Metrika (1997)45 :253-257 A Relationship between Etticiencies of Marginal Designs and the Product Design RAINER SCHWABE 1 Freie Universitfit Berlin 1. Mathematische lnstitut, Arnimallee 2-6, 14 195 Berlin, Germany WENG KEE WONG Department of Biostatistics, UCLA, Los Angeles, CA 90095, USA Abstract: We derive the D- and G-efficienciesof product designs in a multifactor experiment in terms of the D- and G-efficienciesof the designs in the marginal models. Key Words and Phrases: Approximate design, D-, G-optimal design, marginal models, interaction, multi-factor experiments. 1 Introduction Multi-factor experiments are becoming increasingly common in practice. In this paper, we study the relationship between the efficiency of a design in a multi- factor experiment and the marginal efficiencies of each of its factors for a class of linear models. We focus on the D- and G-efficiencies of product designs since these designs are intuitively appealing and easy to construct, see for example bok and Thibodeau (1980), Lau (1988), Lim and Studden (1988), and Wong (1994). Moreover in the settings under consideration every design is dominated by the product of its marginals (see Schwabe, 1996b). All designs considered here are approximate designs or probability measures defined on a given compact design region T. The models are linear with a known regression function F(t) and errors are assumed to be independent and homo- scedastic. Following convention, we measure the worth of the design ~ by its information matrix: M(~) = ~ F(t)F(OT~(dt) , T Work supported by grants Ku 719/2 and 477/645/96 of the Deutsche Forschungsgemeinschaft 0026-1335/97/45:3/253 257 $2.50 © 1997 Physica-Verlag, Heidelberg