MATH408: Probability

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MATH408 Probability StatisticsSummer
1999WEEKS 10 11
Dr. Srinivas R. Chakravarthy Professor of
Mathematics and Statistics Kettering
University (GMI Engineering Management
Institute) Flint, MI 48504-4898 Phone
810.762.7906 Email schakrav_at_kettering.edu Homepag
e www.kettering.edu/schakrav
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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.

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DESIGN OF EXPERIMENTS (contd)

• One of the important tools of statistics that has
  been widely used in evaluating the quality of a
  product, 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.

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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.
• A good design allows for estimation and
  interpretation of these interactions.

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DESIGNED EXPERIMENT (contd)

• 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 optimum 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 PET.
• Both parts are end gated and we also know where
  the areas of failure during a physical testing
  for these two parts.

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STARTING POINT OF DOE (contd)

• 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 DOEs in practice start with such a
  statement.

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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)
  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 DOE

• 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.
• 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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Concepts in DOE (contd)

• 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 factors in the experiment.
• Test run Single combination of factor levels
  that yields an observation on the response.

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SELECTION OF VARIABLES AND FACTORS

• Usually there will be only one response variable
  and the objective of the 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) listing all the factors.

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ILLUSTRATIVE EXAMPLE

• A new brand of printing paper is being considered
  by a leading photographic company.
• The study will be focusing on 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
  developing tray until an image appears.

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EXAMPLE (contd)

• 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 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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DOE STRUCTURE

• 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

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DOE STRUCTURE (contd)

• For example if a teacher wishes to compare the
  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
  minimize this source of bias, randomization
  technique should be adopted in all experimental
  work.

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REPLICATION (Sir R. A. Fisher)

• 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 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.

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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 Taguchis
  orthogonal designs)

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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.

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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

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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.

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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.
• This procedure is called BLOCKING.
• Since the development of RBD in 1925 this design
  has become very popular among all designs.

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RBD (contd)

• 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 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.

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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.

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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.

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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.

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FACTORIAL DESIGNS
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FACTORIAL DESIGNS

• 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
• If n1, we have no 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.

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