Factorial and fractional factorial designs with randomization restrictions - a projective geometric approach

Two-level factorial and fractional factorial designs have played a prominent role in the theory and practice of experimental design. Though commonly used in industrial experiments to identify the significant effects, it is often undesirable to perform the trials of a factorial design (or, fractional factorial design) in a completely random order. Instead, restrictions are imposed on the randomization of experimental runs. In recent years, considerable attention has been devoted to factorial and fractional factorial plans with different randomization restrictions (e.g., nested designs, split-plot designs, split-split-plot designs, strip-plot designs, split-lot designs, and combinations thereof). Bingham et al. (2006) proposed an approach to represent the randomization structure of factorial designs with randomization restrictions. This thesis introduces a related, but more general, representation referred to as randomization defining contrast subspaces (RDCSS). The RDCSS is a projective geometric formulation of randomization defining contrast subgroups (RDCSG) defined in Bingham et al. (2006) and allows for theoretical study. For factorial designs with different randomization structures, the mere existence of a design is not straightforward. Here, the theoretical results are developed for the existence of factorial designs with randomization restrictions within this unified framework. Our theory brings together results from finite projective geometry to establish the existence and construction of such designs. Specifically, for the existence of a set of disjoint RDCSSs, several results are proposed using (t-1)-spreads and partial (t-1)-spreads of PG(p-1; 2). Furthermore, the theory developed here offers a systematic approach for the construction of two-level full factorial designs and regular fractional factorial designs with randomization restrictions. Finally, when the conditions for the existence of a set of disjoint RDCSSs are violated, the data analysis is highly influenced from the overlapping pattern among the RDCSSs. Under these circumstances, a geometric structure called star is proposed for a set of (t-1)-dimensional subspaces of PG(p-1; q), where 1