The Efficiencies of Fractional Factorial Designs

Abstract

A frequently stated advantage of fractional-factorial (FF) designs over one-factor-at-a-time (1FAT) designs is their high relative efficiency. We study k-factor, 2k-run designs, where k is a power of 2 or is divisible by 4, for which the usually stated relative efficiency is k in favor of the resolution IV FF design over an orthogonal 1FAT design. We examine other ways to measure efficiency under main-effects–only modeling in the case when all factors are quantitative, which enables rescaling of their settings. If both designs are restricted to lie in smallest-sized hypercubes or hyperspheres of equal volume, then the relative efficiency of the FF design is 1 under the full model (k effects active). We also show that the designs are identical in an important sense: The 1FAT design can be rescaled and rotated to become a resolution IV FF design. The primary advantage of the FF design lies in its projection properties; if at most k′ factors are expected to be active, then, under equal-volume designs, the efficiency of the FF is at least k/k′. Next, taking a size-of-effects–based approach, we rescale the FF by restricting the design to have the same probability as a 1FAT design of producing too-large swings in the response (i.e., swings that produce unusable runs). Here the efficiency of the FF usually remains greater than 1, but it can reach 1/k in the limit. We show how these ideas may be used to help scale back originally planned settings in a fractional factorial design to reduce the chance of unusable runs while still maintaining the other advantages of FF designs.