Another Look at Projection Properties of Hadamard Matrices

Abstract

Suppose a large number of factors (q) is examined in an experimental situation. It is often anticipated that only a few (k) of these will be important. Usually, it is not known which of the q factors will be the important ones, that is, it is not known which k columns of the experimental design will be of further interest. Screening designs are useful for such situations. It is of practical interest for a given k to know all the classes of inequivalent projections of the design into the k dimensions that have certain statistical properties, since it helps experimenters in selecting a screening design with favorable properties. In this paper we study all the classes of inequivalent projections of certain two-level orthogonal arrays that arise from Hadamard matrices, using well known statistical criteria, such as generalized resolution and generalized minimum aberration. We also pay attention to each projection's distinct runs. Results are given for orthogonal arrays with n = 16, 20 and 24 runs, when they are projected into small dimensions. Useful remarks on design selection are made based on the frequency of appearance of each projection design in every Hadamard matrix.

Key Words:

• Hadamard matrices
• Orthogonal arrays
• Screening designs
• Inequivalent projections
• Factorial designs
• Generalized resolution
• Generalized wordlength pattern
• Distinct runs

Mathematics Subject Classification:

• Primary 62K15
• Secondary 05B20

Acknowledgments

We would like to thank Professor Angela Dean and an anonymous referee for a careful reading of the manuscript and for their useful comments and suggestions that led us to a substantial improvement in the presentation of this paper.