Fractional factorial design

From Wikipedia, the free encyclopedia

In statistics, fractional factorial designs are experimental designs consisting of a carefully chosen subset (fraction) of the experimental runs of a full factorial design. The subset is chosen so as to exploit the sparsity-of-effects principle to expose information about the most important features of the problem studied, while using a fraction of the effort of a full factorial design in terms of experimental runs and resources.

1 Notation
2 Generation
3 Resolution
4 References

[edit] Notation

Fractional designs are expressed as l^k − p, where l is the number of levels of each factor investigated, k is the number of factors investigated, and p is the number of generators. A generator of a fraction determines what effects are combined or confounded with one another. A design with p such generators is called a $\frac{1}{l^p}$ fraction of the full factorial design.

For example, a 2^5 − 2 design is a two level, five factor, 1/4 fractional factorial design. Rather than the 32 runs that would be required for the full 2⁵ factorial experiment, this experiment requires only eight runs. Assuming the higher order interaction terms are negligible, the amount of information that is generated from a fractional factorial experiment is comparable to a full factorial experiment, but the practical cost in time, money, or resources is significantly smaller. The levels of a factor are commonly coded as $+ 1$ for the high level, and $- 1$ for the low level. For a three-level factor, the intermediate value is coded as a $0$ .

Treatment combinations for a 2^5 − 2 design
Treatment combination	I	A	B	C	D = AB	E = AC
de	+	-	-	-	+	+
a	+	+	+	+	-	-
be	+	-	+	+	-	+
abd	+	+	+	-	+	-
cd	+	-	-	+	+	-
ace	+	+	-	+	-	+
bc	+	-	+	+	-	-
abcde	+	+	+	+	+	+

In practice, one only encounters l = 2 levels in factorial designs, since response surface methodology is a much more experimentally efficient way to determine the relationship between factors at multiple levels and the experimental response.

[edit] Generation

In practice, experimenters typically rely on statistical texts to supply the "standard" fractional factorial designs, consisting of the principal fraction. The principal fraction is the set of treatment combinations for which the generators evaluate to + under the treatment combination algebra. However, in some situations, the experimenter may take it upon himself or herself to generate his own fractional design.

A fractional factorial experiment is generated from a full factorial experiment by choosing an alias structure. The alias structure determines which effects are confounded with each other. For example, the five factor 2^5 − 2 can be generated by using a full three factor factorial experiment involving three factors (say A, B, and C) and then choosing to confound the two remaining factors D and E with interactions generated by D = A*B and E = A*C. Thus, when the experiment is run and the experimenter estimates the effects for factor D, he or she really is estimating the combined main effect of D with the two-factor interaction involving A and B.

An important characteristic of a fractional design is the defining relation, which gives the set of interaction columns equal in the design matrix to a column of plus signs, denoted by I. For the above example, since D = AB and E = AC, then ABD and ACE are both columns of plus signs, and consequently so is BDCE. In this case the defining relation of the fractional design is I = ABD = ACE = BCDE. The defining relation allows the alias pattern of the design to be determined.

[edit] Resolution

An important property of a fractional design is its resolution or ability to separate main effects and low-order interactions from one another. Formally, the resolution of the design is the minimum word length in the defining relation excluding (1). The most important fractional designs are those of resolution III, IV, and V: Resolutions below III are impossible or useless and resolutions above V are wasteful in that they can estimate very high-order interactions which rarely occur in practice. The 2^5 − 2 design is resolution III since its defining relation is I = ABD = ACE = BCDE.

Resolution	Ability	Example
I	Impossible	N/A
II	Not useful: main effects are confounded with other main effects	N/A
III	Estimate main effects, but these may be confounded with two-factor interactions	2^3 - 1 with defining relation I = ABC
IV	Estimate main effects unconfounded by higher-order interactions Estimate two-factor interaction effects, but these may be confounded with other two-factor interactions	2^4 - 1 with defining relation I = ABCD
V	Estimate main effects unconfounded by higher-order interactions Estimate two-factor interaction effects unconfounded by higher-order interactions Estimate three-factor interaction effects, but these may be confounded with other three-factor interactions	2^5 - 1 with defining relation I = ABCDE
VI	Estimate main effects unconfounded by higher-order interactions Estimate two-factor interaction effects unconfounded by higher-order interactions Estimate three-factor interaction effects unconfounded by higher-order interactions Estimate four-factor interaction effects, but these may be confounded with other four-factor interactions	2^6 - 1 with defining relation I = ABCDEF

[edit] References

Box,G. E, Hunter,W.G., Hunter, J.S., Hunter,W.G., Statistics for Experimenters: Design, Innovation, and Discovery, 2nd Edition, Wiley, 2005, ISBN: 0471718130

Retrieved from "http://en.wikipedia.org../../../f/r/a/Fractional_factorial_design.html"

Categories: Experimental design | Educational research | Social sciences methodology