ABSTRACT
When the number of factors is large, cost and other practical considerations often call for observing a fraction of all the treatment combinations, in particular if only a subset of the factorial effects is expected to be active. Under such a fractional factorial design, first introduced by Finney (1945), not all the factorial effects can be estimated, and follow-up experiments are often needed to resolve ambiguity. In this and the next three chapters, we assume that the experimental units are unstructured and the experiment is to be conducted with complete randomization. Designs of factorial experiments with more complicated block structures will be discussed in Chapters 13 and 14. We introduce in this chapter an important combinatorial structure called orthogonal arrays and describe how they can be used to run factorial experiments. The estimability of factorial effects under an orthogonal array is discussed. It is shown that when some effects are assumed to be negligible, certain other effects can be uncorrelatedly estimated. Several examples are given to illustrate different types of orthogonal arrays and the distinction between regular and nonregular designs. While the emphasis of this book is on regular designs, we briefly discuss a class of nonregular designs derived from Hadamard matrices. We also present the methods of foldover and difference matrices for constructing fractional factorial designs. The chapter ends with a survey of some variants of orthogonal arrays that were introduced in recent years for applications to the design of computer experiments.
