Because full factorial design experiments are often time- and cost-prohibitive when a number of treatment factors are involved, many people choose to use partial or *fractional* factorial designs. These designs evaluate only a subset of the possible permutations of factors and levels. Generally, a fractional factorial design looks like a full factorial design for fewer factors, with extra factor columns added (but no extra rows).

Using fractional factorial design makes experiments cheaper and faster to run, but can also obfuscate interactions between factors.

For a good explanation of the reasoning behind using fractional factorial design, see Thomas B Barker’s introductory video:

Fractional designs are typically only used for experiments with two-level factors, as designing experiments with more than two levels per factor can get difficult and messy quite fast.

## Fractional factorial design notation

The standard notation for fractional factorial designs is *l*^{k − p}, where:

*l*is the number of levels in each treatment factor.*k*is the number of treatment factors.*p*is the number of interactions that are confounded.

The fraction of trials required is calculated using this formula: 1/(*l*^{p}).

For example, an experiment with two levels per treatment factor and two confounded interactions would require 1/(2^{2}) or 1/4 of the trials required for a full factorial design.

The notation used for the specific combination of factors being tested in a trial uses letters to designate the *high* (or second) level of a specific factor.

For example, in a three-factor experiment, we’d use *a* for the first factor, *b* for the second, and *c* for the third.

In a trial where *a *is at its high level and *b* and *c* are at their low levels, we’d use the notation ** a**.

In a trial where both *a *and* b* are at their high levels and *c* is at its low level, we’d use the notation ** ab**.

When creating a table of trial factors and levels, the low (or first) level is designated with a minus sign, and the high (or second) level is designated with a plus sign. This becomes important when we start generating the fractional design itself.

## Generating the fractional design for an experiment

When we create a fractional factorial design from a full factorial design, the first step is to decide on an alias structure.

Let’s look at a fairly simple experiment model with four factors. We know that to run a full factorial experiment, we’d need at least 2 x 2 x 2 x 2, or 16, trials. That’s too many, so we decide to *confound* one factor. That gives us values of:

*l* = 2

*k* = 4

*p *= 1

This experiment is classed as a *2 ^{4-1}* fractional factorial design.

If the experiment had only three factors, the (full factorial) design table would look like this:

Trial | Factor A | Factor B | Factor C | Notation |
---|---|---|---|---|

1 | – | – | – | (1) |

2 | + | – | – | a |

3 | – | + | – | b |

4 | + | + | – | ab |

5 | – | – | + | c |

6 | + | – | + | ac |

7 | – | + | + | bc |

8 | + | + | + | abc |

Our fractional 4-factor design uses the same table with the same number of trials, but with an extra factor, D. To keep the number of trials the same, we’ll use a level of D in each trial that corresponds with the levels used in factor A and factor B, combined. This is where the plus and minus signs come in! We’ll use this simple formula:

D = A x B

So if A is on its low level (-) and B is on its high level (+), then (remembering back to grade-school math – a negative multiplied by a positive) D will be (-).

The experiment design table then looks like this:

Trial | Factor A | Factor B | Factor C | Factor D | Notation |
---|---|---|---|---|---|

1 | – | – | – | + | d |

2 | + | – | – | – | a |

3 | – | + | – | – | b |

4 | + | + | – | + | abd |

5 | – | – | + | + | cd |

6 | + | – | + | – | ac |

7 | – | + | + | – | bc |

8 | + | + | + | + | abcd |

If we had more treatment factors that we needed to account for, but we still wanted to keep the 8-trial model, we could introduce confounding along the lines of:

E = A x C

F = A x B x C

The Engineering Statistics Handbook has a handy summary table of fractional factorial designs.

## Six Sigma Black Belt Certification Design of Experiment Questions:

**Question: **If the number of runs required for a full-factorial experiment is cost-prohibitive, which of the following experiments would have the same number of variables but fewer runs? (Taken from ASQ sample Black Belt exam.)

(A) Completely randomized factorial

(B) Replicated factorial

(C) Multilevel factorial

(D) Fractional factorial

**Answer:** (d) Fractional factorial. We can eliminate (a) as randomizing keeps all of the runs. Reviewing the Design of Experiments terminology, the other options make little sense. Also see the Design of Experiments study guide.

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