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Factorial Experiments:

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Factorial Experiments:-Blocking, -Confounding, and-Fractional Factorial Designs. Emanuel Msemo Wednesday, July 30, 2014 4:30pm 6:30 pm 1020 Torgersen Hall – PowerPoint PPT presentation

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Title: Factorial Experiments:


1
Factorial Experiments -Blocking,
-Confounding, and -Fractional Factorial
Designs.
Emanuel Msemo
Wednesday, July 30, 2014 430pm 630 pm 1020
Torgersen Hall
2
ABOUT THE INSTRUCTOR
Graduate student in Virginia Tech Department of
Statistics
  • B.A. ECONOMICS AND STATISTICS
  • (UDSM,TANZANIA)
  • MSc. STATISTICS (VT,USA)
  • LEAD/ASSOCIATE COLLABORATOR IN LISA

If your experiment needs a statistician, you
need a better experiment.   Ernest Rutherford
3
MORE ABOUT LISA
www.lisa.vt.edu
What? Laboratory for Interdisciplinary
Statistical Analysis Why? Mission to provide
statistical advice, analysis, and education to
Virginia Tech researchers How? Collaboration
requests, Walk-in Consulting, Short
Courses Where? Walk-in Consulting in GLC and
various other locations Collaboration meetings
typically held in Sandy 312 Who? Graduate
students and faculty members in VT statistics
department
4
HOW TO SUBMIT A COLLABORATION REQUEST
  • Go to www.lisa.stat.vt.edu
  • Click link for Collaboration Request Form
  • Sign into the website using VT PID and password
  • Enter your information (email, college, etc.)
  • Describe your project (project title, research
    goals, specific research questions, if you have
    already collected data, special requests, etc.)
  • Contact assigned LISA collaborators as soon as
    possible to schedule a meeting

5
LISA helps VT researchers benefit from the use of
Statistics
Collaboration Visit our website to request
personalized statistical advice and assistance
with Experimental Design Data Analysis
Interpreting Results Grant Proposals Software
(R, SAS, JMP, SPSS...) LISA statistical
collaborators aim to explain concepts in ways
useful for your research. Great advice right now
Meet with LISA before collecting your data.
Short Courses Designed to help graduate students
apply statistics in their research Walk-In
Consulting M-F 1-3 PM GLC Video
Conference Room
11 AM-1 PM Old Security Building Room
103 For
questions requiring lt30 mins
All services are FREE for VT researchers. We
assist with researchnot class projects or
homework.
6
COURSE CONTENTS
1. INTRODUCTION TO DESIGN AND ANALYSIS OF
EXPERIMENTS
1.1 Introduction
1.2 Basic Principles
1.3 Some standard experimental designs designs
2. INTRODUCTION TO FACTORIAL DESIGNS
2.1 Basic Definitions and Principles
2.2 The advantage of factorials
2.3 The two-Factor factorial designs
2.4 The general factorial designs
2.5 Blocking in a factorial designs
7
3. THE 2K FACTORIAL DESIGNS
3.1 Introduction
3.2 The 22 and 23 designs and the General 2k
designs
3.3 A single replicate of the 2k designs
4. BLOCKING AND CONFOUNDING IN THE 2K FACTORIAL
DESIGNS
4.1 Introduction
4.2 Blocking a replicated 2k factorial design.
4.3 Confounding in the 2k factorial designs.
5. TWO LEVEL FRACTIONAL FACTORIAL DESIGNS
5.1 Why do we need fractional factorial designs?
5.2 The one-half Fraction of the 2k factorial
design
5.3The one-quarter Fraction of the 2k factorial
design
8
INTRODUCTION TO DESIGN AND ANALYSIS OF
EXPERIMENTS
Questions
What is the main purpose of running an experiment
?
What do one hope to be able to show?
Typically, an experiment may be run for one or
more of the following reasons
1. To determine the principal causes of variation
in a measured response
2. To find conditions that give rise to a maximum
or minimum response
3. To compare the response achieved at different
settings of controllable variables
4. To obtain a mathematical model in order to
predict future responses
9
  • An Experiment involves the manipulation of one
  • or more variables by an experimenter in order
    to
  • determine the effects of this manipulation
  • on another variable.
  • Much research departs from this pattern in that
    nature rather than
  • the experimenter manipulates the variables.
    Such research is
  • referred to as Observational studies
  • This course is concerned with COMPARATIVE
  • EXPERIMENTS
  • These allows conclusions to be drawn about
    cause
  • and effect (Causal relationships)

10
Sources of Variation
  • A source of variation is anything that could
  • cause an observation to be different from
  • another observation

Independent Variables
  • The variable that is under the control of the
  • experimenter.
  • The terms independent variables, treatments,
  • experimental conditions, controllable
    variables
  • can be used interchangeably

11
Dependent variable
  • The dependent variable (response) reflects
  • any effects associated with manipulation
  • of the independent variable

Now
Sources of Variation are of two types
  • Those that can be controlled and are of
  • interest are called treatments or treatment
  • factors
  • Those that are not of interest but are
  • difficult to control are nuisance factors

12
Uncontrollable factors
Z1
Z2
ZP
.
PROCESS
INPUTS
OUTPUT (Response)
.
X1
X2
XP
Controllable factors
Adapted from Montgomery (2013)
The primary goal of an experiment is to determine
the amount of variation caused by the treatment
factors in the presence of other sources of
variation
13
The objective of the experiment may include the
following
  • Determine which conditions are most influential
    on the response
  • Determine where to set the influential conditions
    so that the
  • response is always near the desired nominal
    value
  • Determine where to set the influential conditions
    so that variability
  • in the response is small
  • Determine where to set the influential conditions
    so that the effects
  • of the uncontrollable Variables are minimized

14
EXAMPLE
Researchers were interested to see the food
consumption of albino rats when exposed to
microwave radiation
If albino rats are subjected to microwave
radiation, then their food consumption will
decrease
15
TRY!
Independent variable?
.
Dependant variable?
.
Nuisance factor (s)?
.
16
BASIC PRINCIPLES
The three basic principles of experimental
designs are
  • Randomization
  • The allocation of experimental material and the
    order in
  • which the individual runs of the experiment
    are to be
  • performed are randomly determined
  • Replication
  • Independent repeat run of each factor combination
  • Number of Experimental Units to which a treatment
  • is assigned

17
Blocking
  • A block is a set of experimental units sharing a
  • common characteristics thought to affect the
  • response, and to which a separate random
  • assignment is made
  • Blocking is used to reduce or eliminate the
  • variability transmitted from a nuisance factor

18
SOME STANDARD EXPERIMENTAL DESIGNS
The term experimental design refers to a plan of
assigning experimental conditions to subjects and
the statistical analysis associated with the
plan. OR An experimental design is a rule that
determines the assignment of the experimental
units to the treatments.
19
  • Some standard designs that are used frequently
    includes

Completely Randomized design
A completely randomized design (CRD) refer to a
design in which the experimenter assigns the
EUs to the treatments completely at random,
subject only to the number of observations to
be taken on each treatment.
The model is of the form
Response constant effect of a treatment
error
20
Block designs
This is a design in which experimenter partitions
the EUs in blocks, determines the allocation
of treatments to blocks, and assigns the EUs
within each block to the treatments completely
at random
The model is of the form
Response Constant effect of a block
effect of treatment error
21
Designs with two or more blocking factors
These involves two major sources of variation
that have been designated as blocking factors.
The model is of the form
Response Constant effect of row block
effect of column block
effect of treatment error
22
INTRODUCTION TO FACTORIAL DESIGNS
  • Experiments often involves several factors, and
    usually
  • the objective of the experimenter is to
    determine the
  • influence these factors have on the response.
  • Several approaches can be employed to deal when
  • faced with more than one treatments

Best guess Approach
Experimenter select an arbitrary combinations of
treatments, test them and see what happens
23
One - Factor - at - a - time (OFAT)
  • Consists of selecting a starting point, or
    baseline set of
  • levels, for each factor, and then
    successively varying
  • each factor over its range with the other
    factors held
  • constant at the baseline level.

24
  • The valuable approach to dealing with
  • several factors is to conduct a
  • FACTORIAL EXPERIMENT
  • This is an experimental strategy in which
  • factors are varied together, instead of one
  • at a time

25
  • In a factorial design, in each complete trial
  • or replicate of the experiment, all possible
  • combination of the levels of the factors
  • are investigated.

e.g.
If there are a levels of factor A and b levels of
factor B, each replicate contains all ab
treatment combinations
The model is of the form
Response Constant Effect of factor A
Effect of factor B
Interaction effect Error term
26
Consider the following example (adapted from
Montgomery, 2013) of a two-factors (A and B)
factorial experiment with both design factors at
two levels (High and Low)
B High A High
B High A Low
52
30
B Low A Low
B Low A High
20
40
27
Main effect Change in response produced by a
change in the level of a
factor
Factor A
_
Main Effect 40 52
20 30
2
2
21
,Increasing factor A from low level to high
level, causes an average response increase of 21
units
Factor B
?
Main Effect
28
Interaction
A High B High
A Low B High
12
40
50
A Low B Low
20
A High B Low
29
At low level of factor B
The A effect

50 20

30
At high level of factor B
The A effect

12 - 40

-28
The effect of A depends on the level chosen for
factor B
30
If the difference in response between the levels
of one factor is not the same at all levels
of the other factors then we say there is an
interaction between the factors (Montgomery
2013)
The magnitude of the interaction effect is the
average difference in the two factor A effects
AB
(-28 30)

2

-29
A effect
1

In this case, factor A has an effect, but it
depends on the level of factor B be chosen
31
Interaction Graphically
B High
B High
B Low
Response
Response
B Low
Low
High
Low
High
Factor A
Factor A
  • A factorial experiment
  • without interaction
  • A factorial experiment with
  • interaction

32
Factorial designs has several advantages
  • They are more efficient than One Factor at a Time
  • A factorial design is necessary when interactions
  • may be present to avoid misleading conclusions
  • Factorial designs allow the effect of a factor to
    be
  • estimated at a several levels of the other
    factors,
  • yielding conclusions that are valid over a
    range
  • of experimental conditions

33
The two factor Factorial Design
  • The simplest types of factorial design involves
  • only two factors.
  • There are a levels of factor A and b levels of
  • factor B, and these are arranged in a
    factorial
  • design.


  • There are n replicates, and each replicate of the
  • experiment contains all the ab combination.

34
Example
An engineer is designing a battery for use in a
device that will be subjected to some extreme
variations in temperature. The only design
parameter that he can select is the plate
material for the battery. For the purpose of
testing temperature can be controlled in the
product development laboratory (Montgomery, 2013)
Life (in hours) Data
Temperature
Material Type
15
70
125
130 74 150 159 138 168
155 180 188 126 110 160
34 80 136 106 174 150
40 75 122 115 120 139
20 82 25 58 96 82
70 58 70 45 104 60
1
2
3
35
  • The design has two factors each at three levels
    and is
  • then regarded as 32 factorial design.
  • The design is a completely Randomized Design

The engineer wants to answer the following
questions
1. What effects do material type and temperature
have on the life of the battery?
2 .Is there a choice of material that would give
uniformly long life regardless of temperature?
  • Both factors are assumed to be fixed,
  • hence we have a fixed effect model

36
Analysis of Variance for Battery life (in
hours) Source DF
Seq SS Adj SS Adj MS F
P-value Material Type 2
10683.7 10683.7 5341.9 7.91
0.002 Temperature 2
39118.7 39118.7 19559.4 28.97
0.000 Material TypeTemperature 4 9613.8
9613.8 2403.4 3.56 0.019 Error
27
18230.7 18230.7 675.2 Total
35 77647.0
We have a significant interaction between
temperature and material type.
37
Interaction plot
Significant interaction is indicated by the lack
of parallelism of the lines,Longer life is
attained at low temperature, regardless Of
material type
38
The General Factorial Design
  • The results for the two factor factorial
  • design may be extended to the general
  • case where there are a levels of factor A,
  • b levels of factor B, c levels of factor C,
  • and so on, arranged in a factorial
  • experiment.

39
  • Sometimes, it is not feasible or practical
  • to completely randomize all of the runs
  • in a factorial.
  • The presence of a nuisance factor may
  • require that experiment be run in blocks.

The model is of the form
Response Constant Effect of factor A Effect
of factor B interaction
effect Block Effect Error term
40
The 2K Factorial designs
  • This is a case of a factorial design with K
    factors, each
  • at only two levels.
  • These levels may be quantitative or qualitative.
  • A complete replicate of this design requires
  • 2K observation and is called 2K factorial
    design.

Assumptions
1. The factors are fixed.
2. The designs are completely randomized.
3. The usual normality assumptions are satisfied.
41
  • The design with only two factors each at two
    levels is
  • called 22 factorial design
  • The levels of the factors may be arbitrarily
    called
  • Low and High

Factor
A
B
Treatment Combination
- -
- -
A Low, B Low A High, B Low A Low, B High A
High, B High
(1) a b ab
The order in which the runs are made is a
completely randomized experiment
42
  • The four treatment combination in the design can
    be
  • represented by lower case letters
  • The high level factor in any treatment
    combination is
  • denoted by the corresponding lower case letter
  • The low level of a factor in a treatment
    combination is
  • represented by the absence of the
    corresponding letter
  • The average effect of a factor is the change in
    the
  • response produced by a change in the level of
    that
  • factor averaged over the levels of the other
    factor

43
  • The symbols (1), a, b, ab represents the total
  • of the observation at all n replicates
  • taken at a treatment combination

A main effect 1/2nab a b (1)
B main effect 1/2nab b - a (1)
AB effect 1/2nab (1) a b
44
  • In experiments involving 2K designs, it is
  • always important to examine the magnitude
  • and direction of the factor effect to
    determine
  • which factors are likely to be important
  • Effect Magnitude and direction should always
  • be considered along with ANOVA, because the
  • ANOVA alone does not convey this information

45
We define
Contrast A ab a b (1) Total effect of A
  • We can write the treatment combination in the
    order
  • (1), a, b, ab. Also called the standard order
    (or Yates order)

Factorial Effect
Treatment Combination
I
A
B
AB
(1) a b ab

- -
- -
- -
The above is also called the table of plus and
minus signs
46
  • Suppose that three factors, A ,B and C, each at
    two levels
  • are of interest. The design is referred as 23
    factorial design

Treatment Combination
Factorial Effects
I
A
B
AB
C
AC
BC
ABC
- - - -
(1) a b ab c ac bc abc

- - - -
- - - -
- - - -
- - - -
- - - -
- - - -
A contrast ab a ac abc (1) b c - bc
B contrast ?
47
In General
  • The design with K factors each at two levels is
  • called a 2K factorial design
  • The treatment combination are written in
  • standard order using notation introduced
  • in a 22 and 23 designs

48
A single replicate of the 2K Designs
  • For even a moderate number of factors, the total
    number of
  • treatment combinations in a 2K factorials
    designs is large.
  • 25 design has 32 treatment combinations
  • 26 design has 64 treatment combinations
  • Resources are usually limited, and the number of
    replicates that
  • the experimenter can employ may be restricted
  • Frequently, available resources only allow a
    single replicate of the
  • design to be run, unless the experimenter is
    willing to omit some
  • of the original factors

49
  • An analysis of an unreplicated factorials assume
  • that certain high order interaction are
    negligible
  • and combine their means squares to estimate
    the
  • error
  • This is an appeal to sparsity of effect
    principle,
  • that is most systems are dominated by some
    of
  • the main effect and low order interactions,
    and
  • most high order interactions are negligible
  • When analyzing data from unreplicated factorial
  • designs, its is suggested to use normal
    probability
  • plot of estimates of the effects

50
Example
A chemical product is produced in a pressure
vessel. A factorial experiment is carried out in
the pilot plant to study the factors thought to
influence filtration rate of this product. The
four factors are Temperature (A), pressure (B),
concentration of formaldehyde (C), and string
rate (D). Each factor is present at two levels.
The process engineer is interested in maximizing
the filtration rate. Current process gives
filtration rate of around 75 gal/h. The process
currently uses the factor C at high level. The
engineer would like to reduce the formaldehyde
concentration as much as possible but has been
unable to do so because it always results in
lower filtration rates (Montgomery, 2013)
51
The design matrix and response data obtained from
single replicate of the 24 experiment
Factors
Treatment Combination
A B C D Response - - - 48 - - - 43 -
70 - - 80 - - - 68 - - - 71 - - 60 -
- 45 96 - 86 - 65 - -
100 - - 75 - - 65 - - - - 45
- 104
(1) a b ab c ac bc abc d ad bd abd cd acd bcd ab
cd
52
The Normal probability plot is given below
The important effects that emerge from this
analysis are the main effects of A,C and D and
the AC and AD interactions
53
The main effect plot for Temperature
The plot indicate that its better to run the
Temperature at high levels
54
The main effect plot for Concentration of
Formaldehyde
The plot indicate that its better to run the
concentration of formaldehyde at high levels
55
The main effect plot for Stirring rate
The plot indicate that its better to run the
stirring rate at high levels
56
  • However, its necessary to examine any
  • interactions that are important
  • The best results are obtained with low
    concentration of formaldehyde
  • and high temperatures

57
  • The AD interaction indicate that stirring rate D
    has little effect
  • at low temperatures but a very positive
    effects at high temperature
  • Therefore best filtration rates would appear to
    be obtained when A
  • and D are at High level and C is at low level.
    This will allow
  • Formaldehyde to be reduced to the lower levels

58
Factor effect Estimates and sums of squares for
the 24 Design
Model Term Effect Estimates Sum of Squares Percent Contribution
A 21.63 1870.56 32.64
B 3.13 39.06 0.68
C 9.88 390.06 6.81
D 14.63 855.56 14.93
AB 0.13 0.063 1.091E-003
AC -18.13 1314.06 22.93
AD 16.63 1105.56 19.29
BC 2.38 22.56 0.39
BD -0.38 0.56 9.815E-003
CD -1.12 5.06 0.088
ABC 1.88 14.06 0.25
ABD 4.13 68.06 1.19
ACD -1.62 10.56 0.18
BCD -2.63 27.56 0.48
ABCD 1.38 7.56 0.13
59
ANOVA for A, C and D
Source DF Seq SS Adj SS
AdjMS F P A
1 1870.56 1870.56
1870.56 83.36 lt0.0001 C
1 390.06 390.06
390.06 17.38 lt0.0001 D
1 855.56
855.56 855.56 38.13 lt0.0001
AC 1 1314.06
1314.06 1314.06 58.56 lt0.0001
AD 1 1105.56
1105.56 1105.56 49.27 lt0.0001
CD 1 5.06
5.06 5.06 lt1
ACD 1 10.56
10.56 10.56 lt1
Residual Error 8 179.52 22.44
Total 15 5730.94
60
Blocking and Confounding in the 2K factorial
designs
  • There are situations that may hinder the
    experimenter to
  • perform all of the runs in a 2K factorial
    experiment under
  • homogenous conditions
  • A single batch of raw material might not be large
    enough
  • to make all of the required runs
  • An experimenter with a prior knowledge, may
    decide to run a
  • pilot experiment with different batches of
    raw materials
  • The design technique used in this situations is
    Blocking

61
Blocking a Replicated 2K Factorial design
  • Suppose that the 2K factorial design has been
  • replicated n times
  • With n replicates, then each set of homogenous
  • conditions defines a block, and each replicate
    is run in
  • one of the blocks
  • The run in each block (or replicate) will be made
  • in random order

62
Confounding in the 2K Factorial designs
  • Many situations it is impossible to perform a
    complete
  • replicate of a factorial design in one block
  • Confounding is a design technique for arranging a
  • complete factorial experiment in blocks, where
    the block
  • size is smaller than the number of treatment
  • combinations in one replicate
  • The technique causes information about certain
  • treatment effects (usually) higher order
    interactions)
  • to be indistinguishable from or confounded
    with blocks

63
Confounding the 2K Factorial design in two Blocks
  • Suppose we want to run a single replicate of the
    22 design
  • Each of the 22 4 treatment combination requires
    a quantity
  • of raw material
  • Suppose each batch of raw material is only large
    enough for two
  • treatment combination to be tested, thus two
    batches of raw
  • material are required
  • If batches of raw materials are considered as
    blocks, then we must
  • assign two of the four treatment combinations
    to each block

64
Consider table of plus and minus signs for the 22
design
Treatment Combination
Factorial Effect
I
A
B
Block
AB
(1) a b ab

- -
- -
- -
1 2 2 1
Block 1
Block 2
  • The order in which the treatment
  • combination are run within a block
  • are randomly Determined

(1) ab
a b
  • The block effect and the AB interaction are
    identical. That is, AB is
  • confounded with blocks.

65
  • This scheme can be used to confound any 2K design
  • into two blocks
  • Consider a 23 design run into two blocks
  • Suppose we wish to confound the ABC interaction
    with
  • blocks

Factorial Effect
Treatment Combination
I
A
B
AB
C
AC
BC
ABC
Block
(1) a b ab c ac bc abc

- - - -
- - - -
- - - -
- - - -
- - - -
- - - -
- - - -
1 2 2 1 2 1 1 2
66
  • Again, we assign treatment combinations that are
  • minus on ABC to Block 1 and the rest to block 2
  • The treatment combinations within a block are run
  • in a random order

Block 1
Block 2
(1) ab ac bc
a b c abc
  • ABC is confounded with blocks

67
Alternative method for constructing the block
  • The method uses the linear combination

L a1x1 a2x2 ..... akxk
This is called a defining contrast
  • For the 2K ,xi 0 (low level) or xi 1 (high
    level), ai 0 or 1
  • Treatment combination that produces the same
    value of L (mod 2)
  • will be placed in the same block
  • The only possible values of L (mod 2) are 0 and
    1, hence we
  • will have exactly two blocks

68
  • If resources are sufficient to allow the
    replication
  • of confounded designs, it is generally better
    to
  • use a slightly different method of designing
    the
  • blocks in each replicate
  • We can confound different effects in each
    replicate
  • so that some information on all effects is
    obtained
  • This approach is called partial confounding

69
Consider our previous example
Two modification
  • The 16 treatment combination cannot all be run
    using one batch of raw
  • material. Experimenter will use two batches
    of raw material, hence two
  • blocks each with 8 runs

2. Introduce a block effect, by considering one
batch as of poor quality, such That all the
responses will be 20 units less in this block
Experimenter will confound the highest order
interaction ABCD
The defining contrast is
L x1 x2 x3
70
The two resulting blocks are
(1) ab ac bc ad bd cd abcd
a b c d abc bcd acd abd
71
The half Normal plot for the blocked design
72
Source DF Seq SS Adj SS
Adj MS F P-Value Blocks
1 1387.56 1387.56 1387.56 A
1 1870.56
1870.56 1870.56 89.76 lt0.0001 C
1 390.06 390.06
390.06 18.72 lt0.0001 D
1 855.56 855.56
855.56 41.05 lt0.0001 AC
1 1314.06 1314.06 1314.06
63.05 lt0.0001 AD 1
1105.56 1105.56 1105.56 53.05
lt0.0001 Residual Error 9 187.56
187.56 20.8403 Total 15
7110.94
73
  • Similar methods can be used to confound the 2K
    designs
  • to four blocks, and so on, depending on
    requirement

NOTE
Blocking is a noise reduction technique. If we
dont block, then the added variability from the
nuisance variable effect ends up getting
distributed across the other design factors
74
Two level Fractional Factorial Designs
  • As the number of factors in a 2K factorial
    designs increases, the
  • number of runs required for a complete
    replicate of the design
  • rapidly outgrows the resources of most
    experimenters
  • If the experimenter can reasonably assume that
    certain high-order
  • interactions are negligible, information on
    the main effects and
  • lower order interactions may be obtained by
    running a fraction of a
  • complete factorial experiment
  • Fractional factorials designs are widely used for
    product and
  • process designs, process improvement and
    industrial/business
  • experimentation
  • Fractional factorials are used for screening
  • experiments

75
The successfully use of Fractional factorials
designs is based on three key ideas
1. The sparsity of effect principle
2. The projection property
3. Sequential experimentation
76
The one half Fraction of the 2K Design
  • Suppose an experimenter has two factors, each at
    two
  • levels but cannot afford to run all 23 8
    treatment
  • combinations
  • They can however afford four runs
  • This suggests a one half fraction of a 23 design
  • A one half fraction of the 23 design is often
    called a
  • 23-1 design

77
Recall the table of plus and minus signs for a 23
design
  • Suppose we select those treatment combinations
    that have a plus
  • in the ABC column to form 23-1 design, then
    ABC is called a
  • generator of this particular design
  • Usually a generator such as ABC is referred as a
    WORD
  • The identity column is always plus, so we call

I ABC , The defining relation for our design
78
  • Now, It is impossible to differentiate between A
    and BC, B and AC,
  • and C and AB

We say the effects are aliased
  • The alias structure may be easily determined by
    using a defined relation
  • by multiplying any column by the defining
    relation

A I A ABC A2BC BC
A BC
BI B ABC AB2C AC
B AC
This half fraction with I ABC is called the
Principal fraction
79
Design Resolution
  • A design is of resolution R if no p-factor
    effect is
  • aliased with another effect containing less
    than R-p
  • factors
  • Roman numeral subscript are usually used to
    denote
  • design resolutions
  • Designs of resolution III, IV and V are
    particularly
  • important

80
Resolution III designs
  • These are designs in which no main effects are
    aliased with any
  • other main effects, but main effects are
    aliased with two factor
  • interactions and some two factor interactions
    may be aliased with
  • each other

e.g. the 23-1 design with I ABC is of
resolution III
Resolution IV designs
  • No main effects is aliased with any other main
    effect or with any two
  • factor interactions, but two factor
    interactions are aliased with each
  • other

e.g. A 24-1 design with I ABCD is a resolution
IV design
Resolution V designs
  • No main effect or two factor interactions is
    aliased with any other
  • main effect or two factor interaction, but
    two factor interactions are
  • aliased with three factor interactions

e.g. A 25-1 design with I ABCDE is a
resolution V design
81
Construction of One half Fraction
  • A one half fraction of the 2K design is obtained
    by
  • writing down a basic design consisting of the
    runs for
  • the full 2K-1 factorials and then adding the
    kth factor by
  • identifying its plus and minus levels with the
    plus and
  • minus signs of the highest order interactions
    ABC..(K-1)
  • The 23-1 resolution III design is obtained by
    writing
  • down the full 22 factorials as the basic
    design and then
  • equating C to the AB interactions

82
One half fraction of the 23 design
Full 22 Factorial (Basic Design)
Resolution III, I ABC
Run
A
B
A
B
C AB
1 2 3 4
- -
- -
- -
- -
- -
83
Consider the filtration rate example
  • We will simulate what would happen if a half
    fraction
  • of the 24 design had been run instead of the
    full factorial
  • We will use the 24-1 with I ABCD, As this will
    generate
  • the highest resolution possible
  • We will first write down the basic design, which
    is 23 design
  • The basic design has eight runs but with three
    factors
  • To find the fourth factor levels, we solve I
    ABCD for D

D I D ABCD ABCD2 ABC
84
The resolution IV design with I ABCD
Basic Design
Treatment Combination
Run
D ABC
A
C
B
(1) ad bd ab cd ac bc abcd
1 2 3 4 5 6 7 8
- - - -
- - - -
- - - -
- - - -
85
Estimates of Effects
Term Effect A
19.000 B 1.500
C 14.000 D
16.500 AB
-1.000 AC -18.500 AD
19.000
A ,C and D have large effects, and so is the
interactions involving them
86
Thank you
87
Reference
Montgomery, D.C (2013). Design and analysis of
experiments. Wiley, New York.
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