The fractional factorial split-plot (FFSP) design is an important experimental design both in theory and in practice. There is extensive literature on the two-level FFSP design and its various variants. However, there is little work on the s-level FFSP design and its variants in the asymmetrical (i.e., mixed-level) case, where s is any prime or prime power. Such designs are commonly used e.g. in agriculture, medicine and chemistry. This paper provides the necessary and sufficient conditions for the existence of resolution III or IV regular $$s^{(n_1+n_2)-(k_1+k_2)}(s^r)$$ designs which contain clear main effects or two-factor interaction components. In particular, the sufficient conditions are proved through constructing the corresponding designs, and some examples are provided to illustrate the construction methods.

中文翻译：

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具有清晰效果的不对称裂区设计

分数阶乘裂区 (FFSP) 设计在理论和实践中都是重要的实验设计。有大量关于两级 FFSP 设计及其各种变体的文献。然而，在不对称（即混合级）情况下，s 级 FFSP 设计及其变体的工作很少，其中 s 是任何素数或素数幂。这种设计通常用于农业、医学和化学等领域。本文提供了存在分辨率 III 或 IV 正则 $$s^{(n_1+n_2)-(k_1+k_2)}(s^r)$$ 设计的充要条件，其中包含明确的主效应或两个-因素相互作用成分。特别是通过构造相应的设计证明了充分条件，并给出了一些例子来说明构造方法。