Optimal Designs of Two-Level Factorials when N ≡ 1 and 2 (mod 4) under a Baseline Parameterization

This work considers two-level factorial designs under a baseline parameterization where the two levels are denoted by 0 and 1. Orthogonal parameterization is commonly used in two-level factorial designs. But in some cases the baseline parameterization is natural. When only main effects are of interest, such designs are equivalent to biased spring balance weighing designs. Commonly, we assume that the interactions are negligible, but if this is not the case then these non-negligible interactions will bias the main effect estimates. We review the minimum aberration criterion under the baseline parameterization, which is to be used to compare the sizes of the bias among different designs.We define a design as optimal if it has the minimum bias among most efficient designs. Optimal designs for N ≡ 0 (mod 4), where N is the run size, were discussed by Mukerjee & Tang (2011). We continue this line of study by investigating optimal designs for the cases N ≡ 1 and 2 (mod 4). Searching for an optimal design among all possible designs is computationally very expensive, except for small N and m, where m is the number of factors. Cheng’s (2014) results are used to narrow down the search domain. We have done a complete search for small N and m. We have found that one can directly use Cheng’s (2014) theorem to find an optimal design for the case N ≡ 1 (mod 4). But for the case N ≡ 2 (mod 4), a small modification is required.