Open Access Publications from the University of California

Robust and Optimum Fractional Factorial Designs

Abstract

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 2m fractional factorial experiments. We define Si 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, Sm-1, Sm} and D1={S0, S1, S2, Sm} 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 DT be the class of designs with n runs for estimating the main effects only and let FT be the class of foldover designs with 2n runs, n runs from T in DT and another n runs from -T, having full estimation capacity for k=1. We prove that if T* in DT is E-optimum, the foldover design [T*, -T*] is optimum design with respect to AMCR and GMCR in FT. Furthermore, if T* is D- and A- optimum with a special structure for X'1T*X1T* we prove [T*, -T*] is GD, AD, GT, and AT optimal in FT.