Isomorphism Check for $2^n$ Factorial Designs with Randomization Restrictions

Factorial designs with randomization restrictions are often used in industrial experiments when a complete randomization of trials is impractical. In the statistics literature, the analysis, construction and isomorphism of factorial designs has been extensively investigated. Much of the work has been on a case-by-case basis -- addressing completely randomized designs, randomized block designs, split-plot designs, etc. separately. In this paper we take a more unified approach, developing theoretical results and an efficient relabeling strategy to both construct and check the isomorphism of multi-stage factorial designs with randomization restrictions. The examples presented in this paper particularly focus on split-lot designs.