The Abelian group structure of full factorial designs

This study demonstrates formal connections between Abelian group theory and factorial experimental designs. I begin by showing that full-factorial designs are isomorphic to elementary p-groups. I continue with an analysis of the general group and subgroup structure of 3 × 3 full-factorial designs commonly utilized in detection studies, followed by extensions to more general but related designs in the subsequent section, namely, 3 × 3 × 3 designs. I show that the subgroup lattice of 3 × 3 designs decompose the experiment into single and double-target subgroups and a target-absent subgroup. An application of this endeavor is the identification of experimental strata, interactions, interaction contrasts, and fractional designs. By using examples from the reaction-time (RT) and non-parametric modeling literature, it is shown how comparisons across and within these subgroup structures can readily be used to derive predictions about human information processing. Relatedly, I provide predictions derived from subgroup analyses to increase statistical power and simplify data analyses. Finally, some suggestions are provided for analyzing more general designs.