5. Process Improvement
5.3. Choosing an experimental design
5.3.3. How do you select an experimental design?

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-
Burnam 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
Saturated Main Effect designs PB designs also exist for 20-run, 24-run, and 28-run (and higher) designs. With a 20-run design you can run a screening experiment for up to 19 factors, up to 23 factors in a 24-run design, and up to 27 factors in a 28-run design. These Resolution III designs are known as Saturated Main Effect designs because all degrees of freedom are utilized to estimate main effects. The designs for 20 and 24 runs are shown below.
20-Run Plackett-
Burnam design
TABLE 3.19: A 20-Run Plackett-Burman Design
X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19
1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 +1
2 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1
3 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1
4 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1
5 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1
6 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1
7 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1
8 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1
9 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1
10 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1
11 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1 +1
12 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1 -1
13 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1 +1
14 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1 +1
15 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1 +1
16 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1 +1
17 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1 -1
18 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1 -1
19 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1 +1
20 +1 -1 -1 +1 +1 +1 +1 -1 +1 -1 +1 -1 -1 -1 -1 +1 +1 -1 -1
24-Run Plackett-
Burnam design
 X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11 X12 X13 X14 X15 X16 X17 X18 X19 X20 X21 X22 X23 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 3 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 4 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 5 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 6 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 7 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 8 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 9 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 10 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 11 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 12 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 1 13 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 -1 14 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 -1 15 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 1 16 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 17 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 18 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 19 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 20 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 21 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 22 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 23 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 1 24 1 1 1 1 -1 1 -1 1 1 -1 -1 1 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 Xi is either orthogonal to XiXj or identical to plus or minus XiXj. For Plackett-Burman designs, the two-factor interaction column XiXj is correlated with every Xk (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.