The Construction of Saturated $2^{k-p}_R$ Designs

Abstract

If a $2^{k - p}_R$ design, of fixed resolution $R$ and specified number of runs $2^q$, accommodates the maximum possible number of variables, we say that it is saturated. In this paper, we develop a method for constructing saturated designs and apply it to an example. We first show that when $R$ is odd, the set of all distinct $2^{k - p}_R$ designs (where $q = k - p$ is specified) can be obtained easily from a particular class of $2^{(k + 1) - p}_{R + 1}$ designs. We then develop a stage by stage method for constructing this "parent" class of designs of (even) resolution $R + 1$. This class is shown, incidentally, to contain a saturated design. The complete set of $2^{k - p}_R$ designs, which naturally includes all saturated $2^{k - p}_R$ designs, can then be obtained at once. The problem of arranging the designs constructed into blocks of runs, so that the blocked designs have certain desirable confounding properties, is also investigated, and a method for obtaining optimal blocking arrangements is given. As an important part of our method, a "sequential conjecture" procedure is developed and utilized to test the equivalence of any two designs. These procedures have been programmed for the computer, and are illustrated by the example $R = 5, q = 7$.