Roman and Vatican Crossover Designs

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 $\ell t$ subjects, $t$ periods and treatments, and, for each $d$ in the range $1\le d<t$, the number of times that any subject receives treatment $j$ exactly $d$ periods after receiving treatment $i$ is at most $\ell$. Results include showing the existence of Vatican designs when $t+1$ is prime (for any $\ell$), when $5\le t\le 14$ and $\ell >1$, and when $t\in \{3,15\}$ and $\ell$ is even.

12 pages