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## Fractional Factorial designs |

The previous section described the design of full factorial experiments, but noted that even for two-level factors the number of runs required can become excessive in a complete design. One solution to this problem is to only conduct a fraction of the full factorial design, for example one half or one quarter of the full set. Clearly this will result in some information being lost, or hidden in the analysis, but by careful selection of the runs that are included the information loss can be confined to higher order interactions, which are then confounded with the results for other effects. Using the same example as above, if we only had half of the runs available we might select runs 1, 4, 6 and 7 from the table below, for our design. If we then wished to estimate the main effect for factor A we would take the difference between the average of the results for runs 4 and 6 and that obtained for the runs 1 and 7. The figure obtained will generally be similar to, but not the same as, that obtained from a full factorial run of the same experiment.

RUN |
A |
B |
C |
---|---|---|---|

1 |
-1 |
-1 |
-1 |

2 |
+1 |
-1 |
-1 |

3 |
-1 |
+1 |
-1 |

4 |
+1 |
+1 |
-1 |

5 |
-1 |
-1 |
+1 |

6 |
+1 |
-1 |
+1 |

7 |
-1 |
1 |
+1 |

8 |
+1 |
+1 |
+1 |

A power-of-two fractional factorial design that is based on two levels can be denoted by the expression: 2k-f runs, so if f=1 and k=3, the notation 23-1 means that it is a fractional run with half of the number of runs of the full case. This raises the question as to how one should produce fractional designs. The method used is essentially the same as that shown for blocking of a full factorial design in the previous section. In the example above, we select one factor, C say, which we are prepared to accept can be confounded with two-factor interactions of A and B (i.e. terms in AB or x1x2). To do this we replace the column containing C with the product AB, to give:

RUN |
A |
B |
AB or C |
---|---|---|---|

1 |
-1 |
-1 |
+1 |

2 |
+1 |
-1 |
-1 |

3 |
-1 |
+1 |
-1 |

4 |
+1 |
+1 |
+1 |

Note that only the first four rows are included, since these are repeated in the former table. We now have a design comprised of 4 runs, with settings for factors A,B and C in which factors A and B are fully determined but factor C is confounded with the AB interaction effect. As long as this interaction effect is small, relative to the main effect C, then the design will be almost as efficient as a full factorial for the same set of factors. As discussed earlier, this design can be randomized, replicated, and have center points added to the set of runs used.

The term design resolution is used to indicate the nature of the confounding in factional factorial designs. The higher the resolution value the higher the level of interactions that are confounded, but also the larger the number of runs required. A level III resolution indicates that a level 1 factor (main effect) is confounded with at least one two-way interaction (1+2=3), whilst a level IV resolution design will normally only have level 2 interactions confounded or level 1 effects confounded with level 3 interactions. Level III designs and Plackett-Burman designs tend to be used for screening purposes (focusing on main effects) whilst level IV and V designs are used more for investigations of interaction effects.