Some projection properties of orthogonal arrays. (English) Zbl 0838.62061

Summary: The definition of an orthogonal array imposes an important geometric property: the projection of an \(\text{OA} (\lambda 2^t, 2^k, t)\), a \(\lambda 2^t\)-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 \(\text{OA} (N, 2^k, t)\) 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 \(\text{OA} (N,2^k, t)\) 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 \(\text{AO} (N, 2^k, 2)\) 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.


62K15 Factorial statistical designs
05B15 Orthogonal arrays, Latin squares, Room squares
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