Roman and Vatican Crossover Designs
M. A. Ollis

TL;DR
This paper introduces Vatican designs, a new class of crossover experimental designs that generalize Latin squares by balancing treatment sequences across multiple subjects and distances, with proven existence under specific conditions.
Contribution
It defines and constructs Vatican designs, extending Latin square properties to more subjects and greater treatment-distance balance, with existence proofs for various parameter sets.
Findings
Vatican designs exist when t+1 is prime for any ll
Existence of Vatican designs for 5 t 14 and ll > 1
Vatican designs exist for t and ll even
Abstract
Latin squares with a balance property among adjacent pairs of symbols---being "Roman" or "row-complete"---have long been used as uniform crossover designs with the number of treatments, periods and subjects all equal. This has been generalized in two ways: to crossover designs with more subjects and to balance properties at greater distances. We consider both of these simultaneously, introducing and constructing {\em Vatican designs}: these have subjects, periods and treatments, and, for each in the range , the number of times that any subject receives treatment exactly periods after receiving treatment is at most . Results include showing the existence of Vatican designs when is prime (for any ), when and , and when and is even.
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Taxonomy
TopicsOptimal Experimental Design Methods · graph theory and CDMA systems
