Plackett-Burman Designs¶
Plackett-Burman designs
In 1946, R.L. Plackett and J.P. Burman published their now famous paper "The Design of Optimal Multifactorial Experiments" in Biometrika (vol. 33). This paper described the construction of very economical designs with the run number a multiple of four (rather than a power of 2). Plackett-Burman designs are very efficient screening designs when only main effects are of interest.
These designs have run numbers that are a multiple of 4
Plackett-Burman (PB) designs are used for screening experiments because, in a PB design, main effects are, in general, heavily confounded with two-factor interactions. The PB design in 12 runs, for example, may be used for an experiment containing up to 11 factors.
12-Run Plackett-Burman design
TABLE 3.18: Plackett-Burman Design in 12 Runs for up to 11 Factors
| Pattern | X1 | X2 | X3 | X4 | X5 | X6 | X7 | X8 | X9 | X10 | X11 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | +++++++++++ | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 | +1 |
| 2 | -+-+++---+- | -1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 |
| 3 | --+-+++---+ | -1 | -1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 |
| 4 | +--+-+++--- | +1 | -1 | -1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 | -1 |
| 5 | -+--+-+++-- | -1 | +1 | -1 | -1 | +1 | -1 | +1 | +1 | +1 | -1 | -1 |
| 6 | --+--+-+++- | -1 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | +1 | +1 | -1 |
| 7 | ---+--+-+++ | -1 | -1 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | +1 | +1 |
| 8 | +---+--+-++ | +1 | -1 | -1 | -1 | +1 | -1 | -1 | +1 | -1 | +1 | +1 |
| 9 | ++---+--+-+ | +1 | +1 | -1 | -1 | -1 | +1 | -1 | -1 | +1 | -1 | +1 |
| 10 | +++---+--+- | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 | -1 | +1 | -1 |
| 11 | -+++---+--+ | -1 | +1 | +1 | +1 | -1 | -1 | -1 | +1 | -1 | -1 | +1 |
| 12 | ----------- | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
No defining relation
These designs do not have a defining relation since interactions are not identically equal to main effects. With the \(2_{III}^{k-p}\) designs, a main effect column \(X_i\) is either orthogonal to \(X_i X_j\) or identical to plus or minus \(X_i X_j\). For Plackett-Burman designs, the two-factor interaction column \(X_i X_j\) is correlated with every \(X_k\) (for \(k\) not equal to \(i\) or \(j\)).
Economical for detecting large main effects
However, these designs are very useful for economically detecting large main effects, assuming all interactions are negligible when compared with the few important main effects.