 Open Access Publications from the University of California

## Robust and Optimum Fractional Factorial Designs

• Author(s): Huang, fu ze
• Advisor(s): Ghosh, Subir
• et al.
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.