Construction of E(s2) optimal supersaturated designs using cyclic BIBDs☆

https://doi.org/10.1016/S0378-3758(00)00136-1Get rights and content

Abstract

In this paper formulas for computing the E(s2) values of some kinds of E(s2) optimal supersaturated designs are given, and a general algorithm of constructing E(s2) optimal supersaturated designs from cyclic BIBDs is proposed. Within this class of designs, by further discriminating the pairwise correlations, efficient designs of runs from 6 to 24 are constructed and tabulated. Comparisons with other existing designs are made at last, demonstrating the effectiveness of our method.

Introduction

Recently, there has been increasing interest in the study of supersaturated designs. Supersaturated designs are factorial designs in which the number of main effects is greater than the number of experimental runs. Such designs are helpful when experimentation is expensive and the number of factors is large. In practice, the data collected by supersaturated designs are analyzed under the assumption of effect sparsity (Box and Meyer, 1986), i.e., a few dominant factors actually affect the response.

Let X be an n×m design matrix of a supersaturated design with n runs (rows) and m 2-level factors (columns) each with the same number of +1's and −1's (m>n−1). All the columns of X are distinct. Let sij be the (i,j)th entry of X′X. Booth and Cox (1962), in the first systematic construction of supersaturated designs, proposed the following criterion for comparing designs:E(s2)=∑1⩽i<j⩽msij2m2.Because it is a measure of non-orthogonality under the assumption that only two out of the m factors are active, it should be minimized, i.e., an E(s2) optimal design minimizes the E(s2) value over all possible designs.

After Booth and Cox (1962), the subject of supersaturated designs remained dormant until the appearance of Lin (1993). Other recent works include, e.g., Wu (1993), Lin (1995), Nguyen (1996), Tang and Wu (1997), Li and Wu (1997) and Cheng (1997). Lin (1993) used half fractions of Hadamard matrices (HFHM) to construct supersaturated designs of size (n,m)=(2t,4t−2). Nguyen (1996) described a method of constructing supersaturated designs from cyclic BIBDs that is a generalization of the method of Lin (1993). And Tang and Wu (1997) proposed a general method also through the use of Hadamard matrices. Then Cheng (1997) gave a unified treatment of Tang and Wu's (1997) optimality result and the optimality of HFHM.

We recall that a balanced incomplete block design, denoted BIBD(v,b,r,k,λ), is an arrangement of v treatments into b blocks of size k, where k<v, such that each treatment appears in r blocks, and every pair of treatments appear together in λ blocks.

In this paper, the equivalence of the existence of an E(s2) optimal supersaturated design of size (n,m)=(2t,c(2t−1))(t⩾3,c⩾2, and when t is odd, c is even) and that of a BIBD(2t−1,c(2t−1),c(t−1),t−1,c(t−2)/2) is introduced firstly, and formulas for computing the E(s2) values of E(s2) optimal supersaturated designs of size (n,m)=(2t,c(2t−1)±e)(e=1 and 2, and when t is odd, c is even) are given in Section 2. In Section 3, we present an algorithm of constructing E(s2) optimal supersaturated designs from cyclic BIBDs, and designs of runs from 6 to 24 for different factors are constructed and tabulated. Comparisons with other existing designs are made in Section 4, demonstrating the effectiveness of our method.

Section snippets

E(s2) optimal designs and their E(s2) values

It is known that for any supersaturated design X, by changing the signs of all the entries in the same column if necessary, we may normalize it so that all the entries of its first row are equal to 1. So without loss of generality, we denote X=(1m′X1), where 1m′ is a row vector of m 1's. All the X's below are of this form.

Here are some existing results from Nguyen (1996) and Cheng (1997).

Lemma 2.1

(i) For any m⩾n and the design X, we haveE(s2)⩾n2(m−n+1)/((n−1)(m−1)).

(ii) For m=c(n−1), if there exists X∗

Constructing designs from cyclic BIBDs

For given (n,m),E(s2) optimal supersaturated designs are not unique. Tang and Wu (1997) had made further discrimination within E(s2) optimal designs by minimizing the pairwise correlations. Here, as there are different BIBDs with the same parameters, in order to select one to construct an E(s2) optimal supersaturated design, we introduce the following criterion for further discrimination.

Definition 3.1

For a supersaturated design X of size (n,m), let frmax(X)=∑1⩽i<j⩽mI(|rij|=rmax)/(m2), where rmax=max1⩽i<j⩽m|r

Comparisons with other methods

In Section 3, supersaturated designs for different t's and c's are constructed and their initial blocks are listed. Note that the designs whose m values are marked with “b” in the tables above are all new designs, and the majority of these new designs have rmax⩾0.5. These designs are more of theoretical interest than of practical use. For existing designs, those of Lin (1993), Nguyen (1996) and Tang and Wu (1997) are all E(s2) optimal, and those of Lin (1993) and Nguyen (1996) also have the

Acknowledgements

The authors cordially thank Professor C.F. Jeff Wu, Doctor Boxin Tang and Doctor William W. Li for sending us related materials. And we are grateful to the associate editor and the referees for their valuable comments and suggestions on our paper.

References (11)

  • K.H.V. Booth et al.

    Some systematic supersaturated designs

    Technometrics

    (1962)
  • G.E.P. Box et al.

    An analysis for unreplicated fractional factorial

    Technometrics

    (1986)
  • C.-S. Cheng

    E(s2)-optimal supersaturated designs

    Statist. Sinica

    (1997)
  • A. Dey

    Theory of Block Designs.

    (1986)
  • J.A. John

    Cyclic Designs.

    (1987)
There are more references available in the full text version of this article.

Cited by (0)

☆

This work was supported by NNSF project 19771049 of China.

View full text
Copyright © 2000 Elsevier Science B.V. All rights reserved.