Latin square

A Latin square of order $q$ is a $q\times q$ array of $q$ symbols, each occurring once in each row and column. Used as a row-column design, its symbols label treatments. Centered treatment indicators are orthogonal to both row and column indicators, because every row-treatment and column-treatment pair occurs once. This balance separates additive effects from two nuisance directions.

For one response per cell in an order-$q$ Latin square, fit an intercept and additive row, column and treatment effects with sum-to-zero constraints. Each factor contributes $q-1$ statistical degrees of freedom; the residual has $q^2-1-3(q-1)=(q-1)(q-2)$. The centered factor spaces are mutually orthogonal, so sequential sums of squares do not depend on fitting order. With an additional orthogonal symbol factor from a Graeco-Latin square, residual statistical degrees of freedom become $(q-1)(q-3)$. Exact F-tests require independent equal-variance normal errors; unmodeled interactions can invalidate the error term.

Latin squares on the same array are mutually orthogonal when every ordered pair of symbols from any two squares occurs exactly once on superposition. Pairwise balance makes the centered symbol-indicator spaces orthogonal in an additive linear regression model.

A Graeco-Latin square is the superposition of two orthogonal Latin squares of the same order. Each symbol of either square occurs once per row and column, and every cross-square pair occurs once. It permits four mutually orthogonal additive factor spaces: rows, columns, and the two symbol factors.