Roman and Vatican Crossover Designs

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 $\ell t$ subjects, $t$ periods and treatments, and, for each $d$ in the range $1 \leq d < t$, 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 $5 \leq t \leq 14$ and $\ell >1$, and when $t \in \{ 3,15 \}$ and $\ell$ is even.