Optimal blocking schemes and projection properties for fractional factorial designs.

Abstract

This dissertation documents the results of studies of two research topics for fractional factorial designs: (i) Blocking is commonly used in design of experiments to reduce the systematic variation and increase the precision of estimation of effects. For 2<italic><super>n -- k</super></italic> designs in 2<italic><super>p</super></italic> blocks, Sun, Wu and Chen (1997), and Sitter, Chen and Feder (1997) studied the aliasing and confounding patterns between blocking factors and treatment factors and proposed criteria to choose optimal blocking schemes. Their criteria are primarily based on the lengths of the words in the defining contrast subgroup. We point out some weakness of their criteria and suggest several modifications which allow more lower-order effects to be estimable. Extensions to 3-level fractional factorial designs are also considered. (ii) For studying quadratic response surfaces, a prevailing experimental plan in the literature is the central composite design, proposed by Box and Wilson (1951). In their paper, it was also shown that the 3<italic><super>n -- k</super></italic> designs are inappropriate for quadratic response surface study because of their uneconomical run size. We take a different approach and propose, for the 3<italic><super>n -- k</super></italic> designs, a <italic>two-stage data analysis strategy</italic>, i.e., first screening and then fitting quadratic response surfaces on the projected design of the factors identified as important in screening. Under the strategy, the run size of the 3<italic><super>n -- k</super></italic> designs can be utilized efficiently for two objectives, screening and fitting quadratic response surfaces, <italic>without</italic> adding more runs. Based on this analysis strategy, we study projection properties of the 3<italic><super> n -- k</super></italic> designs. The projected designs are classified into different types in terms of combinatorial isomorphism and model isomorphism. Then, they are compared with some second-order designs, such as the central composite designs, in terms of <italic>D</italic>- and <italic>G</italic>-efficiencies. In addition to the 3<italic><super>n -- k</super></italic> designs, we also study the projection properties of some nonregular three-level designs, such as the <italic>OA</italic>(18, 3<super>7</super>) and the <italic>OA </italic>(36, 3<super>12</super>) . These three-level designs are evaluated according to an eligible projection criterion, which is based on the total of projected designs that can be used to fit quadratic response surfaces. A new method to construct designs that can significantly improve the eligible projection property of the 3<italic><super>n -- k</super></italic> designs is also presented.