Generating alias relationships for two-level Plackett and Burman designs

When the number of runs N in a Plackett and Burman design is a power of two, the design is a 2^{k - p}_III fractional factorial design, and the alias relationships are easily obtained. When N is a multiple of four but not a power of two, the alias relationships are extremely complicated. When only a few factors are expected to be relevant, and if all interactions involving three or more factors are tentatively assumed to be zero, knowledge of the alias relationships is valuable. It is then often possible to disentangle, either completely or partially, the main effects and two-factor interactions of those factors that appear to be of most importance in the initial analysis. In this article, we consider cases for N ≤ 100 and explain how to sequentially construct the alias table for Plackett and Burman designs generated by cyclic generation and foldover, as well as the N = 28 run design generated by block permutation. (This excludes only the N = 52, 76 and 100 designs generated via block permutation and the N = 92 run design, which is of a special construction type.) Applications of these tables are briefly discussed.