Note: This page has been translated by MathWorks. Please click here

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

To view all translated materals including this page, select Japan from the country navigator on the bottom of this page.

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 character vector 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 `0`

s
and `1`

s 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 2

-runs,
if possible. If ^{k}`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`

.

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

Was this topic helpful?