Invariance of generalized wordlength patterns

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Abstract

The generalized wordlength pattern (GWLP) introduced by Xu and Wu [2001. Generalized minimum aberration for asymmetrical fractional factorial designs. Ann. Statist. 29, 1066–1077] for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang [2004. Projection justification of generalized minimum aberration for asymmetrical fractional factorial designs. Metrika 60, 279–285] defined the J-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the J-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the J-characteristics are not. We briefly discuss some implications of these results.

Introduction

In a regular fractional factorial design D, the quantities Ai(D)= the number of defining words of length icontain useful information about the design. In particular, the smallest index i for which Ai(D)>0 is the resolution of the design. Moreover, one way of comparing two designs having k factors and equal resolution is to compare their wordlength patterns (A1,A2,,Ak) (Franklin, 1984, Fries and Hunter, 1980). The better design is said to have less aberration.

While nonregular designs no longer have defining words as such, a generalized wordlength pattern (GWLP) can be defined for them combinatorially. This was done for two-level designs by Tang and Deng (1999), and was generalized to arbitrary (possibly mixed-level) designs by Xu and Wu (2001) using group characters.

An intermediate computation in the two-level case gives a set of values that Tang and Deng called J-characteristics (first introduced in Deng and Tang, 1999), and Tang (2001) showed that these numbers completely determine the design D, somewhat analogous to the way that a defining subgroup determines a regular design. Ai and Zhang (2004) generalized this to arbitrary designs by looking closely at the corresponding computation in Xu and Wu (2001).

In defining GWLPs of arbitrary designs, Xu and Wu (2001) assigned to the ith factor the cyclic group Zsi, where si is the number of levels of the factor. While this choice is a computational convenience, it is also arbitrary, and in fact the calculation of the GWLP can be carried through using other abelian groups as well, as we indicate below.

This, however, raises the following question for non-prime si. Since the (irreducible) characters of two groups of equal order will generally be different, does the choice of group affect either the J-characteristics or the GWLP of a given design? Certainly any dependence of the GWLP on an arbitrary choice would raise a serious question about its use in comparing designs using relative aberration. It will be clearly seen that the J-characteristics do depend on this choice. However, perhaps surprisingly, this does not affect the values of the GWLP. That is our main result.

There are many excellent expositions of character theory, such as Isaacs (1976), Ledermann (1987) and Serre (1977). In general, we will mention known results without citation. We will also use a number of results from multilinear algebra (the theory of tensor products). These are collected in Appendix A.

Notation: We will denote the integers by Z, and the integers modulo s by Zs as above. The complex numbers will be denoted by C and complex Euclidean space by Cs. Vectors in Cs will be viewed as columns. The conjugate of zC will be denoted by z¯, the transpose of a vector or matrix by a prime (), and the adjoint (or conjugate transpose) of a matrix or linear transformation A by A*. The inner product of v=[v1,,vs] and w=[w1,,ws]Cs is given byv,w=i=1sviw¯i.The cardinality of a set E will be written |E|.

The Hamming weight of u=(u1,,uk), wt(u), is the number of nonzero components of u. (In Section 2 we will replace “nonzero” by “nonidentity” in order to deal with groups whose identity element is not 0.)

We alert the reader to the fact that we will use G (or Gi) as an index set, with elements g or h. Sometimes such sets will be groups, but often they will be viewed just as sets. We will try to make absolutely clear from context when a result requires a group structure and when it does not.

Section snippets

Definitions

A fractional factorial design on k factors is a multisubset D of a finite Cartesian product G=G1××Gk, that is, the set G with the element g repeated O(g) times, O(g)0. The set Gi indexes the si levels of factor i, and we let s=s1sk. We will refer to O as the counting or multiplicity function of D. The elements (k-tuples) of the design are referred to as runs, and the number of runs in the design, counting multiplicities, isN=|D|=gGO(g).The design D may also be viewed as an orthogonal array

J-characteristics. The character table

We see that the irreducible characters of a finite abelian group G of order s may be written χg1,,χgs, where gi are the elements of G in some order. The values χgi(gj) form the character table of G, the columns of which are mutually orthogonal and of norm s (with respect to the inner product (1)). Another way to say this is that the s×s matrix H formed by this table has the property that H*H=HH*=sI, where H* is the adjoint of H (H is thus a complex Hadamard matrix).

Let G=G1××Gk where each Gi

Independence of group structure

By imposing a group structure on the set G=G1××Gk, we define the irreducible characters χg. We want to show that the numbers wt(g)=j|χg(D)|2,j=1,,k,appearing in (6) are independent of the group structure chosen. This sum is somewhat unwieldy, and so we will break it into smaller sums over elements g which are not only of weight j but also differ from the identity in exactly the same components.

To begin with, we fix an order of the elements in each set Gi, with the understanding that whenever

Conclusion

The definition of the generalized wordlength pattern (GWLP) given by Xu and Wu (2001) makes sense if one chooses abelian rather than cyclic groups to index the levels of each factor. The choice to use cyclic groups there is arbitrary, and we have shown that while it does affect the so-called J-characteristics of a design, it does not affect the GWLP. This removes a possible ambiguity in the definition of the GWLP, and therefore in the use of minimum aberration as an optimality criteria for

Acknowledgments

We thank Dan Lutter for some useful discussions, and the referee and editor for some helpful suggestions.

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