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Fractional factorial design generators
generators = fracfactgen(terms)
generators = fracfactgen(terms,k)
generators = fracfactgen(terms,k,R)
generators = fracfactgen(terms,k,R,basic)
generators = fracfactgen(terms) uses the Franklin-Bailey algorithm to find generators for the smallest two-level fractional-factorial design for estimating linear model terms specified by terms. terms is a string consisting of words formed from the 52 case-sensitive letters a-Z, separated by spaces. Use 'a'-'z' for the first 26 factors, and, if necessary, 'A'-'Z' for the remaining factors. For example, terms = 'a b c ab ac'. Single-character words indicate main effects to be estimated; multiple-character words indicate interactions. Alternatively, terms is an m-by-n matrix of 0s and 1s where m is the number of model terms to be estimated and n is the number of factors. For example, if terms contains rows [0 1 0 0] and [1 0 0 1], then the factor b and the interaction between factors a and d are included in the model. generators is a cell array of strings with one generator per cell. Pass generators to fracfact to produce the fractional-factorial design and corresponding confounding pattern.
generators = fracfactgen(terms,k) returns generators for a two-level fractional-factorial design with 2^{k}-runs, if possible. If k is [], fracfactgen finds the smallest design.
generators = fracfactgen(terms,k,R) finds a design with resolution R, if possible. The default resolution is 3.
A design of resolution R is one in which no n-factor interaction is confounded with any other effect containing less than R – n factors. Thus a resolution III design does not confound main effects with one another but may confound them with two-way interactions, while a resolution IV design does not confound either main effects or two-way interactions but may confound two-way interactions with each other.
If fracfactgen is unable to find a design at the requested resolution, it tries to find a lower-resolution design sufficient to calibrate the model. If it is successful, it returns the generators for the lower-resolution design along with a warning. If it fails, it returns an error.
generators = fracfactgen(terms,k,R,basic) also accepts a vector basic specifying the indices of factors that are to be treated as basic. These factors receive full-factorial treatments in the design. The default includes factors that are part of the highest-order interaction in terms.
Suppose you wish to determine the effects of four two-level factors, for which there may be two-way interactions. A full-factorial design would require 2^{4} = 16 runs. The fracfactgen function finds generators for a resolution IV (separating main effects) fractional-factorial design that requires only 2^{3} = 8 runs:
generators = fracfactgen('a b c d',3,4) generators = 'a' 'b' 'c' 'abc'
The more economical design and the corresponding confounding pattern are returned by fracfact:
[dfF,confounding] = fracfact(generators) dfF = -1 -1 -1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 -1 1 1 1 1 confounding = 'Term' 'Generator' 'Confounding' 'X1' 'a' 'X1' 'X2' 'b' 'X2' 'X3' 'c' 'X3' 'X4' 'abc' 'X4' 'X1*X2' 'ab' 'X1*X2 + X3*X4' 'X1*X3' 'ac' 'X1*X3 + X2*X4' 'X1*X4' 'bc' 'X1*X4 + X2*X3' 'X2*X3' 'bc' 'X1*X4 + X2*X3' 'X2*X4' 'ac' 'X1*X3 + X2*X4' 'X3*X4' 'ab' 'X1*X2 + X3*X4'
The confounding pattern shows, for example, that the two-way interaction between X1 and X2 is confounded by the two-way interaction between X3 and X4.