Fractional factorial designs
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.
s are confounded, i.e., cannot be estimated independently of each other (see below). A design with p such generators is a 1/(lp) fraction of the full factorial design.
For example, a 25 − 2 design is 1/4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 25 factorial experiment, this experiment requires only eight runs.
In practice, one rarely encounters l > 2 levels in fractional factorial designs, since response surface methodology
is a much more experimentally efficient way to determine the relationship between the experimental response and factors at multiple levels. In addition, the methodology to generate such designs for more than two levels is much more cumbersome.
The levels of a factor are commonly coded as +1 for the higher level, and −1 for the lower level. For a three-level factor, the intermediate value is coded as 0.
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 25 − 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. These two expressions are called the generators of the design. So for example, when the experiment is run and the experimenter estimates the effects for factor D, what is really being estimated is a combination of the main effect of D and 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.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, 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
Sparsity-of-effects principle
The sparsity-of-effects principle states that a system is usually dominated by main effects and low-order interactions. Thus it is most likely that main effects and two-factor interactions are the most significant responses . In other words, higher order interactions such as three-factor...
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.
Notation
Fractional designs are expressed using the notation lk − p, where l is the number of levels of each factor investigated, k is the number of factors investigated, and p describes the size of the fraction of the full factorial used. Formally, p is the number of generators, assignments as to which effects or interactionInteraction (statistics)
In statistics, an interaction may arise when considering the relationship among three or more variables, and describes a situation in which the simultaneous influence of two variables on a third is not additive...
s are confounded, i.e., cannot be estimated independently of each other (see below). A design with p such generators is a 1/(lp) fraction of the full factorial design.
For example, a 25 − 2 design is 1/4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 25 factorial experiment, this experiment requires only eight runs.
In practice, one rarely encounters l > 2 levels in fractional factorial designs, since response surface methodology
Response surface methodology
In statistics, response surface methodology explores the relationships between several explanatory variables and one or more response variables. The method was introduced by G. E. P. Box and K. B. Wilson in 1951. The main idea of RSM is to use a sequence of designed experiments to obtain an...
is a much more experimentally efficient way to determine the relationship between the experimental response and factors at multiple levels. In addition, the methodology to generate such designs for more than two levels is much more cumbersome.
The levels of a factor are commonly coded as +1 for the higher level, and −1 for the lower level. For a three-level factor, the intermediate value is coded as 0.
Generation
In practice, experimenters typically rely on statistical reference books 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 25 − 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. These two expressions are called the generators of the design. So for example, when the experiment is run and the experimenter estimates the effects for factor D, what is really being estimated is a combination of the main effect of D and the two-factor interaction involving A and B.
An important characteristic of a fractional design is the defining relation
Relation (mathematics)
In set theory and logic, a relation is a property that assigns truth values to k-tuples of individuals. Typically, the property describes a possible connection between the components of a k-tuple...
, 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.
| Treatment combination | I | A | B | C | D = AB | E = AC |
|---|---|---|---|---|---|---|
| de | − | − | − | |||
| a | − | − | − | − | ||
| be | − | − | − | |||
| abd | − | − | ||||
| cd | − | − | − | |||
| ace | − | − | ||||
| bc | − | − | − | |||
| abcde |
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 not useful and resolutions above V are wasteful in that they can estimate very high-order interactions which rarely occur in practice. The 25 − 2 design above is resolution III since its defining relation is I = ABD = ACE = BCDE.| Resolution | Ability | Example |
|---|---|---|
| II | Not useful: main effects are confounded with other main effects | 22 − 1 with defining relation I = AB |
| III | Estimate main effects, but these may be confounded with two-factor interactions | 23 − 1 with defining relation I = ABC |
| IV | Estimate main effects unconfounded by two-factor interactions Estimate two-factor interaction effects, but these may be confounded with other two-factor interactions |
24 − 1 with defining relation I = ABCD |
| V | Estimate main effects unconfounded by three-factor (or less) interactions Estimate two-factor interaction effects unconfounded by two-factor interactions Estimate three-factor interaction effects, but these may be confounded with other two-factor interactions |
25 − 1 with defining relation I = ABCDE |
| VI | Estimate main effects unconfounded by four-factor (or less) interactions Estimate two-factor interaction effects unconfounded by three-factor (or less) interactions Estimate three-factor interaction effects, but these may be confounded with other three-factor interactions |
26 − 1 with defining relation I = ABCDEF |