ABSTRACT
As we pointed out in Chapter 7, when the number of factors k in a 2k or 3k factorial design becomes large, the number of treatment combinations becomes large and unwieldy. Even when 2k factorial experiments involving such number of factors are performed, there is the cumbersome task of identifying and estimating the k main effects, the kC2 two-factor interactions, the kC3 threefactor interactions, and the k one-factor interaction. There is also the task of calculating their sums of squares for use in ANOVA. The same is true of the 3k factorial design. Resources and convenience may dictate that even a single replicate of the 2k factorial design, involving a large number of factors, is not a feasible proposition. In such situations, fractional replication is used for the study of the factors. An attempt is then made to design an experiment to use a fraction of a single replicate of the 2k or 3k factorial design to elicit information on the effects of the factors on the response of the system. Such studies of a fraction of a 2k or 3k design would eventually lead to the whittling down of the number of factors that should be thoroughly investigated. The theory of fractional replication in these designs is based on the fact that sometimes it is possible to identify a high-order interaction of factors, whose effect is known prior to experimentation to be negligible. This high-order interaction is then used as the identity, I, in the proposed design. If the principal fraction is chosen, all of its contrasts that appear with the + sign are used as the defining relation or defining contrasts. This high-order interaction (I) is used to define the fractional replicate of the design to be carried out in actual experimentation. It is also possible to choose the complementary fraction as the defining relation. In this case, all the members of the relation will have contrasts with a negative sign.
