Fractional Factorial Designs With Admissible Sets of Clear Two-Factor Interactions

摘要

We consider the problem of selecting two-level fractional factorial designs that allow joint estimation of all main effects and some specified two-factor interactions (2fi's) without aliasing from other 2fi's. This problem is to find, among all 2~(m-p) designs with given m and p, those resolution IV designs whose sets of clear 2fi's contain the specified 2fi's as subsets. We use a linear graph to represent the set of clear 2fi's for a resolution IV design, where each line connecting two vertexes represents a clear 2fi between the factors represented by the two vertexes. We call a 2~(m-p) resolution IV design admissible if its graph is not isomorphic to any proper subgraph of the graph of any other 2~(m-p) resolution IV design. We show that all even resolution IV designs are inadmissible. We then use a classical subgraph-isomorphism algorithm to determine all admissible designs of 32, 64, and 128 runs. This leads to a concise catalog of all admissible designs of 32 and 64 runs, and a lengthy but substantially reduced list (compared to the number of non-isomorphic designs) for 128 runs. All admissible designs of 128 runs are included in supplementary materials that are available online. A file of R code is available on the first author's website for generating the actual design tables for any 2~(m-p) designs of 32, 64, and 128 runs.