Transcript 2014-07-30-Factorial-Experiments - LISA

Factorial Experiments:
-Blocking,
-Confounding, and
-Fractional Factorial
Designs.
Wednesday, July 30, 2014
4:30pm – 6:30 pm
1020 Torgersen Hall
Emanuel Msemo
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
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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
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
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
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)
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
Dependent variable
The dependent variable (response) reflects
any effects associated with manipulation
of the independent variable
Sources
of
Variation
are
of
two
types:
Now
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
Uncontrollable factors
Z1
Z2
ZP
…….
INPUTS
PROCESS
…….
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
OUTPUT (Response)
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
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”
TRY!
Independent variable?
……………………….
Dependant variable?
……………………….
Nuisance factor (s)?
……………………….
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
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
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.
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 EU’s 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
Block designs
This is a design in which experimenter partitions the EU’s
in blocks, determines the allocation of treatments to
blocks, and assigns the EU’s 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
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
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
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.
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
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
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
30
A Low
B Low 20
A Low
52
B High
A High
40 B Low
A High
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 = ?
Interaction
A Low
40
B High
A High
12 B High
A Low 20
B Low
50
A High
B Low
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
“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
A effect
=
AB
=
=
(-28 – 30)
2
-29
1
In this case, factor A has an effect, but it depends on the
level of factor B be chosen
Interaction Graphically
B High
B High
B Low
B Low
Low
High
Factor A
A factorial experiment
without interaction
Low
High
Factor A
A factorial experiment with
interaction
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
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.
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
1
2
3
15
130
74
150
159
138
168
70
155
180
188
126
110
160
34
80
136
106
174
150
125
40
75
122
115
120
139
20
82
25
58
96
82
70
58
70
45
104
60
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
Analysis of Variance for Battery life (in hours)
Source
DF Seq SS
Material Type
2 10683.7
Temperature
2
39118.7
Material Type*Temperature 4
9613.8
Error
27 18230.7
Total
35 77647.0
Adj SS
10683.7
39118.7
9613.8
18230.7
Adj MS
F
P-value
5341.9 7.91
0.002
19559.4 28.97 0.000
2403.4
3.56 0.019
675.2
We have a significant interaction between temperature
and material type.
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
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.
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
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.
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
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
The symbols (1), a, b, ab represents the total
of the observation at all n replicates
taken at a treatment combination
A main effect = 1/2n[ab + a – b – (1)]
B main effect = 1/2n[ab +b - a – (1)]
AB effect = 1/2n{[ab + (1) – a – b]
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
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)
Treatment
Combination
(1)
a
b
ab
Factorial Effect
I
A
B
AB
+
+
+
+
+
+
+
+
+
+
The above is also called the table of plus and minus signs
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
(1)
a
b
ab
c
ac
bc
abc
Factorial Effects
I
A
B
AB
C
AC
BC
ABC
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
A contrast = [ab + a + ac + abc – (1) – b – c - bc
B contrast = ?
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
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
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
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)
The design matrix and response data obtained from single
replicate of the 24 experiment
Treatment
Combination
(1)
a
b
ab
c
ac
bc
abc
d
ad
bd
abd
cd
acd
bcd
abcd
Factors
A
B
C
D
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
-
+
+
+
+
+
+
+
+
Response
48
43
70
80
68
71
60
45
96
86
65
100
75
65
45
104
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
The main effect plot for Temperature
The plot indicate that its better to run the Temperature at high levels
The main effect plot for Concentration of
Formaldehyde
The plot indicate that its better to run the concentration of formaldehyde
at high levels
The main effect plot for Stirring rate
The plot indicate that its better to run the stirring rate at
high levels
However, its necessary to examine any
interactions that are important
The best results are obtained with low concentration of formaldehyde
and high temperatures
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
Factor effect Estimates and sums of squares for the 24 Design
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
Model Term
ANOVA for A, C and D
Source
A
C
D
A*C
A*D
C*D
A*C*D
Residual Error
Total
DF
Seq SS
Adj SS
AdjMS
F
P
1
1
1
1
1
1
1
8
15
1870.56
390.06
855.56
1314.06
1105.56
5.06
10.56
179.52
5730.94
1870.56
390.06
855.56
1314.06
1105.56
5.06
10.56
22.44
1870.56
390.06
855.56
1314.06
1105.56
5.06
10.56
83.36
17.38
38.13
58.56
49.27
<1
<1
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
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
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
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
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
Consider table of plus and minus signs for the 22 design
Treatment
Combination
(1)
a
b
ab
Factorial Effect
I
A
B
AB
Block
+
+
+
+
+
+
+
+
+
+
1
2
2
1
Block 1 Block 2
(1)
ab
a
b
The order in which the treatment
combination are run within a block
are randomly Determined
The block effect and the AB interaction are identical. That is, AB is
confounded with blocks.
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
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
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
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
Consider our previous example;
Two modification;
1. 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
The two resulting blocks are;
(1)
ab
ac
bc
ad
bd
cd
abcd
a
b
c
d
abc
bcd
acd
abd
The half Normal plot for the blocked design
Source
DF
Seq SS
Adj SS
Adj MS
F
Blocks
A
C
D
A*C
A*D
Residual Error
Total
1
1
1
1
1
1
9
15
1387.56
1870.56
390.06
855.56
1314.06
1105.56
187.56
7110.94
1387.56 1387.56
1870.56 1870.56 89.76
390.06
390.06
18.72
855.56
855.56
41.05
1314.06 1314.06 63.05
1105.56 1105.56 53.05
187.56
20.8403
P-Value
<0.0001
<0.0001
<0.0001
<0.0001
<0.0001
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 don’t
block, then the added variability from the nuisance
variable effect ends up getting distributed across the
other design factors
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
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
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
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
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
B*I = B * ABC = AB2C = AC
B = AC
This half fraction with I = ABC is called the
Principal fraction
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
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
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
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
+
+
+
+
+
+
+
+
+
+
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
The resolution IV design with I = ABCD
Basic Design
Run
A
B
C
1
2
3
4
5
6
7
8
+
+
+
+
+
+
+
+
+
+
+
+
Treatment
D = ABC Combination
(1)
ad
+
bd
+
ab
cd
+
ac
bc
abcd
+
Estimates of Effects
Term
A
B
C
D
A*B
A*C
A*D
Effect
19.000
1.500
14.000
16.500
-1.000
-18.500
19.000
A ,C and D have large effects, and so is the interactions
involving them
Thank you
Reference
Montgomery, D.C (2013). Design and analysis of experiments. Wiley, New York.