## 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.

### MSC:

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