## Abstract

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.

## Keywords

Main effect Mixed-level Regular fractional factorial design Two-factor interaction component## Mathematics Subject Classification

Primary 62K15 Secondary 62K05## Notes

### Acknowledgements

The authors thank Editor Professor Hajo Holzmann, and two anonymous referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11431006, 11771220, 11771250 and 11801308), National Ten Thousand Talents Program, Tianjin Development Program for Innovation and Entrepreneurship, Tianjin “131” Talents Program, Natural Science Foundation of Shandong Province (Grant No. ZR2018BA013), and the PhD Candidate Research Innovation Fund of Nankai University. The first two authors contributed equally to this work.

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