Some problems in the theory and construction of factorial designs.
Qu, Xianggui
2002
Abstract
At the beginning of an investigation there may be many conceivably important factors. It is often reasonable to assume that only a few of them are important, but their identities are not known. Therefore, screening designs are needed. Screening designs can be broadly classified into two categories: orthogonal and nonorthogonal designs. Nonorthogonal designs can be popular if run size economy and flexibility with respect to level combinations are desired. One important sub-category of nonorthogonal designs are the one-factor-at-a-time (OFAT) designs. Strict and standard OFAT designs are studied in this thesis. Besides discussing the construction and statistical properties of OFAT designs of resolution III and IV, OFAT designs of resolution V are constructed for the first time. All the main effects and two-factor interactions (2fi's) are estimable in these designs. Strict OFAT designs of resolution V are saturated and standard OFAT designs have two more runs than saturated designs. The constructed designs have attractive features such as run size economy and robustness to premature termination. They are useful in computer modeling as well as in physical experiments. Comparisons with other designs show that, if the response error is small, there are no particular disadvantages in running experiments using OFAT designs. A fundamental and practically important question in the theory of fractional factorial designs is the issue of optimal factor assignment to columns of the design matrix. A new criterion called maximum estimability (maxest) is proposed as a solution to this problem for nonorthogonal designs as well as for regular and nonregular designs. The maxest criterion is a refinement of Webb's resolution and can further distinguish designs of the same resolution by using the numbers of clear or strongly clear main effects and 2fi's. It also extends the maximum resolution and the minimum aberration criteria for regular designs. For nonregular orthogonal or nonorthogonal designs, the maxest criterion focuses on low-order effects and is coding-dependent, which is different from other criteria such as the various generalized minimum aberration criteria and the minimum moment criterion. As an application, the maxest criterion is used to study the projections of nonregular two-level, three-level and mixed-level designs such as the Plackett-Burman and related designs, <italic>OA</italic>(18, 3<super>7</super>, 2), <italic> OA</italic>(36, 3<super>12</super>, 2), <italic>OA</italic>(18, 2<super>1 </super>3<super>7</super>, 2) and <italic>OA</italic>(36, 3<super>12</super>2<super> 11</super>, 2). Projections of these designs are classified using the estimability vector associated with the maxest criterion and new geometric projections are discovered. Non-isomorphic designs are compared and ranked in terms of their projection properties. Their rankings under the maxest criterion are different from those under other criteria. In comparison with other classifications, the proposed classifications have fewer classes and provide more insight for statistical modeling.Subjects
Aberration Criteria Construction Factorial Designs Nonorthogonal Plackett-burman Problems Screening Designs Some Theory
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