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.
[5] The resulting matrix, minus that column, is a "supersaturated design"[6] for finding significant first order effects, under the assumption that few exist.
Box–Behnken designs can be made smaller, or very large ones constructed, by replacing the fractional factorials and incomplete blocks traditionally used for plan and seed matrices, respectively, with Plackett–Burmans.
For example, a quadratic design for 30 variables requires a 30 column PB plan matrix of zeroes and ones, replacing the ones in each line using PB seed matrices of −1s and +1s (for 15 or 16 variables) wherever a one appears in the plan matrix, creating a 557 runs design with values, −1, 0, +1, to estimate the 496 parameters of a full quadratic model.
To estimate the 105 parameters in a quadratic model of 13 variables, one must formally exclude from consideration or compute |X'X| for well over 106C102, i.e. 313C105, or roughly 10484 matrices.