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Factorial Designs: Why and How

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Test each of k factors at 2 levels ... You run the experiment by setting the factor levels shown in columns A, B, and C ... Half-Factorial Design, k=4 ... – PowerPoint PPT presentation

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Title: Factorial Designs: Why and How


1
Factorial Designs Why and How
  • ChE 408 Engineering Experimental Design
  • Valerie L. Young

2
Review of Experimental Design Vocabulary
  • Response
  • Factor
  • Level

3
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

4
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.

5
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

6
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)

7
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.

8
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

9
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.

10
Example 23 Factorial Design
  • Response Yield,
  • Factors

11
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

12
Standard 23 Layout
13
Our 23 Layout (with Results)
14
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

15
Expanded 23 Layout
16
Calculating Effects
17
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).

18
Interpretation Scree Plot
19
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

20
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

21
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

22
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.

23
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.

24
Half-Factorial 24 Layout
25
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 at home 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

26
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

27
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
  • If you need to test more than 2 levels of each
    factor, a Response Surface design may be more
    efficient
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