Orthogonal designs of parallel flats type

Abstract

Consider an s n factorial experiment, where s is such that GF( s) exists. Let T i be the set of s n− r treatments (often called ‘assemblies’ or ‘runs’) t satisfying the equations At = c i over GF( s), where A ( r × n) is of rank r, t is ( n × 1), and c i is an ( r × 1) vector for i = 1, …, f. Let T be obtained by taking all the T i together so that T has N = fs n− r treatments. Let C = ( c 1, …, c f ). Thus, T is determined by the pair ( A, C). Assume that F is any vector of distinct sets of factorial effects, each set (except for μ, the general mean) carrying ( s − 1) degres of freedom. Let M be the information matrix for estimating all the parameters in F by using T, assuming that the effects not included in F are negligible. In this paper, we give a simple necessary and sufficient condition on A and C such that M is diagonal. We also give a large set of examples for the case when s = 2, and F consists of the general mean, main effects, and certain classes of two-factor interactions. (Readers who are interested only in the examples may go directly to Section 3.)