Post on 18-Oct-2021
LECTURE 8(Winter'99)1
MATH602: APPLIED STATISTICS
Dr. Srinivas R. ChakravarthyDepartment of Science and Mathematics
KETTERING UNIVERSITYFlint, MI 48504-4898
Lecture 8 Winter 1999
LECTURE 8(Winter'99) 2
LECTURE 8
DESIGN OF EXPERIMENTS
-Earlier we talked about the quality of a product and
how statistics is used to continuously to improve the
quality of a product. We saw a number of statistical
methods to analyze the data and make interpretations.
-One of the important tools of statistics that has been
widely used in evaluating the quality of a product,
LECTURE 8(Winter'99) 3
identifying the sources that affect the quality, setting up
the values of the parameters that will optimize the
response variable, is the Design of Experiments.
-Designing an experiment is like designing a product.
-The purpose should be clearly defined to begin with.
-The experiment should be set up to answer a specific
question or a set of questions.
LECTURE 8(Winter'99) 4
WHAT IS A DESIGNED EXPERIMENT?
- Enables us to observe the behavior of a particular
aspect of reality.
-Experimental design is an organized approach to the
collection of information.
-In most practical problems, many variables influence
the outcome of an experiment.
-Usually these interact in very complex ways.
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-A good design allows for estimation and interpretation
of these interactions.
-An experimenter chooses certain factors and in a
controlled environment varies these factors so as to
observe the effects.
-No statistical tool can come to rescue data obtained
from designs conducted haphazardly.
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OBJECTIVES
- Maximize the amount of information
- Identify factors that (a) affect the average response; (b) affect
the variability; (c) do not contribute significantly.
- Identify the mathematical model relating the response to the
factors
- Identify Aoptimum@ settings for the factors
- CONFIRM the settings
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STARTING POINT OF DOE: Consider the
following scenario: A process engineer in the
manufacture of reinforced pet moldings using injection-
molding process asks the following question: We are
manufacturing two different parts using two-cavity
injection molds. One part, the shaft, is molded in a 55%
glass fiber reinforced PET polyester, while the other
part, the tube is produced from a 45% fiber reinforced
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PET. Both parts are end gated and we also know where
the areas of failure during a physical testing for these
two parts. We want to find the optimum molding
process. That is, what should be the levels of the
factors: melt temperature, mold pressure, hold time,
injection speed, and hold pressure that will optimize the
strength of the reinforced pet moldings?
-Almost all DOE’s in practice start with such a
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statement.
MAJOR STEPS IN DOE: Design of experiment
(DOE) is an iterative decision-making process. Like
any area of applied science, the steps involved in DOE
can be grouped into three stages: analysis, synthesis,
and evaluation. These phases are characterized as:
Analysis: (a) Recognition of the problem; (b)
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formulating the experimental problem; (c) analysis of
the experiment.
Synthesis: (a) Designing the experimental model; (b)
designing the analytical model.
Evaluation: (a) Conducting the experiment; (b)
Deriving solution(s) from the model; (c) Make
appropriate conclusions and recommendations.
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Basic concepts in Design of Experiments: Factor,
level, treatment, effect, response, test run, interaction,
blocking, confounding, experimental unit, replication,
randomization, and covariate. Some of these were seen
in our lecture on ANOVA.
Block: A factor that has influence on the variability of
the response variable.
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Randomization: This refers to assigning the
experimental units randomly to treatments.
Replication: This refers to the repetition of an
experiment. This should be practiced in all
experimental work in order to increase the precision.
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Block: A group of homogeneous experimental units.
Confounding: When one or more effects that cannot
be unambiguously be attributed to a single factor or
interaction.
Covariate: An uncontrollable variable that influences
the response but is unaffected by any other
experimental factors. Covariates are not additional
responses and hence their values are not affected by the
LECTURE 8(Winter'99) 14
factors in the experiment.
Test run: Single combination of factor levels that
yields an observation on the response.
SELECTION OF RESPONSE (or dependent)
VARIABLES AND FACTORS: Usually there will be
only one response variable and the objective of the
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experiment will indicate the response variable. The
response variable can be qualitative or quantitative.
-The selection of factors is a critical one and involves
a detailed plan. At first all possible factors, irrespective
whether they are practical to be measured or not,
should be included in the experiment.
-A common approach is to use a cause-and-effect
diagram (refer to Lecture 1 notes for details on this)
LECTURE 8(Winter'99) 16
listing all the factors.
-To better understand these concepts, let us look at
some illustrative examples.
ILLUSTRATIVE EXAMPLE 1: A new brand of
printing paper is being considered by a leading
photographic company. The study will be focusing on
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the effects of various factors on the development time.
So, the response variable for this is the development
time. The experiment will consists of the following
steps: (i) a test negative will be placed on the glass top
of a contact printer; (ii) a sample of printing paper will
be placed on top of the negative; (iii) the light on the
contact printer will be turned on for a specific amount
of time; and (iv) the printing paper will be placed on a
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developing tray until an image appears. The following
factors are considered to play a role: (1) exposure time;
(2) density of test negative; (3) temperature of the
laboratory where the developing is done; (4) intensity
of exposing light; (5) types of developer; (6) amount of
developer; (7) grade of printing paper; (8) condition of
printing paper; (9) voltage fluctuations during the
experiment; (10) humidity; (11) number of times the
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developer will be used; (12) size of printing paper; and
(13) operator. After careful study, the company decided
to use three factors: exposure time, type of developer,
and grade of printing paper in the experiment and the
remaining factors are either controlled or made as
experimental error.
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ILLUSTRATIVE EXAMPLE 2: An experiment is
conducted to study the flow of suspended particles in
two types of coolants used in industrial equipment. The
coolants are to be forced through a slit aperture in the
middle of a fixed length pipe. This experiment is
conducted with three different flow rates and four
different angles of inclination at which the pipe will be
kept. The study will focus on the buildup of the
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particles in the coolant on the edge of the aperture.
For this experiment, the response variable is the flow
rate of suspended particles; the factors are: coolant, a
qualitative factor at three levels (1 and 2); pipe angle,
a quantitative factor at four levels (15, 25, 45, 60
degrees from horizontal position); and flow rate, a
quantitative factor at three levels (60, 90 and 120
ft/sec). All 24 combinations of the factor levels are to
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be included in the experiment. All the test runs are to
be conducted on a particular so as to eliminate day-to-
day variation in the response variable. The sequence of
tests was determined randomly to minimize any bias in
the experimentation. Since the temperature will vary
from early morning to late evening (during the time of
the test runs), this may affect the test results and so the
temperature is taken to be a covariate.
LECTURE 8(Winter'99) 23
ILLUSTRATIVE EXAMPLE 3: The engineering
application of a particular printed circuit board requires
that the variability of the thickness of solder coating on
these boards be as small as possible. In order to
determine what factors may cause the variability in the
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thickness an experiment was proposed. The response
variable is solder coating thickness and the following
factors were identified as key ones: (1) tool type-
measuring instrument in measuring the solder
thickness; (2) inspectors-persons performing the
inspection of the boards; (3) position of measurment-
past experience has shown that error measurement
tends to be larger at position close to the component
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part; and (4) boards.
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The design of experiments refers to the structure of the
experiment with reference to
- the set of treatments included
- the set of experimental units
- the rules by which the treatments are assigned to the
units
- the measurements taken
-For example if a teacher wishes to compare the
LECTURE 8(Winter'99) 27
relative merits of four teaching aids: text book only,
text book and class notes, text book and lab manual,
text book, lab manual and class notes.
-Treatments: four teaching aids
-Experimental units: participating students (or classes)
Rules: Once the treatments and the experimental units
are selected the rules are required for assigning the
treatments to the experimental units.
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RANDOMIZATION: (Sir R. A. Fisher) assigning the
units randomly to treatments. This tends to eliminate
the influence of external factors (or noise factors) not
under the direct control of the experimenter; avoid any
selection bias. Also the variation from these noise
factors can bias the estimated effects. Hence in order to
LECTURE 8(Winter'99) 29
minimize this source of bias, randomization technique
should be adopted in all experimental work.
REPLICATION: repetition of an experiment.
For example if we have 3 treatments and 6 units, the
assignment of 3 units at random to the 3 treatments
constitute one replication and the assignment of the
remaining 3 units to the 3 treatments constitute another
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replication of the experiment.
-Replication should be practiced in all DOE work.
-Also replication is used to assess the error mean square
as well as to increase the precision.
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SOME COMMON PROBLEMS IN DOE
(a) experimental variation hides true factor effects;
(b) uncontrolled factors compromise experimental
conclusions;
(c) one-factor-at-a-time designs will not give a true
picture of many-factor experiments.
LECTURE 8(Winter'99) 32
Experimental variation hides true factor effects
CASE 1:
CASE 2:
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Uncontrolled factors compromise experimental conclusions
Suppose that methods (such as weight-loss, groove-
depth, etc) of determining wear and tear of tires are
under study. In order to find a relationship between
various methods (for calibration purposes), one has to
be aware of large variation associated with the
"uncontrolled" factors such as road conditions,
vehicles, drivers, and weather.
LECTURE 8(Winter'99) 34
One-factor-at-a-time
While it looks that this approach requires very minimum
number of experimental runs, the following example will
illustrate how one can be way off from the optimum.
Suppose that two factors: A (temperature) and B (time) are
under study to look at the effect of yield in a chemical
experiment.
LECTURE 8(Winter'99) 35
ILLUSTRATIVE EXAMPLE
(ONE-FACTOR-AT-A-TIME)
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COMMONLY USED DESIGNS
- Completely Randomized Designs (CRD)
- Randomized Block Designs (RBD)
- Latin Square Designs (LSD)
- 2n Factorial Designs.
- Fractional Factorial Designs (including Taguchi=s
orthogonal designs)
LECTURE 8(Winter'99) 37
Completely Randomized Design (CRD)
-This is the basic design.
-All other randomized designs stem from it by imposing
restrictions upon the allocation of the treatments to the units.
- The units are assigned to treatments at random.
-Thus every unit chosen for the study has an equal chance of
being assigned to any treatment.
-This is useful when the units are homogeneous.
-Most useful in laboratory techniques.
LECTURE 8(Winter'99) 38
Advantages and Disadvantages of a CRD
(1) it is felxible
(2) its MSE has a larger degrees of freedom
(3) it allows for missing observations
(4) it has fewer assumptions
-Heterogeneous; # of treatments is large.
LECTURE 8(Winter'99) 39
ANALYSIS OF A CRD
-The analysis of single-factor studies that we discussed
in ANOVA is applicable and there is no need to repeat
the analysis here.
LECTURE 8(Winter'99) 40
Randomized Block Design (RBD)
-When experimental units are heterogeneous to reduce
experimental error variability we need to sort the units
into homogeneous groups called blocks.
-The treatments are then randomly assigned within
blocks.
-That is, randomization is restricted.
LECTURE 8(Winter'99) 41
-This procedure is called BLOCKING.
-Since the development of RBD in 1925 this design has
become very popular among all designs.
-As an example of this design, suppose that a company
is considering buying one of 5 word processors for use
in its offices. In order to study the average time for its
employees to learn the word processors, if all have the
same ability we could use a CRD. However this will be
LECTURE 8(Winter'99) 42
the case. We can sort the employees into blocks of 5
and assign randomly the 5 word processors for
learning. If we had used a CRD any effect that should
have been attributed to blocks would end up in the error
term. By blocking we remove a source of variation
from the error term.
LECTURE 8(Winter'99) 43
Advantages and Disadvantages of a RBD
(1) provides precise results with proper blocking
(2) No need to have equal sample sizes
(3) the analysis is simple
(4) one can bring in more variability among the
experimental units, which usually is the case in practice.
(1) missing observations; (2) DF are not as large as with a
CRD; (3) Need more assumptions.
LECTURE 8(Winter'99) 44
ANALYSIS OF A RBD
The analysis of multi-factor studies that we discussed in
ANOVA is applicable and there is no need to repeat the
analysis here.
LECTURE 8(Winter'99) 45
Latin Square Design (LSD)
-RBD is one of many block designs. In it one source of
variation is blocked.
-If there are two sources of variations that need to be
blocked we need to use different design called the Latin
Square design.
-The treatments are grouped into two blocks, once in
rows and once in columns.
LECTURE 8(Winter'99) 46
-Through this, the error variance is reduced.
-This design has a wide variety of applications in industrial,
field, laboratory, greenhouse, educational, marketing,
medicine, and sociology.
-According to Fisher, if experimentation were only
concerned with the comparison of four to eight treatments
or varieties, LSD would therefore be not merely the
principal but almost the universal design employed.
LECTURE 8(Winter'99) 47
Example of a LSD
-Testing 3 electronic components (A-C) for its
fascimile telephone to determine whether or not the
average transmission speed of a page was
approximately the same for the 3 components.
-Three different kinds of pages: text only, picture only
and text and picture
LECTURE 8(Winter'99) 48
-Three kinds of transmitting were used.
-The design that blocks using two variables (page type
and device type) and tests for differences among the
same number of treatments (electronic components) is
a 3x3 LSD.
-The rows of the square stand for the page type and the
columns stand for the device type. The Latin square is:
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DEVICEPAGE
1 2 3
Text B A C
Picture A C B
Text & Picture C B A
LECTURE 8(Winter'99) 50
-Thus, a LSD has r treatments; two blocking variables, each
having r levels and each row and each column in the design
square contains all treatments.
Advantages and Disadvantages of a LSD:
(1) reduces the variability of experimental error.
(2) effects studied from a small-scale experiment.
# of levels for each blocking variable must equal the
number of treatments and the assumptions of the model are
restrictive.
LECTURE 8(Winter'99) 51
STATISTICAL MODEL
.εγβτµ ijkkjiijk + + + + = y
yijk = obs. of the j-th row and k-th column for the i-th
treatment effect.
ANOVA:
STOT = SSTR + SSROW + SSCOL + SSE
These are calculated as follows:
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2...
1 1 1
2 YNYSSTr
i
r
j
r
kijk −= ∑∑∑
= = =
2...
1
2.. YN
r
TSS
r
i
iTr −= ∑
=
2...
1
2.. YN
r
TSS
r
j
jRow −= ∑
=
2...
1
2.. YNr
TSS
r
k
kCol −= ∑
=
LECTURE 8(Winter'99) 53
EXAMPLE 1: Suppose that four cars and four drivers
are used in the study of possible differences between
four gasoline additives in reducing the oxides of
nitrogen in the auto emissions. Because of the
possibility of systematic differences among the cars and
among the drivers, it was decided to use LSD. Table 1
gives the (coded) results for the reduction in the oxides
of nitrogen.
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CarDriver
1 2 3 4
I A
21
B
26
D
20
C
25
II D
23
C
26
A
20
B
27
III B
15
D
13
C
16
A
16
IV C
17
A
15
B
20
D
20
LECTURE 8(Winter'99) 55
FACTORIAL DESIGNS• In many experiments the success or failure may depend
more on the selection of treatments for comparisons to bemade than on the design itself.
• Thus an experimenter should be careful not only in theselection of the design but also on the treatments.
• Factorial experiments are very commonly used inmanufacturing and engineering experiments.
• Here several factors at two or more levels, are controlledto carefully measure the effects.
• The effects in a factorial experiment are composed of maineffects and interactions.
LECTURE 8(Winter'99) 56
• All the units are used to evaluate main effects andinteractions.
• The main effect of a factor is composed of a set ofsingle-degree of freedom contrasts among the totalnumber of levels of that factor.
• The interaction of two factors is the failure of the levels ofone factor to retain the same order and magnitude ofperformance throughout all levels of the second factor.
• Full factorial designs are used to assess all possiblecombinations of the factor levels under study.
• For example, a full factorial experiment consisting of 8two-level factors require 256 trials.
LECTURE 8(Winter'99) 57
• This will have information not only on the main effects ofall 8 factors, but also on the interaction of the factorsincluding whether all 8 factors work in conjunction toaffect the response variable.
• Usually in practice, one will be interested in the maineffects and the interaction of 2 factors at a time, as thehigher order interaction will be negligibly small.
LECTURE 8(Winter'99) 58
Advantages and Disadvantages of a Factorial Design:The advantages of a factorial design are:(1) all experimental units are utilized in evaluating the main
effects and interactions(2) the effects are evaluated over a wide range of conditions
with the minimum resources(3) a factorial set of treatments is optimum for estimating the
main effects and interactionsThe main disadvantage is the large number of combinationsneeded to study the effects.However, an alternative approach to use fractional factorialdesigns. We will see this later.
LECTURE 8(Winter'99) 59
• For a general factorial design an experimenter selects afixed number of levels for each of the factors that areunder study and then runs experiments with all possiblecombinations.
• For example, if there are 3 factors with levels 3, 4, and 6,then the experiment requires a minimum of 3x4x6 = 72runs. This design is called a 3 x 4 x 6 factorial design.
• A factorial design with k factors, all of which have only 2levels is a 2k factorial design.
• 2k factorial designs play an important role for a number ofreasons.
LECTURE 8(Winter'99) 60
• First of all they require only few runs per factor studied.These can be used to see whether there are any majortrends and to determine the direction for further study.
• When a more thorough study is needed they could besupplemented to form composite designs. Also where weneed to determine whether there are any interactionspresent among the factors we could quickly use thisdesign.
• When the factors to be studied are large, because of thesize of the design we could either have only onereplication for the experiment or obtain most of theinformation by looking only at a fraction of the design.
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- Effect of a factor: Change in response produced by a
change in the level of that factor averaged over the levels
of the other factor(s).
- Magnitude and direction of factor effects are to be
examined to see which are likely to be important.
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INTERACTION
- Exists if the difference in response between the levels of
one factor is not the same at all levels of the other
factor(s).
- Calculated as the average difference between the effect of
A at high level of B and the effect of A at the low level of
B.
LECTURE 8(Winter'99) 63
In 2k design:
• All factor effects will have 1 d.f
• If there are n replicates, SSE will have (n-1)2k d.f.
• Replicates are very important in testing for lack of fit [
Recall this from Regression Analysis]
• If n=1, we have estimate for error [Why?]
• Use higher order interactions to get an estimate.
- Plot the estimates on a normal probability paper. All
effects that are insignificant will fall on a line.
LECTURE 8(Winter'99) 64
22 FACTORIAL DESIGNS
• Two factors, say, A and B, at two levels.
• A full factorial design will consist of 4 runs.
• We can estimate two main effects and one two-factor
effect from these 4 trials.
• The total sum of squares will be split as: SSA, SSB and
SSAB.
LECTURE 8(Winter'99) 65
The Regression Model: It is easy to see how the effect
estimates in a 2k factorial design into a regression model, which
can be used to predict the response for the factor space. For
example, a 22 design to estimate linear effects is transformed to
a multiple linear regression model as:
Y = b0 + b1 X1 + b2 X2 + e,
where X1 and X2 are coded variables.
2/)(
2/)(
LowHigh
LowHigh
AA
AAAX
−+−
=
LECTURE 8(Winter'99) 66
The fitted model is:
22110ˆˆˆˆ XXY βββ ++=
which in our case reduces to
21 22ˆ XXYY BA ∆
+∆
+=
[Why?]
LECTURE 8(Winter'99) 67
n
b
n
aabYY
AAA 2
)1(
2
+−
+=−=∆ −+
n
a
n
babYY
BBB 2
)1(
2
+−
+=−=∆ −+
n
ba
n
abYY
ABABAB 22
)1( +−
+=−=∆ −+
LECTURE 8(Winter'99) 68
Note: The numerator terms in the above average effects are
refered as CONTRASTS. A contrast is a linear combination
of the parameters:
iicL µ∑=
.0such that =∑ ic
In general, SS (due to a contrast) = (contrast)2/ n 2k.
LECTURE 8(Winter'99) 69
EXAMPLE 2: A study was conducted to determine the effectof mixing time and the temperature on the yellowness index ofa paint to be used on commercial products. Let y = yellownessindex, x1 = mixing temperature, 50oF and 70oF; x1 = mixingtime, 0.5 and 2 hours. The study yielded the following data.
Obs. No. A B Index
1 50 0.5 6.0
2 70 0.5 8.03 50 2.0 7.5
4 70 2.0 8.5
LECTURE 8(Winter'99) 70
23 FACTORIAL DESIGNS
-Three factors, say, A, B and C, at two levels.
-A full factorial design will consist of 8 runs.
-We can estimate three main effects, three two-factor
effects and one three-factor effect from these 8 trials.
-The total sum of squares will be split similar to 22
design.
LECTURE 8(Winter'99) 71
EXAMPLE 3: Suppose that a chemist is interested in
the yield of a chemical process that has three factors:
temperature (A), concentration (B) and catalyst (C).
The first two variables are quantitative while the
variable catalyst is qualitative. The data for these
variables and the yield are given below. Note that there
are 2 replications.
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Yield(in gms)
Obs.No.
A B C
1 2 TOTAL
1 160 20 I 60 58
2 180 20 I 72 68
3 160 40 I 54 64
4 180 40 I 68 62
5 160 20 II 52 45
6 180 20 II 83 76
7 160 40 II 45 53
LECTURE 8(Winter'99) 73
8 180 40 II 80 76
USE OF MINITAB IN 2k DESIGNS