Statistics Toolbox

fracfact

Generate fractional factorial design from generators

Syntax

• x = fracfact('gen') 
  [x,conf] = fracfact('gen') 
  

Description

x = fracfact('gen') generates a fractional factorial design as specified by the generator string gen, and returns a matrix x of design points. The input string gen is a generator string consisting of "words" separated by spaces. Each word describes how a column of the output design should be formed from columns of a full factorial. Typically gen will include single-letter words for the first few factors, plus additional multiple-letter words describing how the remaining factors are confounded with the first few.

The output matrix x is a fraction of a two-level full-factorial design. Suppose there are m words in gen, and that each word is formed from a subset of the first n letters of the alphabet. The output matrix x has 2ⁿ rows and m columns. Let F represent the two-level full-factorial design as produced by ff2n(n). The values in column j of x are computed by multiplying together the columns of F corresponding to letters that appear in the jth word of the generator string.

[x,conf] = fracfact('gen') also returns a cell array, conf, that describes the confounding pattern among the main effects and all two-factor interactions.

Examples

Example 1

We want to run an experiment to study the effects of four factors on a response, but we can only afford eight runs. (A run is a single repetition of the experiment at a specified combination of factor values.) Our goal is to determine which factors affect the response. There may be interactions between some pairs of factors.

A total of sixteen runs would be required to test all factor combinations. However, if we are willing to assume there are no three-factor interactions, we can estimate the main factor effects in just eight runs.

• [x,conf] = fracfact('a b c abc')
  x =
      -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
  conf = 
      '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 first three columns of the x matrix form a full-factorial design. The final column is formed by multiplying the other three. The confounding pattern shows that the main effects for all four factors are estimable, but the two-factor interactions are not. For example, the X1*X2 and X3*X4 interactions are confounded, so it is not possible to estimate their effects separately.

After conducting the experiment, we may find out that the 'ab' effect is significant. In order to determine whether this effect comes from X1*X2 or X3*X4 we would have to run the remaining eight runs. We can obtain those runs by reversing the sign of the final generator.

• fracfact('a b c -abc')
  ans =
      -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
  

Example 2

Suppose now we need to study the effects of eight factors. A full factorial would require 256 runs. By clever choice of generators, we can find a sixteen-run design that can estimate those eight effects with no confounding from two-factor interactions.

• [x,c] = fracfact('a b c d abc acd abd bcd');
  c(1:10,:)
  ans = 
      'Term'     'Generator'    'Confounding'                  
      'X1'       'a'            'X1'                           
      'X2'       'b'            'X2'                           
      'X3'       'c'            'X3'                           
      'X4'       'd'            'X4'                           
      'X5'       'abc'          'X5'                           
      'X6'       'acd'          'X6'                           
      'X7'       'abd'          'X7'                           
      'X8'       'bcd'          'X8'                           
      'X1*X2'    'ab'           'X1*X2 + X3*X5 + X4*X7 + X6*X8'
  
  This confounding pattern shows that the main effects are not confounded with two-factor interactions. The final row shown reveals that a group of four two-factor interactions is confounded. Other choices of generators would not have the same desirable property.
  [x,c] = fracfact('a b c d ab cd ad bc');
  c(1:10,:)
  ans = 
      'Term'     'Generator'    'Confounding'       
      'X1'       'a'            'X1 + X2*X5 + X4*X7'
      'X2'       'b'            'X2 + X1*X5 + X3*X8'
      'X3'       'c'            'X3 + X2*X8 + X4*X6'
      'X4'       'd'            'X4 + X1*X7 + X3*X6'
      'X5'       'ab'           'X5 + X1*X2'        
      'X6'       'cd'           'X6 + X3*X4'        
      'X7'       'ad'           'X7 + X1*X4'        
      'X8'       'bc'           'X8 + X2*X3'        
      'X1*X2'    'ab'           'X5 + X1*X2' 
  

Here all the main effects are confounded with one or more two-factor interactions.

References

[1]  Box, G. A. F., W. G. Hunter, and J. S. Hunter (1978), Statistics for Experimenters, Wiley, New York.

See Also
ff2n, fullfact, hadamard

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