UC Riverside

This thesis is devoted to the study of robust and optimum fractional factorial designs. We consider models that contain the general mean, main effects, and k two-factor interactions for 2^m fractional factorial experiments. We define S_i to be the set of all (1 × m) vectors, with elements 1 and -1 of weight i, where the weight of a vector is the number of nonzero elements in it. We present the robustness property of two classes of designs D={S₀, S₁, S_m-1, S_m} and D₁={S₀, S₁, S₂, S_m} with respect to any t runs as well as a specific set of t runs in the sense that the full estimation capacity of the designs remain when we delete any t runs as well as specific t runs from the designs D and D₁. The number of runs are (2+m) and [2+m+.5(m)(m-1)] in D and D₁ respectively.

We introduce a general structure M for the information matrices of a class of models possibly describing the data from a fractional factorial experiment with m factors each at two levels and n runs. We characterize all the eigenvalues and eigenvectors for such matrices M. For m=4 we establish the robustness property of the design D₇={S₀, S₁, S₃}. The runs of D₇ are contained in design D when m=4. We show all the information matrices from design D₇ and designs obtained from D₇ by deleting some runs are special cases of M.

Let D_T be the class of designs with n runs for estimating the main effects only and let F_T be the class of foldover designs with 2n runs, n runs from T in D_T and another n runs from -T, having full estimation capacity for k=1. We prove that if T* in D_T is E-optimum, the foldover design [T*, -T*] is optimum design with respect to AMCR and GMCR in F_T. Furthermore, if T* is D- and A- optimum with a special structure for X'_1T*X_1T* we prove [T*, -T*] is GD, AD, GT, and AT optimal in F_T.