# Plackett-Burman design

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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. [R.L. Plackett and J.P. Burman, "The Design of Optimum Multifactorial Experiments", "Biometrika" 33 (4), pp. 305-25, June 1946] 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 in which each combination of levels for any pair of factors appears the same number of times. 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 4 times an integer. In particular, it worked for all such N up to 100 except N=92. 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. [R. C. Bose & K. Kishen, "On the problem of confounding in the general symmetrical factorial design", "Sankhya" 5, 21 (1940)] 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.

When interactions between factors are not negligible, they are often confounded in Plackett-Burman designs with the main effects, meaning that the designs do not permit one to distinguish between certain main effects and certain interactions. This is called aliasing or confounding.

References

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