Documentation

fracfactgen

Fractional factorial design generators

Syntax

generators = fracfactgen(terms)
generators = fracfactgen(terms,k)
generators = fracfactgen(terms,k,R)
generators = fracfactgen(terms,k,R,basic)

Description

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 character vector or string scalar 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-letter words indicate main effects to be estimated; multiple-letter 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 character vectors 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 2k-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 Rn 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.

Examples

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 24 = 16 runs. The fracfactgen function finds generators for a resolution IV (separating main effects) fractional-factorial design that requires only 23 = 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.

References

 Box, G. E. P., W. G. Hunter, and J. S. Hunter. Statistics for Experimenters. Hoboken, NJ: Wiley-Interscience, 1978.