Factorial Designs: Why and How

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Factorial Designs Why and How

• Engineering Experimental Design
• Valerie L. Young

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Review of Experimental Design Vocabulary

• Response
• Factor
• Level

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Review of Experimental Design Vocabulary

• Response Dependent variable
• The process output
• The value we want to predict or optimize
• Factor Independent variable
• The process inputs
• Things we can adjust to affect the response
• Level
• A specific value of a factor that we will test

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Experimental Design Means
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Experimental Design Means

• Specify experimental objective
• Specify experimental conditions
• Factors (number, identity)
• Response (and method of measurement)
• Levels (number, identity, values, method of
  determination)
• Number of trials
• Specify analysis methods

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Elements of Experimental Design
Specify Experimental Objective
Specify Analysis Methods
Specify Experimental Conditions
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Elements of Experimental Design

• This part of the process is simplified if you use
  a standard experimental design, a template
• Factorial Designs are a class of standard
  designs that are good for screening factors

Specify Experimental Conditions
Specify Analysis Methods
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Factorial Design means . . .

• You choose 2 or more factors to test
• You choose 2 or more levels for each factor
• You measure the response using various
  combinations of factors and levels
• Not vary-one-factor-and-keep-others-constant
• You determine which factors have the largest
  effects on the response, and whether there are
  interactions between factors

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What about OFAT?

• When we first learn about the scientific
  method, we are taught that good scientists hold
  everything else constant while they test one
  factor at a time (OFAT).
• OFAT is actually an inefficient, inadequate
  method of experimentation for identifying
  significant factors.
• OFAT requires more experiments than factorial
  design to test the same number of factors.
• OFAT cannot reveal interactions between factors
  factorial design can.

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Word to the Wise

• Factorial design is NOT an uncontrolled
  experiment
• You dont just let everything vary willy-nilly
• You must test specific levels of each factor
• You must test each level of each factor more than
  once

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Factorial Design Can Be Used To . .

• Identify factors with significant effects on the
  response
• Identify interactions among factors
• Identify which factors have the most important
  effects on the response
• Decide whether further investigation of a
  factors effect is justified
• Investigate the functional dependence of a
  response on multiple factors simultaneously (if
  and only if you test many levels of each factor)

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When is OFAT Appropriate?

• When you are interested in developing a
  functional relationship between a factor and the
  response, and you know that interactions between
  that factor and others are unimportant.
• When there is only one factor of interest or
  importance.

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2k Factorial Designs

• Test each of k factors at 2 levels
• Great for screening to see which effects and
  interactions are important
• Cannot identify nonlinear effects
• Do NOT analyze by regression
• Use ANOVA and scree plots
• With only two levels of each factor, all
  relationships are linear

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Example 23 Factorial Design

• We are interested in optimizing the yield from a
  particular reactor. There are three factors that
  may be important reaction temperature, reaction
  time, and catalyst concentration. We set up a 23
  factorial design to determine the relative
  importance of the factors and their interactions.
• This example is from R.J. DelVecchio,
  Understanding Design of Experiments, Hanser
  Gardner, 1997.

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Example 23 Factorial Design

• Response Yield,
• Factors

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A Word About Levels

• Levels need not be numeric
• Factors can be variable data (numbers) or
  attribute data (on/off, male/female,
  Ford/Subaru/Mercedes)
• For example, if we were testing two types of
  catalyst (say alumina and silica), we could
  arbitrarily designate one as high and one as
  low

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Standard 23 Layout
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Our 23 Layout (with Results)
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Randomization

• Although we write our test matrix in this
  standard order, we should actually perform the 8
  experiments in RANDOM order
• Randomization will make any factor we overlooked
  likely to contribute to random uncertainty rather
  than systematic error

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Expanded 23 Layout
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Expanded 23 Layout
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Calculating Effects
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Calculating Effects
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Reminder about Running the Experiment

• When we write the expanded layout, we are only
  putting in the interaction columns to make it
  easy to calculate interaction effects.
• You run the experiment by setting the factor
  levels shown in columns A, B, and C
• The interaction levels are set automatically
• When A and B are both high or both low, their
  interaction is automatically high (positive).
• When A and B are set one high and one low, their
  interaction is automatically low (negative).

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Interpretation Scree Plot
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Interpretation Scree Plot

• The scree plot gives us qualitative information
• The scree plot can be used to justify treating
  the effects of temperature, catalyst , and
  temperature-time interaction as significant
• Are the other effects significant?
• If we had replicate runs, we would have a direct
  measure of experimental variability and could use
  ANOVA to decide.
• Since we dont have replicate runs, we could use
  ANOVA assuming that the effect of the 3-way
  interaction is about the same as the effect of
  experimental variability

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Interpretation Multifactor ANOVA

• Excel cannot do ANOVA for k gt 2
• Matlab can do ANOVA for k gt 2
• Beyond the scope of this course
• The Matlab help isnt bad if you need to learn
  how to do it
• Statistics packages such as Minitab and SAS can
  do ANOVA for k gt 2

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Fractional Factorial Designs

• To do a full 2k factorial experiment, you must do
  at least 2k runs
• Testing 4 factors costs twice as much as testing
  3 factors.
• You can test more factors in the same 8 runs if
  you settle for a fractional factorial design

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Expanded 23 Layout Revisited
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Half-Factorial Design, k4

• A half-factorial design means we will test 4
  factors in 8 experiments instead of 2416.
• Often, the effect of ABC is very small.
• If we want to test four factors, we could set
  high and low levels of Factor D according to the
  s and s in column ABC
• Whatever effect we calculate for D will really be
  a combination of the effect of D and the effect
  of ABC.
• Interactions between D and the other factors will
  not be identified.

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Confounding of Effects

• Confounding when a calculated effect is
  actually a combination of the effects of multiple
  factors and/or interactions.
• When we replace interaction ABC with a plan for
  testing D,
• we confound the effect of D with the three-way
  interaction.
• we confound the effects of A, B, and C with
  three-way interactions that include D
• we confound the two-way interactions with other
  two-way interactions that include D
• Only use fractional factorial design if you are
  pretty sure that the additional factors dont
  interact with the original ones.

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Half-Factorial 24 Layout
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Interpreting the Half-Factorial Design

• As before, we total the responses when each
  factor is and when each factor is -, take the
  difference, and then divide by 4 (because there
  are 4 and 4 in each column) to calculate the
  effects.
• A scree plot will indicate which effects are
  important.
• Try this later for the following vector of
  responses 65.6 79.3 51.3 69.6 59.8 77.7
  74.2 87.9
• Assume factor D is reaction vessel size 100 L
  or 200 L

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Calculating Effects
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Where are the values of the levels of each factor?

• Note that the values you pick for each level of
  each factor are NOT used in the analysis
• Calculation of effects based only on values of
  response
• Calculations for ANOVA based only on values of
  response
• Although they do not appear in the analysis, the
  values of the levels are important
• Too small difference means too small change in
  response to see
• Too large difference means failed experiment

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More Factors in the Same Number of Runs

• You can look at 5, 6, even 7 factors in 8 runs if
  you are willing to assume that interactions are
  negligible compared to experimental error.
• Normally, for 5-6 factors, the additional factors
  replace two-way interactions and the ABC column
  is assumed to be an estimate of experimental
  error.
• Note that any effect you see for an additional
  factor is really a combination of its effect and
  any effect of the interaction that it replaced

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Word to the Wise . . .

• When you say you are going to do a factorial
  design, many people assume you mean a 2k
  factorial design.
• You should be specific about the number of levels
  of each factor you will test
• e.g., a 3k design tests each factor at 3 levels
• If you need to test more than 2 levels of each
  factor, a Response Surface design may be more
  efficient