Plackett–Burman design

Plackett–Burman designs are experimental designs presented in 1946 by Robin L. Plackett and J. P. Burman while working in the British Ministry of Supply. Their goal was to find experimental designs for investigating the dependence of some measured quantity on a number of independent variables (factors), each taking L levels, in such a way as to minimize the variance of the estimates of these dependencies using a limited number of experiments. Interactions between the factors were considered negligible. The solution to this problem is to find an experimental design where each combination of levels for any pair of factors appears the same number of times, throughout all the experimental runs (refer to table). A complete factorial design would satisfy this criterion, but the idea was to find smaller designs.

For the case of two levels (L=2), Plackett and Burman used the method found in 1933 by Raymond Paley for generating orthogonal matrices whose elements are all either 1 or -1 (Hadamard matrices). Paley's method could be used to find such matrices of size N for most N equal to a multiple of 4. In particular, it worked for all such N up to 100 except N = 92. If N is a power of 2, however, the resulting design is identical to a fractional factorial design, so Plackett–Burman designs are mostly used when N is a multiple of 4 but not a power of 2 (i.e. N = 12, 20, 24, 28, 36 …). If one is trying to estimate less than N parameters (including the overall average), then one simply uses a subset of the columns of the matrix.

For the case of more than two levels, Plackett and Burman rediscovered designs that had previously been given by Raj Chandra Bose and K. Kishen at the Indian Statistical Institute. Plackett and Burman give specifics for designs having a number of experiments equal to the number of levels L to some integer power, for L = 3, 4, 5, or 7.

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