The performance of orthogonal arrays with adjoined or unavailable runs
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Orthogonal arrays are a class of fractional factorial designs that are optimal according to a range of optimality criteria. This makes it tempting to construct fractional factorial designs by adjoining additional runs to an OA, or by removing runs from an OA, when the number of runs available for the experiment is only slightly larger or smaller than the number in the OA. In this thesis we examine the performance of OA plus p runs designs and OA minus t runs designs in the context of D-optimality and model-robustness. Although we attempt to make general observations where possible, our primary goal is to inform the use of quantitative factors, hence we focus on factors at more than two levels to allow for some curvature in the model. We begin by considering the performance of these designs under a main effects only model, and show that optimality depends only on the pairwise Hamming distance of the adjoined (or removed) runs. We present an algorithm for finding optimal Hamming distances and provided general methods for constructing optimal sets of runs once the optimal pairwise Hamming distances have been identified. In order to consider the performance of these designs when interaction terms are included in the model, it transpired that we require a complete set of geometrically non-isomorphic designs to study. Thus, we enumerate all geometric isomorphism classes for symmetric OAs with ternary factors and 18 runs, and prove that these classes cover the entire OA[18; 3m] space. We then consider the inclusion of a subset of linear-by-linear interactions in the model, and derive matrices to be optimised under this setting. We conduct an empirical study on the OA[18; 3m] we have enumerated and give examples of D-optimal and model-robust designs for each of the design spaces, that is, for each of m = 3, 4, 5, 6 and 7.
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