Nearly Orthogonal Designs

Orthogonal arrays (OA's) are very useful as factorial designs in experiments. When used as factorial designs, OA's have many nice properties. However, the usefulness of OA's is limited by the crucial condition required for run size. Another problem is that: most of mixed-level OA's are unsaturated, leaving quite a number of degrees of freedom unused. Nearly orthogonal arrays (NOA's) are a new class of factorial designs in the recent 12years, which can be used as good factorial designs when there are not suitable OA's. Although there are already several pieces of work on NOA in the literature, it is still not systematically studied. We studied the problem systematically in this paper.Firstly, we give a rigorous definition for an NOA of strength m, which was not clearly presented in the literature. It depends on two conditions: 1) the m–projection property, and 2) a minimal value, where the m–projection property implies that, for every group of m factors, the design includes a full factorial; is a measure of closeness to orthogonality of strength m, the smaller, the better. For the construction of NOA's, we developed efficient algorithms. For NOA's of small run sizes, we developed an algorithm of complete search to guarantee the designs constructed being global optimal under our definition. For NOA's of mediate or large run sizes, we developed an algorithm of finite search with many random starts. In the algorithm, we used many new computational skills, such as the columnwise–pairwise skill, the threshold accepting skill, etc. These algorithms are of high efficiency. Based on the definition, and using the algorithms, we constructed many NOA's of strength 2 or 3, with small or mediate run sizes. Among the NOA's constructed, the most useful ones are those with two and three level factors. Most of the NOA's of strength 2 are saturated with a small number of them being supersaturated. The NOA's of strength 3 have a nice property similar to OA's of strength 3. That is: every sub-model of three factors can be fully analyzed. Moreover, they can be transformed into NOA of strength 2 including high level factors. Three-level NOA's are constructed so that, together with the three widely used OA's of 9, 18, and 27 runs, they form a series of three-level designs with run sizes differing from each other by only 3. This offers experimenters much more choices in selecting three-level designs of suitable sizes. We also constructed NOA's including small number of high level factors, which may occasionally be used.