Constructing Two-Level Designs by Concatenation of Strength-3 Orthogonal Arrays

Two-level orthogonal arrays of <i>N</i> runs, <i>k</i> factors and a strength of 3 provide suitable fractional factorial designs in situations where many of the main effects are expected to be active, as well as some two-factor interactions. If they consist of <i>N</i>/2 mirror image pairs, these designs are fold-over designs. They are called even and provide at most <i>N</i>/2 − 1 degrees of freedom to estimate interactions. For <i>k</i> &lt; <i>N</i>/3 factors, there exist strength-3 designs that are not fold-over designs. They are called even-odd designs and they provide many more degrees of freedom to estimate interactions. For <i>N</i> ≤ 48, attractive even-odd designs can be extracted from complete catalogs of strength-3 orthogonal arrays. However, for larger run sizes, no complete catalogs exist. In order to construct even-odd designs with <i>N</i> &gt; 48, we develop an algorithm for an optimal concatenation of strength-3 designs involving <i>N</i>/2 runs. Our approach involves column permutations of one of the concatenated designs, as well as sign switches of the elements of one or more columns of that design. We illustrate the potential of the algorithm by generating two-level even-odd designs with 64 and 128 runs involving up to 33 factors, because this allows a comparison with benchmark designs from the literature. With a few exceptions, our even-odd designs outperform or are competitive with the benchmark designs in terms of the aliasing of two-factor interactions and in terms of the available degrees of freedom to estimate two-factor interactions.