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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 `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 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

`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.

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