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| {{multiple issues|expert=statistics|date=April 2012|refimprove = April 2012| technical = April 2012
| | She is known by the title of Myrtle Shryock. Playing baseball is the pastime he will by no means stop doing. California is our birth location. Hiring has been my occupation for some time but I've currently applied for another 1.<br><br>My page ... [http://apcbook.com.ng/index.php?do=/profile-446/info/ apcbook.com.ng] |
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| In [[statistics]], '''fractional factorial designs''' are [[experimental design]]s 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.
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| ==Notation==
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| Fractional designs are expressed using the notation ''l''<sup>k − p</sup>, 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 [[interaction (statistics)|interaction]]s are ''confounded'', ''i.e.'', cannot be estimated independently of each other (see below). A design with ''p'' such generators is a 1/(''l<sup>p</sup>'') fraction of the full factorial design.
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| For example, a 2<sup>5 − 2</sup> design is 1/4 of a two level, five factor factorial design. Rather than the 32 runs that would be required for the full 2<sup>5</sup> factorial experiment, this experiment requires only eight runs.
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| 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.
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| 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.
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| To save space, the points in a two-level factorial experiment are often abbreviated with strings of plus and minus signs. The strings have as many symbols as factors, and their values dictate the level of each factor: conventionally, <math>-</math> for the first (or low) level, and <math>+</math> for the second (or high) level. The points in this experiment can thus be represented as <math>--</math>, <math>+-</math>, <math>-+</math>, and <math>++</math>.
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| The factorial points can also be abbreviated by (1), a, b, and ab, where the presence of a letter indicates that the specified factor is at its high (or second) level and the absence of a letter indicates that the specified factor is at its low (or first) level (for example, "a" indicates that factor A is on its high setting, while all other factors are at their low (or first) setting). (1) is used to indicate that all factors are at their lowest (or first) values.
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| ==Generation==
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| 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, experimenters may take it upon themselves to generate their own fractional design.
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| 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<sup>5 − 2</sup> 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''.
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| An important characteristic of a fractional design is the defining [[Relation (mathematics)|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.
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| {| class="wikitable" border="1" style="text-align:center; background:#9090ff"
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| |+Treatment combinations for a 2<sup>5 − 2</sup> design
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| !Treatment combination
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| !width="50"|I
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| !width="50"|A
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| !width="50"|B
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| !width="50"|C
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| !width="50"|D = AB
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| !width="50"|E = AC
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| |-
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| ! de
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| | {{no|+}} || − || − || − || {{no|+}} || {{no|+}}
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| |-
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| ! a
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| | {{no|+}} || {{no|+}} || − || − || − || −
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| |-
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| ! be
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| | {{no|+}} || − || {{no|+}} || − || − || {{no|+}}
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| |-
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| ! abd
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| | {{no|+}} || {{no|+}} || {{no|+}} || − || {{no|+}} || −
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| |-
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| ! cd
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| | {{no|+}} || − || − || {{no|+}} || {{no|+}} || −
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| |-
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| ! ace
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| | {{no|+}} || {{no|+}} || − || {{no|+}} || − || {{no|+}}
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| |-
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| ! bc
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| | {{no|+}} || − || {{no|+}} || {{no|+}} || − || −
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| |-
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| ! abcde
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| | {{no|+}} || {{no|+}} || {{no|+}} || {{no|+}} || {{no|+}} || {{no|+}}
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| |}
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| ==Resolution==
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| 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 the expanded experimentation has no practical benefit in most cases—the bulk of the additional effort goes into the estimation of very high-order interactions which rarely occur in practice. The 2<sup>5 − 2</sup> design above is resolution III since its defining relation is I = ABD = ACE = BCDE.
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| {| class="wikitable"
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| ! Resolution
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| ! Ability
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| ! Example
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| |-
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| | I
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| | Not useful: an experiment of exactly one run only tests one level of a factor and hence can't even distinguish between the high and low levels of that factor
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| | 2<SUP>1 − 1</SUP> with defining relation I = A
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| |-
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| | II
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| | Not useful: main effects are confounded with other main effects
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| | 2<SUP>2 − 1</SUP> with defining relation I = AB
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| |-
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| | III
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| | Estimate main effects, but these may be confounded with two-factor interactions
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| | 2<SUP>3 − 1</SUP> with defining relation I = ABC
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| |-
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| | IV
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| Estimate main effects unconfounded by two-factor interactions<BR>
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| Estimate two-factor interaction effects, but these may be confounded with other two-factor interactions
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| | 2<SUP>4 − 1</SUP> with defining relation I = ABCD
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| |-
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| | V
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| Estimate main effects unconfounded by three-factor (or less) interactions<BR>
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| Estimate two-factor interaction effects unconfounded by two-factor interactions<BR>
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| Estimate three-factor interaction effects, but these may be confounded with other three-factor interactions
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| | 2<SUP>5 − 1</SUP> with defining relation I = ABCDE
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| |-
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| | VI
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| Estimate main effects unconfounded by four-factor (or less) interactions<BR>
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| Estimate two-factor interaction effects unconfounded by three-factor (or less) interactions<BR>
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| Estimate three-factor interaction effects, but these may be confounded with other three-factor interactions
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| | 2<SUP>6 − 1</SUP> with defining relation I = ABCDEF
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| |}
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| The resolution described is only used for regular designs. Regular designs have run size that equal a power of two, and only full aliasing is present. Nonregular designs are designs where run size is a multiple of 4; these designs introduce partial aliasing, and generalized resolution is used as design criteria instead of the resolution described previously.
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| == References ==
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| {{cite book | last = Box | first = G.E. | coauthors = Hunter, J.S., Hunter,W.G. | title = Statistics for Experimenters: Design, Innovation, and Discovery, 2nd Edition | publisher = Wiley | year = 2005 | isbn = 0-471-71813-0}}
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| ==External links==
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| *[http://methodology.psu.edu/ra/most/fefaq Full Factorial and Fractional Factorial Experiments: Frequently Asked Questions (The Methodology Center, Penn State University)]
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| *[http://www.itl.nist.gov/div898/handbook/pri/section3/pri334.htm Fractional Factorial Designs (National Institute of Standards and Technology)]
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| ==See also==
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| [[Robust Parameter Design (RPD)|Robust Parameter Designs]]
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| {{Experimental design}}
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| {{Statistics}}
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| {{DEFAULTSORT:Fractional Factorial Design}}
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| [[Category:Design of experiments]]
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| [[Category:Quality control]]
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| [[Category:Quality]]
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She is known by the title of Myrtle Shryock. Playing baseball is the pastime he will by no means stop doing. California is our birth location. Hiring has been my occupation for some time but I've currently applied for another 1.
My page ... apcbook.com.ng