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## Plackett-Burman designs |

Plackett-Burman (PB) designs (also known as Hadamard matrix designs) are a special case of the fractional factorial design in which the number of runs is a multiple of 4, e.g. 12, 16, 20 or 24. A PB design can require as few as k+1 runs to determine the main effects for k factors although these factors will be heavily confounded with two-factor and higher interactions. As an example, the 11 factor PB design with 12 runs is shown below. For simplicity we have simply used + and — rather than +1 and -1, in accordance with the format of Plackett and Burman's original 1946 paper [PLA1]. The pattern for the first row (or column) determines the entire design. Each subsequent row (or column) is simply the previous row, say, shifted one step to the right, with the final symbol from the previous row being placed at the start of the next row. As such it is simply a cyclical arrangement of the first row (or column). The final row (in the example below) is set to all minus (-).

Run |
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
K |

1 |
+ |
+ |
- |
+ |
+ |
+ |
- |
- |
- |
+ |
- |

2 |
- |
+ |
+ |
- |
+ |
+ |
+ |
- |
- |
- |
+ |

3 |
+ |
- |
+ |
+ |
- |
+ |
+ |
+ |
- |
- |
- |

4 |
- |
+ |
- |
+ |
+ |
- |
+ |
+ |
+ |
- |
- |

5 |
- |
- |
+ |
- |
+ |
+ |
- |
+ |
+ |
+ |
- |

6 |
- |
- |
- |
+ |
- |
+ |
+ |
- |
+ |
+ |
+ |

7 |
+ |
- |
- |
- |
+ |
- |
+ |
+ |
- |
+ |
+ |

8 |
+ |
+ |
- |
- |
- |
+ |
- |
+ |
+ |
- |
+ |

9 |
+ |
+ |
+ |
- |
- |
- |
+ |
- |
+ |
+ |
- |

10 |
- |
+ |
+ |
+ |
- |
- |
- |
+ |
- |
+ |
+ |

11 |
+ |
- |
+ |
+ |
+ |
- |
- |
- |
+ |
- |
+ |

12 |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |
- |

Plackett and Berman included the first line patterns for all cases up to 100 factors at the end of their paper. For example, for 8 runs the pattern is: +++-+--, and for 16 runs, ++++-+-++--+---

As is apparent, this type of design is best suited for problems for which the number of factors is large (perhaps very large, e.g. 20+ factors), two-level (factorial) designs are appropriate, interaction effects are of no real concern, and the number of runs needs to be kept as small as possible. PB designs are discussed in detail in Box et al. (2005, section 7.1, [BOX1]).

References

[BOX1] Box G E P, Hunter J S, Hunter W G (1978, 2005) Statistics for Experimenters: An Introduction to Design, Data Analysis and Model Building. J Wiley & Sons, New York.

[PLA1] Plackett R L, Burman J P (1946) The Design of Optimal Multifactorial Experiments. Biometrika, 33(4), 305-325