Latin Square and Greco-Latin Square Design

A randomised complete block design controls one nuisance factor by grouping experimental units into blocks. A Latin square controls two nuisance factors simultaneously — the row variable and the column variable — at the cost of requiring that the number of treatments equal the number of row levels and the number of column levels. When you have only one blocking dimension, the randomised block design is simpler and wastes fewer degrees of freedom on the second blocking factor.

For an n×n Latin square the error (residual) has (n−1)(n−2) degrees of freedom. For n = 3 that is only 2, for n = 4 it is 6, and for n = 5 it is 12. Very small squares therefore have limited power, and a power analysis should be conducted to verify that the chosen n provides adequate sensitivity before the experiment is run.

The standard Latin square model assumes additivity — that row, column, and treatment effects combine without interaction. If subject-matter knowledge or preliminary data suggest that the treatment effect differs across rows or columns, the additive model is inappropriate. In that case a full factorial design with replication is the correct choice, as it explicitly estimates the interaction terms.

Use a Greco-Latin square when you have three nuisance factors — rows, columns, and a third factor such as operator, batch, or time period — each with exactly n levels, and you want to remove all three simultaneously. The Greco-Latin square is formed by superimposing two mutually orthogonal Latin squares, ensuring that every treatment appears exactly once in combination with every level of every blocking factor. Orthogonal Latin squares of order 6 do not exist, so the Greco-Latin square cannot be constructed for n = 6.