Some Projection Properties of Orthogonal Arrays

The definition of an orthogonal array imposes an important geometric property: the projection of an OA(λ 2ᵗ, 2ᵏ, t), a λ 2ᵗ-run orthogonal array with k two-level factors and strength t, onto any t factors consists of λ copies of the complete 2ᵗ factorial. In this article, projections of an OA(N, 2ᵏ, 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 2ᵗ⁺¹ factorial, one or more copies of a half-replicate of 2ᵗ⁺¹ or a combination of both. It is also shown that for k ≥ t + 2, only when N is a multiple of 2ᵗ⁺¹ can the projection onto some $t + 1$ factors be copies of a half-replicate of 2ᵗ⁺¹. Therefore, if N is not a multiple of 2ᵗ⁺¹, then the projection of an OA(N, 2ᵏ, t) with k ≥ t + 2 onto any $t + 1$ factors must contain at least one complete 2ᵗ⁺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, 2ᵏ, 2) with k ≥ 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.