Title: Factorial Designs: Why and How
1Factorial Designs Why and How
- ChE 408 Engineering Experimental Design
- Valerie L. Young
2Review of Experimental Design Vocabulary
3Factorial 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
4What 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.
5Word 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
6Factorial 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)
7When 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.
82k 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
9Example 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.
10Example 23 Factorial Design
11A 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
12Standard 23 Layout
13Our 23 Layout (with Results)
14Randomization
- 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
15Expanded 23 Layout
16Calculating Effects
17Reminder 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).
18Interpretation Scree Plot
19Interpretation 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
20Interpretation 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
21Fractional 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
22Half-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.
23Confounding 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.
24Half-Factorial 24 Layout
25Interpreting 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
26More 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
27Word 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