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Fractional Factorial Experimental Designs

KEY WORDS alias structure, confounding, defining relation, dissolved oxygen, factorial design, frac-
tional factorial design, half-fraction, interaction, main effect, reference distribution, replication, ruggedness
testing, t distribution, variance.

Two-level factorial experimental designs are very efficient but the number of runs grows exponentially
as the number of factors increases.

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3 factors at 2 levels 2 = 8 runs
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4 factors at 2 levels 2 = 16 runs
5
5 factors at 2 levels 2 = 32 runs
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6 factors at 2 levels 2 = 64 runs
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7 factors at 2 levels 2 = 128 runs
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8 factors at 2 levels 2 = 256 runs
Usually your budget cannot support 128 or 256 runs. Even if it could, you would not want to commit your
entire budget to one very large experiment. As a rule-of-thumb, you should not commit more than 25% of the
budget to preliminary experiments for the following reasons. Some of the factors may be inactive and you
will want to drop them in future experiments; you may want to use different factor settings in follow-up
experiments; a two-level design will identify interactions, but not quadratic effects, so you may want to augment
the design and do more testing; you may need to repeat some experiments; and/or you may need to replicate
the entire design to improve the precision of the estimates. These are reasons why fractional factorial designs
are attractive. They provide flexibility by reducing the amount of work needed to conduct preliminary exper-
iments that will screen for important variables and guide you toward more interesting experimental settings.
Fractional means that we do a fraction or a part of the full factorial design. We could do a half-fraction,
a quarter-fraction, or an eighth-fraction. A half-fraction is to do half of the full factorial design, or
4 5
(1/2)2 = (1/2)16 = 8 runs to investigate four factors; (1/2)(2 ) = (1/2)32 = 16 runs to investigate five factors;
5
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and so on. Examples of quarter-fractions are (1/4)2 = (1/4)32 = 8, or (1/4)2 = (1/4)128 = 32 runs. An
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example eighth-fraction is (1/8)2 = (1/8)256 = 32 runs. These five examples lead to designs that could
investigate 4 variables in 8 runs, 5 factors in 16 runs or 8 runs, 7 factors in 32 runs, or 8 factors in 32 runs.
Of course, some information must be sacrificed in order to investigate 8 factors in 32 runs, instead of
the full 256 runs, but you will be surprised how little is lost. The lost information is about interactions,
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if you select the right 32 runs out of the possible 2 = 256. How to do this is explained fully in Box et al.
(1978) and Box and Hunter (1961a, 1961b).

Case Study: Sampling High Dissolved Oxygen Concentrations

Ruggedness testing is a means of determining which of many steps in an analytical procedure must be
carefully controlled and which can be treated with less care. Each aspect or step of the technique needs
checking. These problems usually involve a large number of variables and an efficient experimental
approach is needed. Fractional factorial designs provide such an approach.

© 2002 By CRC Press LLC