Some Projection Properties of Orthogonal Arrays

Abstract

The definition of an orthogonal array imposes an important geometric property: the projection of an $\mathrm{OA}(\lambda 2^t, 2^k, t)$, 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 $\mathrm{OA}(N, 2^k, t)$ 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 $k \geq t + 2$, 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 $\mathrm{OA}(N, 2^k, t)$ with $k \geq t + 2$ 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 $\mathrm{OA}(N, 2^k, 2)$ with $k \geq 4$, 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.