembly
operations.
I The quality improvement team runs an experiment to
investigate considering 6 factors, each at two levels:
A: mold temperature
B: screw speed
C: holding time
D: cycle time
E: gate size
F: holding pressure
I The team decides to run 16 runs of the same design shown
in Table 8.9.
学霸联盟
Experiments
Michelle F. Miranda
University of Victoria
michellemiranda@uvic.ca Office: DTB 543
Chapter 8- Fractional Factorial Designs 2
Two-level Fractional Factorial Design
I For a 2k factorial designs, 2k increases very fast as k
increases
22 = 4, 23 = 8, 24 = 16, 25 = 32, 26 = 64 runs
.
I Sometimes there is not enough resources to run a complete
replicate for a 2k design.
I We can run a fraction of 2k runs, and this yields fractional
factorial designs.
I Major use is in screening experiments (when many factors
are considered with the goal of identifying those with large
effects).
Fractional Factorial Designs 3
One-half fraction of the 2k design
I Example: Consider the case when the experimenter cannot
run all 8 combinations of the 23 design, but can afford 4
runs.
I This is a one-half fraction of a 23 design
I This is called a 23−1 design
One-half fraction of the 2k design 4
I Suppose we select 4 treatments: a, b, c, and abc as our
one-half fraction
I Notice: the 23−1 design is formed by selecting the
combination with plus on ABC
I I = ABC is called the generator and this one-half fraction
is usually called the principal fraction . We can also refer
to ABC as a word or the defining relation for our design.
Effect Estimation 5
I From the table, we notice that the contrasts to estimate
the main effects are
[A] =
1
2
(a− b− c+ abc)
[B] =
1
2
(−a+ b− c+ abc)
[C] =
1
2
(−a− b+ c+ abc)
I Notation [A], [B], and [C] indicates the linear combinations
associated with the main effects
Aliases 6
I We can also verify that the linear combination of the
observations used to estimate the two-factor interactions
are
[BC] =
1
2
(a− b− c+ abc)
[AC] =
1
2
(−a+ b− c+ abc)
[AB] =
1
2
(−a− b+ c+ abc)
I Thus when we estimate [A] we are in fact estimating
[A] + [BC]; [A] and [BC] are aliases
I Notation to indicate aliases [A]→ A+BC, [B]→ B +AC,
[C]→ C +AB
One-half fraction of the 2k design 7
I Now suppose we select the last 4 treatments: ab, acb, bc,
and (1) as our one-half fraction
I In this case I = −ABC and this one-half fraction is called
the alternate or complementary fraction
One-half fraction of the 2k design 8
I When we estimate A,B, and C with this particular
fraction, we are estimating A−BC, B−AC, AND C −AB
I The alternate fraction gives us
[A′]→ A−BC
[B′]→ B −AC
[C ′]→ C −AB
One-half fraction of the 2k design 9
I When we estimate A,B, and C with this particular
fraction, we are estimating A−BC, B−AC, AND C −AB
I The alternate fraction gives us
[A′]→ A−BC
[B′]→ B −AC
[C ′]→ C −AB
I The two one-half fractions form a complete 23 design
One-half fraction of the 2k design 10
I If we run both halves of the design, we can obtain
de-aliased estimates of all the effects
1
2
([A] + [A′]) =
1
2
(A+BC +A−BC)→ A
1
2
([A]− [A′]) = 1
2
(A+BC −A+BC)→ BC
I By assembling the full 23 design with I = +ABC in the
first group of runs and I = −ABC in the second, the 23
confounds ABC with blocks.
Design resolution 11
I The previous 23−1 is called a resolution III design ( 23−1III )
I Definitions of designs with resolution III, IV, and V
1. Resolution III designs. No main effect is aliased with
any other main effect, but main effects are aliased with
two-factor interactions, and some two-factor interactions
may be aliased with each other.
2. Resolution IV designs. No main effect is aliased with
any other main effect or with any two-factor interactions,
but two-factor interactions are aliased with each other.
Example: A 24−1 design with I = ABCD is a resolution
IV design (24−1IV ).
3. Resolution V designs. No main effect or two-factor
interactions are aliased with any other main effect or
two-facvtor interactions, but two-factor interactions are
aliased with three-factor interactions.
Example: A 25−1 design with I = ABCDE is a
resolution V design (25−1V ).
Constructing 2k−1 with the highest resolution 12
I A one-half fraction of the 2k design of the highest
resolution can be constructed by writing down a basic
design consisting of the runs of a full 2k−1 factorial and
then adding the kth factor the following way:
I Identify the plus and minus signs of the kth factor with the
plus and minus signs of the highest order interaction
ABC . . . (K − 1)
I Example: The 23−1III fractional factorial is obtained by
writing down the full 22 factorial as the basic design and
then equating factor C to the AB interaction. The
alternate fraction is obtained by equating factor C to the
−AB interaction.
Constructing 2k−1 with the highest resolution 13
Constructing 2k−1 with the highest resolution 14
In general:
I The basic design always has the right number of runs
(rows) but it is missing one column.
I The missing column is K = ABC . . . (K − 1).
I Any interaction effect could be used to generate the
columns for the kth factor
I Using any effect other than K = ABC . . . (K − 1) will not
produce a design of the highest possible resolution.
I Another way of looking at it: to partition the runs into
two-blocks with the highest order interaction
K = ABC . . .K confounded. Each block is a 2k−1
fractional factorial design of the highest resolution.
Projection of fractions into factorials 15
I Any fractional design of resolution R contains complete
factorial designs in any subset of R− 1 factors.
I If several factors are of interest but only R− 1 have
important effects, then a fractional factorial design of
resolution R is the appropriate choice.
I The fractional factorial design of resolution R will project
into a full factorial in the R− 1 significant factors
Projection of fractions into factorials 16
I The maximum resolution of a one-half fraction of the 2k
design is R = k, every 2k−1 design will project into a full
factorial in any of the (k − 1) original k factors.
I A 2k−1 design may be projected into two replicates of a
full factorial in any subset of k − 2 factors.
I A 2k−1 design may be projected into four replicates of a
full factorial in any subset of k − 3 factors and so on.
Example 8.1 17
Consider the filtration rate experiment in Example 6.2
Example 8.1 18
What have happened it a half-fraction of the 24 design had
been run instead of the full factorial?
I To run the half-fraction of the 24 design we will use
I = ABCD as the generator.
I This choice of generator will produce the highest possible
resolution (IV) for the design.
I First step: to write down the basic design , which is a 23
design.
Example 8.1 19
I The basic design has the necessary number of runs (eight)
but only three columns, A,B, and C.
I To find the fourth factor, solve I = ABCD for D giving us
D = ABC
Aliases relations 20
I Using the defining relation I = ABCD we can find that
each main effect is aliased with a three-factor interaction.
Example:
AI = A(ABCD) = A2BCD = BCD
I Every two-factor interaction is aliased with another
two-factor interaction. Example:
ABI = ABABCD = CD
I The four main effects plus the three two-factor interaction
alias pair account for seven degrees of freedom.
Example 8.1 21
I The last two columns of Table 8.3 correspond to the runs
of the 24−1IV design
I The linear combination of observations associated with the
A effect is
[A] =
1
4
(−(1) + ad− bd+ ab− cd+ ac− bc+ abcd)
=
1
4
(−45 + 100− 45 + 65− 75 + 60− 80 + 96)
= 19→ A+BCD
I AB + CD alias chain has a small estimate, so either both
interactions are negligible, or both are high with opposite
signs.
I If A,C,and D are significant, then the significant
interactions are most likely AC and AD.
I Ockham’s principle: scientific principle that when one is
confronted with several different possible
interpretations of a phenomena, the simplest
interpretation is usually the correct one.
Final model 23
I Factor B is not significant so it can be dropped.
I Consequently, we can project this 24−1IV design into a single
replicate 23 design in factors A, C, and D.
I Finally, we can do further analysis such as regression
analysis.
I In this example, nothing is significant. We learned from the
full factorial model that this is not the case. Running the
fraction was not enough to find the important effects.
Sequences of Fractional Factorials 24
I Great economy if the runs can be made sequentially
I Suppose k = 4 (16 runs).
I It is preferable to run a 24−1IV fractional design (8 runs),
then decide on the best set of runs to perform next.
I If necessary, we can always run the alternate fraction and
complete the 24 design.
I Both one-half fractions represents blocks of the complete
design with the highest order interaction confounded with
blocks.
I Advantage: in many cases we learn enough from the first
fraction and can make adjustments if necessary.
The one-quarter fraction of the 2k design 25
I Useful if we have a large number of factors
I The design contains 2k−2 runs, called a 2k−2 fractional
design.
I Construct the design by writing down the basic design of a
full factorial in k − 2 factors.
I A one-quarter fraction of the 2k design has two generators.
I If P and Q are the chosen generators, then I = P and
I = Q are called the generating relations for the design.
I The signs of P and Q will determine which fraction is
produced.
I The four fractions P+Q+, P−Q+, P+Q−, and P−Q− are
members of the same family.
The one-quarter fraction of the 2k design 26
I P+Q+ is the principal fraction.
I The complete defining relation consists of all columns
that are equal to the identity column I: i.e. P , Q and PQ
(generalized interaction).
I P , Q and PQ in the defining relation are called words
I Each effect has three aliases (careful when choosing the
generator so important effects are not aliased with each
other)
Example: Consider the 26 design 27
I Suppose we choose I = ABCE and I = BCDF . The
generalized interaction of the generators is ADEF ,
therefore the complete defining relation for this design is:
I = ABCE = BCDF = ADEF
I To find the aliases of any effect, multiply that effect by
each word in the defining relation
A = BCE = ABCDF = DEF
Example - Alias Structure 28
I Every main effect is aliased by three- and five-factor
interactions
I Two-factor interactions are aliased with each other and
with higher order interactions
I This is a resolution IV design
Design construction 29
I Write down the basic design, which consists of 16 runs of
the 24 design in A,B,C, and D
I The two factors E and F are added by associating their
plus and minus levels with the plus and minus signs of the
interactions ABC and BCD, respectively
Example 30
Another way to construct the design 31
I Another way to construct this design is to derive the 4
blocks of the 26 design with ABCE and BCDF
confounded with blocks then choose the treatment
combinations that are positive on ABCE and BCDF .
I This would be a 26−2 fractional factorial with generating
relations I = ABCE and I = BCDF and because both
generators are positive, this is the principal fraction.
The alternate fraction 32
I There are three alternate fractions of this particular 26−2IV
coming from I = ABCE and I = −BCDF ; I = −ABCE
and I = BCDF ; I = −ABCE and I = −BCDF
I We can use similar strategy as Table 8.9. For example, if
we wish to find the fraction for I = ABCE and
I = −BCDF , we set E as before and F = −BCD
I The complete defining relation for this alternate fraction is
I = ABCE = −BCDF = −ADEF .
I The aliases for A are A = BCE = −DEF = −ABCDF
and this the linear combination of the observations [A]
actually estimates A+BCE −DEF −ABCDF
Projections 33
I The 26−2IV design will project into a single replicate of a 24
design in any subset of four factors that is not a word in
the defining relation
I It collapses to a replicated one-half fraction of a 24 in any
subset of four factors that is a word in the defining relation.
In this case, the design in Table 8.9 becomes two replicates
of a 24−1 in the factors ABCE, BCDF , and ADEF
I There are 12 other combinations of the 6 factors such as
ABCD,ABCF for which the design projects to a single
replicate of the 24.
I This designs also collapses to two replicates of a 23 in any
subset of three of the six factors or four replicates of a 22 in
any subset of two factors.
Example 8.4 34
I Parts manufactured in a molding process are showing
excessive shrinkage, causing problems in assembly
operations.
I The quality improvement team runs an experiment to
investigate considering 6 factors, each at two levels:
A: mold temperature
B: screw speed
C: holding time
D: cycle time
E: gate size
F: holding pressure
I The team decides to run 16 runs of the same design shown
in Table 8.9.
学霸联盟
