On the Identity Relationship for Fractional Replicates of the $2^n$ Series

Abstract

The paper considers $\frac{1}{2}_r$ fractional replication designs of the factorial series with $n$ factors each at two levels. The identity relationship for such designs is often written in terms of a symbol $I$ and collections of letters which denote interactions among the factors. These collections may conveniently be called "words," and the number of letters in a collection, the "length" of the word. The problem considered is that of the existence of an identity relationship which contains words of specified lengths. It is known that the words of an identity, together with the symbol $I$, form an Abelian group. The group contains sets of independent generators, and the products of such generators. Necessary and sufficient conditions are developed for the existence of an identity relationship for which the lengths of a set of independent generators and their products are specified. Further, it is shown how to construct such an identity relationship, and it is proved that the identity relationship is unique, apart from renaming the letters. For the more general case in which the lengths of the words are given--but are not associated with particular generators and products--a necessary condition is developed for the existence of the identity relationship. It is shown by example that this condition is not sufficient.