DESIGN OF EXPERIMENTS by R. C. Baker

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DESIGN OF EXPERIMENTSbyR. C. Baker

• How to gain 20 years of experience in one short
  week!

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Role of DOE in Process Improvement

• DOE is a formal mathematical method for
  systematically planning and conducting scientific
  studies that change experimental variables
  together in order to determine their effect of a
  given response.
• DOE makes controlled changes to input variables
  in order to gain maximum amounts of information
  on cause and effect relationships with a minimum
  sample size.

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Role of DOE in Process Improvement

• DOE is more efficient that a standard approach of
  changing one variable at a time in order to
  observe the variables impact on a given
  response.
• DOE generates information on the effect various
  factors have on a response variable and in some
  cases may be able to determine optimal settings
  for those factors.

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Role of DOE in Process Improvement

• DOE encourages brainstorming activities
  associated with discussing key factors that may
  affect a given response and allows the
  experimenter to identify the key factors for
  future studies.
• DOE is readily supported by numerous statistical
  software packages available on the market.

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BASIC STEPS IN DOE

• Four elements associated with DOE
• 1. The design of the experiment,
• 2. The collection of the data,
• 3. The statistical analysis of the data, and
• 4. The conclusions reached and recommendations
  made as a result of the experiment.

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TERMINOLOGY

• Replication repetition of a basic experiment
  without changing any factor settings, allows the
  experimenter to estimate the experimental error
  (noise) in the system used to determine whether
  observed differences in the data are real or
  just noise, allows the experimenter to obtain
  more statistical power (ability to identify small
  effects)

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TERMINOLOGY

• .Randomization a statistical tool used to
  minimize potential uncontrollable biases in the
  experiment by randomly assigning material,
  people, order that experimental trials are
  conducted, or any other factor not under the
  control of the experimenter. Results in
  averaging out the effects of the extraneous
  factors that may be present in order to minimize
  the risk of these factors affecting the
  experimental results.

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TERMINOLOGY

• Blocking technique used to increase the
  precision of an experiment by breaking the
  experiment into homogeneous segments (blocks) in
  order to control any potential block to block
  variability (multiple lots of raw material,
  several shifts, several machines, several
  inspectors). Any effects on the experimental
  results as a result of the blocking factor will
  be identified and minimized.

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TERMINOLOGY

• Confounding - A concept that basically means that
  multiple effects are tied together into one
  parent effect and cannot be separated. For
  example,
• 1. Two people flipping two different coins would
  result in the effect of the person and the effect
  of the coin to be confounded
• 2. As experiments get large, higher order
  interactions (discussed later) are confounded
  with lower order interactions or main effect.

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TERMINOLOGY

• Factors experimental factors or independent
  variables (continuous or discrete) an
  investigator manipulates to capture any changes
  in the output of the process. Other factors of
  concern are those that are uncontrollable and
  those which are controllable but held constant
  during the experimental runs.

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TERMINOLOGY

• Responses dependent variable measured to
  describe the output of the process.
• Treatment Combinations (run) experimental trial
  where all factors are set at a specified level.

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TERMINOLOGY

• Fixed Effects Model - If the treatment levels
  are specifically chosen by the experimenter, then
  conclusions reached will only apply to those
  levels.
• Random Effects Model If the treatment levels
  are randomly chosen from a population of many
  possible treatment levels, then conclusions
  reached can be extended to all treatment levels
  in the population.

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PLANNING A DOE

• Everyone involved in the experiment should have a
  clear idea in advance of exactly what is to be
  studied, the objectives of the experiment, the
  questions one hopes to answer and the results
  anticipated

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PLANNING A DOE

• Select a response/dependent variable (variables)
  that will provide information about the problem
  under study and the proposed measurement method
  for this response variable, including an
  understanding of the measurement system
  variability

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PLANNING A DOE

• Select the independent variables/factors
  (quantitative or qualitative) to be investigated
  in the experiment, the number of levels for each
  factor, and the levels of each factor chosen
  either specifically (fixed effects model) or
  randomly (random effects model).

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PLANNING A DOE

• Choose an appropriate experimental design
  (relatively simple design and analysis methods
  are almost always best) that will allow your
  experimental questions to be answered once the
  data is collected and analyzed, keeping in mind
  tradeoffs between statistical power and economic
  efficiency. At this point in time it is
  generally useful to simulate the study by
  generating and analyzing artificial data to
  insure that experimental questions can be
  answered as a result of conducting your
  experiment

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PLANNING A DOE

• Perform the experiment (collect data) paying
  particular attention such things as randomization
  and measurement system accuracy, while
  maintaining as uniform an experimental
  environment as possible. How the data are to be
  collected is a critical stage in DOE

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PLANNING A DOE

• Analyze the data using the appropriate
  statistical model insuring that attention is paid
  to checking the model accuracy by validating
  underlying assumptions associated with the model.
  Be liberal in the utilization of all tools,
  including graphical techniques, available in the
  statistical software package to insure that a
  maximum amount of information is generated

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PLANNING A DOE

• Based on the results of the analysis, draw
  conclusions/inferences about the results,
  interpret the physical meaning of these results,
  determine the practical significance of the
  findings, and make recommendations for a course
  of action including further experiments

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SIMPLE COMPARATIVE EXPERIMENTS

• Single Mean Hypothesis Test
• Difference in Means Hypothesis Test with Equal
  Variances
• Difference in Means Hypothesis Test with Unequal
  Variances
• Difference in Variances Hypothesis Test
• Paired Difference in Mean Hypothesis Test
• One Way Analysis of Variance

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CRITICAL ISSUES ASSOCIATED WITH SIMPLE
COMPARATIVE EXPERIMENTS

• How Large a Sample Should We Take?
• Why Does the Sample Size Matter Anyway?
• What Kind of Protection Do We Have Associated
  with Rejecting Good Stuff?
• What Kind of Protection Do We Have Associated
  with Accepting Bad Stuff?

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Single Mean Hypothesis Test

• After a production run of 12 oz. bottles, concern
  is expressed about the possibility that the
  average fill is too low.
• Ho m 12
• Ha m ltgt 12
• level of significance a .05
• sample size 9
• SPEC FOR THE MEAN 12 .1

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Single Mean Hypothesis Test

• Sample mean 11.9
• Sample standard deviation 0.15
• Sample size 9
• Computed t statistic -2.0
• P-Value 0.0805162
• CONCLUSION Since P-Value gt .05, you fail to
  reject hypothesis and ship product.

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Single Mean Hypothesis Test Power Curve
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Single Mean Hypothesis Test Power Curve
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Single Mean Hypothesis Test Power Curve -
Different Sample Sizes
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DIFFERENCE IN MEANS - EQUAL VARIANCES

• Ho m1 m2
• Ha m1 ltgt m2
• level of significance a .05
• sample sizes both 15
• Assumption s1 s2
  
• Sample means 11.8 and 12.1
• Sample standard deviations 0.1 and 0.2
• Sample sizes 15 and 15

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DIFFERENCE IN MEANS - EQUAL VARIANCES Can you
detect this difference?
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DIFFERENCE IN MEANS - EQUAL VARIANCES
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DIFFERENCE IN MEANS - unEQUAL VARIANCES

• Same as the Equal Variance case except the
  variances are not assumed equal.
• How do you know if it is reasonable to assume
  that variances are equal OR unequal?

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DIFFERENCE IN VARIANCE HYPOTHESIS TEST

• Same example as Difference in Mean
• Sample standard deviations 0.1 and 0.2
• Sample sizes 15 and 15
• Null Hypothesis ratio of variances 1.0
• Alternative not equal
• Computed F statistic 0.25
• P-Value 0.0140071
• Reject the null hypothesis for alpha 0.05.

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DIFFERENCE IN VARIANCE HYPOTHESIS TEST Can you
detect this difference?
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DIFFERENCE IN VARIANCE HYPOTHESIS TEST -POWER
CURVE
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PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST

• Two different inspectors each measure 10 parts on
  the same piece of test equipment.
• Null hypothesis DIFFERENCE IN MEANS 0.0
• Alternative not equal
• Computed t statistic -1.22702
• P-Value 0.250944
• Do not reject the null hypothesis for alpha
  0.05.

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PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST -
POWER CURVE
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ONE WAY ANALYSIS OF VARIANCE

• Used to test hypothesis that the means of several
  populations are equal.
• Example Production line has 7 fill needles and
  you wish to assess whether or not the average
  fill is the same for all 7 needles.
• Experiment sample 20 fills from each of the 9
  needles and test at 5 level of sign.
• Ho m1 m2 m3 m4 m5 m6 m7

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RESULTS ANALYSIS OF VARIANCE TABLE
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SINCE NEEDLE MEANS ARE NOT ALL EQUAL, WHICH ONES
ARE DIFFERENT?

• Multiple Range Tests for 7 Needles

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VISUAL COMPARISON OF 7 NEEDLES
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FACTORIAL (2k) DESIGNS

• Experiments involving several factors ( k of
  factors) where it is necessary to study the joint
  effect of these factors on a specific response.
• Each of the factors are set at two levels (a
  low level and a high level) which may be
  qualitative (machine A/machine B, fan on/fan off)
  or quantitative (temperature 800/temperature 900,
  line speed 4000 per hour/line speed 5000 per
  hour).

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FACTORIAL (2k) DESIGNS

• Factors are assumed to be fixed (fixed effects
  model)
• Designs are completely randomized (experimental
  trials are run in a random order, etc.)
• The usual normality assumptions are satisfied.

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FACTORIAL (2k) DESIGNS

• Particularly useful in the early stages of
  experimental work when you are likely to have
  many factors being investigated and you want to
  minimize the number of treatment combinations
  (sample size) but, at the same time, study all k
  factors in a complete factorial arrangement (the
  experiment collects data at all possible
  combinations of factor levels).

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FACTORIAL (2k) DESIGNS

• As k gets large, the sample size will increase
  exponentially. If experiment is replicated, the
  runs again increases.

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FACTORIAL (2k) DESIGNS (k 2)

• Two factors set at two levels (normally referred
  to as low and high) would result in the following
  design where each level of factor A is paired
  with each level of factor B.

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FACTORIAL (2k) DESIGNS (k 2)

• Estimating main effects associated with changing
  the level of each factor from low to high. This
  is the estimated effect on the response variable
  associated with changing factor A or B from their
  low to high values.

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FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

• Neither factor A nor Factor B have an effect on
  the response variable.

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FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

• Factor A has an effect on the response variable,
  but Factor B does not.

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FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

• Factor A and Factor B have an effect on the
  response variable.

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FACTORIAL (2k) DESIGNS (k 2) GRAPHICAL OUTPUT

• Factor B has an effect on the response variable,
  but only if factor A is set at the High level.
  This is called interaction and it basically means
  that the effect one factor has on a response is
  dependent on the level you set other factors at.
  Interactions can be major problems in a DOE if
  you fail to account for the interaction when
  designing your experiment.

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EXAMPLEFACTORIAL (2k) DESIGNS (k 2)

• A microbiologist is interested in the effect of
  two different culture mediums medium 1 (low) and
  medium 2 (high) and two different times 10
  hours (low) and 20 hours (high) on the growth
  rate of a particular CFU.

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EXAMPLEFACTORIAL (2k) DESIGNS (k 2)

• Since two factors are of interest, k 2, and we
  would need the following four runs resulting in

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EXAMPLEFACTORIAL (2k) DESIGNS (k 2)

• Estimates for the medium and time effects are
• Medium effect (1539)/2 (17 38)/2
  -0.5
• Time effect (3839)/2 (17 15)/2 22.5

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EXAMPLEFACTORIAL (2k) DESIGNS (k 2)
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EXAMPLEFACTORIAL (2k) DESIGNS (k 2)

• A statistical analysis using the appropriate
  statistical model would result in the following
  information. Factor A (medium) and Factor B
  (time)

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EXAMPLECONCLUSIONS

• In statistical language, one would conclude that
  factor A (medium) is not statistically
  significant at a 5 level of significance since
  the p-value is greater than 5 (0.05), but factor
  B (time) is statistically significant at a 5
  level of significance since this p-value is less
  than 5.

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EXAMPLECONCLUSIONS

• In layman terms, this means that we have no
  evidence that would allow us to conclude that the
  medium used has an effect on the growth rate,
  although it may well have an effect (our
  conclusion was incorrect).

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EXAMPLECONCLUSIONS

• Additionally, we have evidence that would allow
  us to conclude that time does have an effect on
  the growth rate, although it may well not have an
  effect (our conclusion was incorrect).

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EXAMPLECONCLUSIONS

• In general we control the likelihood of reaching
  these incorrect conclusions by the selection of
  the level of significance for the test and the
  amount of data collected (sample size).

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2k DESIGNS (k gt 2)

• As the number of factors increase, the number of
  runs needed to complete a complete factorial
  experiment will increase dramatically. The
  following 2k design layout depict the number of
  runs needed for values of k from 2 to 5. For
  example, when k 5, it will take 32 experimental
  runs for the complete factorial experiment.

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2k DESIGNS (k gt 2)
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Interactions for 2k Designs (k 3)

• Interactions between various factors can be
  estimated for different designs above by
  multiplying the appropriate columns together and
  then subtracting the average response for the
  lows from the average response for the highs.

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Interactions for 2k Designs (k 3)
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2k DESIGNS (k gt 2)

• Once the effect for all factors and interactions
  are determined, you are able to develop a
  prediction model to estimate the response for
  specific values of the factors. In general, we
  will do this with statistical software, but for
  these designs, you can do it by hand calculations
  if you wish.

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2k DESIGNS (k gt 2)

• For example, if there are no significant
  interactions present, you can estimate a response
  by the following formula. (for quantitative
  factors only)

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ONE FACTOR EXAMPLE

• Simple one factor example where the factor is
  the number of hours a student studies for an exam
  (LOW 10 HRS, HIGH 20 HRS) and the response
  variable is their grade. Estimate the model for
  prediction a students grade as a function of the
  number of hours they study.

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ONE FACTOR EXAMPLE
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ONE FACTOR EXAMPLE

• The output shows the results of fitting a
  general linear model to describe the relationship
  between GRADE and HRS STUDY. The equation of
  the fitted general model is
• GRADE 29.3 3.1 (HRS STUDY)
• The fitted orthogonal model is
• GRADE 75 15 (SCALED HRS)

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Two Level Screening Designs

• Suppose that your brainstorming session resulted
  in 7 factors that various people think might
  have an effect on a response. A full factorial
  design would require 27 128 experimental runs
  without replication. The purpose of screening
  designs is to reduce (identify) the number of
  factors down to the major role players with a
  minimal number of experimental runs. One way to
  do this is to use the 23 full factorial design
  and use interaction columns for factors.

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Note that Any factor d effect is now
confounded with the ab interaction Any factor
e effect is now confounded with the ac
interaction etc. What is the de interaction
confounded with????????
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Problems that Interactions Cause!

• Interactions If interactions exist and you fail
  to account for this, you may reach erroneous
  conclusions. Suppose that you plan an experiment
  with four runs and three factors resulting in the
  following data

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Problems that Interactions Cause!

• Factor A Effect 0
• Factor B Effect 0
• Factor C Effect 5
• In this example, if you were assuming that
  larger is better then you would set Factor C at
  the high level and it appears to make no
  difference where you set factors A and B. In
  this case there is a factor A interaction with
  factor B and this interaction is confounded with
  the factor C effect.

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Problems that Interactions Cause!
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Resolution of a Design

• The above design would be called a resolution III
  design because main effects are aliased
  (confounded) with two factor interactions.

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Resolution of a Design

• Resolution III Designs No main effects are
  aliased with any other main effect BUT some (or
  all) main effects are aliased with two way
  interactions
• Resolution IV Designs No main effects are
  aliased with any other main effect OR two factor
  interaction, BUT two factor interactions may be
  aliased with other two factor interactions
• Resolution V Designs No main effect OR two
  factor interaction is aliased with any other main
  effect or two factor interaction, BUT two factor
  interactions are aliased with three factor
  interactions.

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Common Screening Designs

• Fractional Factorial Designs the total number
  of experimental runs must be a power of 2 (4, 8,
  16, 32, 64, ). If you believe first order
  interactions are small compared to main effects,
  then you could choose a resolution III design.
  Just remember that if you have major
  interactions, it can mess up your screening
  experiment.

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Common Screening Designs

• Plackett-Burman Designs Two level, resolution
  III designs used to study up to n-1 factors in n
  experimental runs, where n is a multiple of 4 (
  of runs will be 4, 8, 12, 16, ). Since n may
  be quite large, you can study a large number of
  factors with moderately small sample sizes. (n
  100 means you can study 99 factors with 100 runs)

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Other Design Issues

• May want to collect data at center points to
  estimate non-linear responses
• More than two levels of a factor no problem
  (multi-level factorial)
• What do you do if you want to build a non-linear
  model to optimize the response. (hit a target,
  maximize, or minimize) called response surface
  modeling

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Other Design Issues

• What do you do if the factors levels are
  categorical and not quantitative, or some are
  categorical and some are quantitative?
• What do you do if the structure of you experiment
  is nested? These are called heirarchical
  designs and will allow you to partition the total
  variability among the different levels of the
  design (called variance components)

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Response Surface Designs Box-BehnkenAfter
screening designs identify major factors Next
step.

• Design class Response Surface
• Design name Box-Behnken design
  
• Base Design
• -----------
• Number of experimental factors 3 Number of
  blocks 1
• Number of responses 1
• Number of runs 15 Error degrees
  of freedom 5
• Randomized No
• Factors Low High
  Units Continuous
• --------------------------------------------------
  ----------------------
• Factor_A -1.0 1.0
  Yes
• Factor_B -1.0 1.0
  Yes
• Factor_C -1.0 1.0
  Yes

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Response Surface Designs Box-Behnken
FACTOR A FACTOR B FACTOR C
0 0 0
-1 -1 0
1 -1 0
-1 1 0
1 1 0
-1 0 -1
1 0 -1
0 0 0
-1 0 1
1 0 1
0 -1 -1
0 1 -1
0 -1 1
0 1 1
0 0 0
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Response Surface Designs Central Composite

• Design class Response Surface
• Design name Central composite blocked cube-star
  
• Number of experimental factors 3 Number of
  blocks 2
• Number of responses 1
• Number of runs 16 Error degrees
  of freedom 5
• Randomized No
• Factors Low High
  Units Continuous
• --------------------------------------------------
  ----------------------
• Factor_A -1.0 1.0
  Yes
• Factor_B -1.0 1.0
  Yes
• Factor_C -1.0 1.0
  Yes

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Response Surface Designs Central Composite
FACTOR A FACTOR B FACTOR C
-1 -1 -1
1 -1 -1
-1 1 -1
1 1 -1
0 0 0
-1 -1 1
1 -1 1
-1 1 1
1 1 1
-1.76383 0 0
1.76383 0 0
0 -1.76383 0
0 0 0
0 1.76383 0
0 0 -1.76383
0 0 1.76383
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Multilevel Factorial Designs

• Design class Multilevel Factorial
• Number of experimental factors 3 Number of
  blocks 1
• Number of responses 1
• Number of runs 27 Error degrees
  of freedom 17
• Randomized No
• Factors Low High
  Levels Units
• --------------------------------------------------
  -----------------------
• Factor_A -1.0 1.0
  3
• Factor_B -1.0 1.0
  3
• Factor_C -1.0 1.0
  3

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Multilevel Factorial Designs
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Nested Design

• Design class Variance Components
• Number of experimental factors 3
• Number of responses 1
• Number of runs 27
• Randomized No
• Factors Levels Units
• -----------------------------------------------
• Factor_A 3
• Factor_B 3
• Factor_C 3
• You have created a variance components design
  which will estimate the contribution of 3 factors
  to overall process variability. The design is
  hierarchical, with each factor nested in the
  factor above it. A total of 27 measurements are
  required.

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Nested Design
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Response Surface Designs Box-BehnkenEXAMPLE -
RECAP

• Design class Response Surface
• Design name Box-Behnken design
  
• Base Design
• -----------
• Number of experimental factors 3 Number of
  blocks 1
• Number of responses 1
• Number of runs 15 Error degrees
  of freedom 5
• Randomized No
• Factors Low High
  Units Continuous
• --------------------------------------------------
  ----------------------
• Factor_A 10 30
  Yes
• Factor_B 30 60
  Yes
• Factor_C 40 60
  Yes

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Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1 (NOISE)
RUN F1 F2 F3
1 10 45 60
2 30 45 40
3 20 30 40
4 10 30 50
5 20 45 50
6 30 60 50
7 20 45 50
8 30 45 60
9 20 45 50
10 20 60 40
11 10 45 40
12 30 30 50
13 20 60 60
14 10 60 50
15 20 30 60
90
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
RUN F1 F2 F3 Y0
1 10 45 60 11800
2 30 45 40 8800
3 20 30 40 8400
4 10 30 50 9300
5 20 45 50 9400
6 30 60 50 8300
7 20 45 50 9400
8 30 45 60 10800
9 20 45 50 9400
10 20 60 40 8400
11 10 45 40 9800
12 30 30 50 11300
13 20 60 60 10400
14 10 60 50 12300
15 20 30 60 10400
91
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
92
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
93
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 0
94
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
RUN F1 F2 F3 Y100
1 10 45 60 11825
2 30 45 40 8781
3 20 30 40 8413
4 10 30 50 9216
5 20 45 50 9288
6 30 60 50 8261
7 20 45 50 9329
8 30 45 60 10855
9 20 45 50 9205
10 20 60 40 8538
11 10 45 40 9718
12 30 30 50 11308
13 20 60 60 10316
14 10 60 50 12056
15 20 30 60 10378
95
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
96
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
97
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100
98
Response Surface Designs Box-BehnkenREAL
MODEL Y 40F1200F2100F3-10F1F29F1F1
(NOISE)Example std. dev. of noise 100

• Optimize Response
• -----------------
• Goal maximize Y
• Optimum value 13139.4
• Factor Low High
  Optimum
• --------------------------------------------------
  ---------------------
• Factor_A 10.0 30.0
  10.1036
• Factor_B 30.0 60.0
  60.0
• Factor_C 40.0 60.0
  60.0

99
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
RUN F1 F2 F3 Y100
1 10 30 40 8270
2 20 30 40 8272
3 30 30 40 10324
4 10 45 40 9928
5 20 45 40 8520
6 30 45 40 8973
7 10 60 40 11082
8 20 60 40 8377
9 30 60 40 7410
10 10 30 50 9191
11 20 30 50 9331
12 30 30 50 11131
13 10 45 50 10615
100
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
RUN F1 F2 F3 Y100
14 20 45 50 9302
15 30 45 50 9723
16 10 60 50 12088
17 20 60 50 9343
18 30 60 50 8260
19 10 30 60 10313
20 20 30 60 10363
21 30 30 60 12267
22 10 45 60 11763
23 20 45 60 10534
24 30 45 60 10791
25 10 60 60 13281
26 20 60 60 10349
27 30 60 60 9497
101
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
102
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100

• Optimize Response
• -----------------
• Goal maximize Y
• Optimum value 13230.6
• Factor Low High
  Optimum
• --------------------------------------------------
  ---------------------
• Factor_A 10.0 30.0
  10.0
• Factor_B 30.0 60.0
  60.0
• Factor_C 40.0 60.0
  60.0

103
Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
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Response Surface Designs Three Level Factorial
Design (33)REAL MODEL Y 40F1200F2100F3-10F1F
29F1F1 (NOISE)Example std. dev. of noise
100
105
CLASSROOM EXERCISE

• STUDENT IN-CLASS EXPERIMENT Collect data for
  experiment to determine factor settings (two
  factors) to hit a target response (spot on wall).
• Factor A height of shaker (low and high)
• Factor B location of shaker (close to hand and
  close to wall)
• Design experiment would suggest several
  replications

106
CLASSROOM EXERCISE

• Conduct Experiment student holds 3 foot pin
  the tail on the donkey stick and attempts to hit
  the target. An observer will assist to mark the
  hit on the target.
• Collect data students take data home for week
  and come back with what you would recommend AND
  why.
• YOU TELL THE CLASS HOW TO PLAY THE GAME TO WIN.

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CLASSROOM EXERCISE
108
CLASSROOM EXERCISE
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CLASSROOM EXERCISE

• HOMEWORK
• .Determine the effects marker stick and
  vertical pole have on the mean location of the
  hit.
• .Determine the effects marker stick and
  vertical pole have on the standard deviation
  of the hit.
• .Which factor would you say affects the mean
  location of the hit?
• .Which factor would you say affects the standard
  deviation of the hit?
• OPTIMAL SETTINGS Where would you recommend we
  locate the vertical pole and the marker stick
  IF we wish to (a) MINIMIZE THE VARIABILITY OF THE
  HIT and (b) HIT THE TARGET LOCATED AT 0?

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PIN THE TAIL DATA INPUT
111
ESTIMATE OF EFFECTS (MEAN HIT)

• Estimated effects for MEAN
• --------------------------------------------------
  --------------------
• average 0.875
• AMARKER STICK 1.906
• BVERTICAL POLE 12.969
• AB 4.625
• --------------------------------------------------
  --------------------
• No degrees of freedom left to estimate standard
  errors.

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ESTIMATE OF EFFECTS (MEAN HIT)
113
ESTIMATE OF EFFECTS (MEAN HIT)
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INTERACTION PLOT (MEAN HIT)
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3-D PLOT OF RESPONSE (MEAN HIT)
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CONTOUR PLOT OF RESPONSE (MEAN HIT)
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ANALYSIS OF VARIANCE TABLE (MEAN HIT)

• Analysis of Variance for MEAN
• --------------------------------------------------
  ------------------------------
• Source Sum of Squares Df
  Mean Square F-Ratio P-Value
• --------------------------------------------------
  ------------------------------
• AMARKER STICK 3.63284 1
  3.63284 0.17 0.7511
• BVERTICAL POLE 168.195 1
  168.195 7.86 0.2181
• Total error 21.3906 1
  21.3906
• --------------------------------------------------
  ------------------------------

118
ESTIMATED LINEAR RESPONSE MODEL (MEAN HIT)

• Regression coeffs. for MEAN
• --------------------------------------------------
  --------------------
• constant 0.875
• AMARKER STICK 0.953
• BVERTICAL POLE 6.4845
• --------------------------------------------------
  --------------------
• The StatAdvisor
• ---------------
• This pane displays the regression equation
  which has been fitted to
• the data. The equation of the fitted model is
• MEAN 0.875 0.953MARKER STICK
  6.4845VERTICAL POLE

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OPTIMAL FACTOR SETTINGS (MEAN HIT)

• Optimize Response
• -----------------
• Goal maintain MEAN at 0.0
• Optimum value 0.0
• Factor Low High
  Optimum
• --------------------------------------------------
  ---------------------
• MARKER STICK -1.0 1.0
  0.03311
• VERTICAL POLE -1.0 1.0
  -0.139803

120
ESTIMATE OF EFFECTS (STD DEV HIT)

• Estimated effects for STD DEV
• --------------------------------------------------
  --------------------
• average 2.63275
• AMARKER STICK 2.7605
• BVERTICAL POLE 0.3735
• AB -0.0895

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ESTIMATE OF EFFECTS (STD DEV HIT)

• Analysis of Variance for STD DEV
• --------------------------------------------------
  ------------------------------
• Source Sum of Squares Df
  Mean Square F-Ratio P-Value
• --------------------------------------------------
  ------------------------------
• AMARKER STICK 7.62036 1 7.62036
  951.33 0.0206
• BVERTICAL POLE 0.139502 1 0.139502
  17.42 0.1497
• Total error 0.00801025 1
  0.00801025
• --------------------------------------------------
  ------------------------------
• Total (corr.) 7.76787 3

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OPTIMAL FACTOR SETTINGS (STD DEV HIT)

• Optimize Response
• -----------------
• Goal minimize STD DEV
• Optimum value 1.06575
• Factor Low
  High Optimum
• --------------------------------------------------
  ---------------------
• MARKER STICK -1.0 1.0
  -1.0
• VERTICAL POLE -1.0 1.0
  -1.0

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INTERACTION (STD DEV HIT)
124
CONTOUR PLOT OF RESPONSE (STD DEV HIT)
125
SO, WHATS THE ANSWER?

• I WOULD
• 1. SET THE MARKER STICK AT LOW (CLOSE TO THE
  WALL)
• 2. SET THE VERTICAL POLE AT A VALUE THAT WILL
  HIT THE TARGET.

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SO, WHATS THE ANSWER?

• FROM REGRESSION FOR MEAN HIT, SET MARKER STICK
  AT -1, HIT AT 0, AND SOLVE FOR VP
• HIT .0875 .953MS 6.4845VP
• 0 .875 .953(-1) 6.4845VP
• Resulting in
• VP .012 and MS -1

127
Contour Plots for Mean and Std. Dev.