SOME PROJECTION PROPERTIES OF ORTHOGONAL ARRAYS

The definition of an orthogonal array imposes an important geometric property: the projection of an OA(lambda 2(t), 2(k), t), a lambda 2(t)-run orthogonal array with k two-level factors and strength t, onto any t factors consists of lambda copies of the complete 2(t) factorial. In this article, projections of an OA(N, 2(k), t) onto t + 1 and t + 2 factors are considered. The projection onto any t + 1 factors must be one of three types: one or more copies of the complete 2(t+1) factorial, one or more copies of a half-replicate of 2(t+1) or a combination of both. It is also shown that for k greater than or equal to t + 2, only when N is a multiple of 2(t+1) can the projection onto some t + 1 factors be copies of a half-replicate of 2(t+1). Therefore, if N is not a multiple of 2(t+1), then the projection of an OA(N, 2(k), t) with k greater than or equal to t + 2 onto any t + 1 factors must contain at least one complete 2(t+1) factorial. Some properties of projections onto t + 2 factors are established and are applied to show that if N is not a multiple of 8, then for any OA(N, 2(k), 2) with k greater than or equal to 4, the projection onto any four factors has the property that all the main effects and two-factor interactions of these four factors are estimable when the higher-order interactions are negligible.