**Latin squares** are balanced variants of the randomized
complete block design, with treatment factor(s) replicated in two
cross-factored blocks. The two blocking factors each have the same number of
blocks as there are levels of the treatment factor(s). The defining feature of
a Latin square is that treatment factor levels are randomly allocated to cells
within the square grid of column and row blocks in such a way as to have each
factor level represented once in every column block and every row block (making
it a balanced and complete design). These designs are particularly useful for
field experiments that benefit from blocking random variation in two
dimensions. They are also used in crossover trials, where the orthogonal
blocking factors are Subject and Order of treatment. Unless replicated, Latin
squares do not allow testing of interactions with blocks which must be assumed
to have negligible effects.

For example, a treatment factor A with four levels (1, 2, 3, 4) is randomly allocated amongst 16 plots, such that each level of A appears once in each of four column blocks and once in each of four row blocks. A map of the layout of treatment levels might look like this:

3 |
4 |
1 |
2 |

2 |
1 |
4 |
3 |

4 |
2 |
3 |
1 |

1 |
3 |
2 |
4 |

This example illustrates one of the 576 possible Latin
squares for a 4-by-4 layout; larger squares have many orders of magnitude more
combinations (e.g., 161,280 for a 5-by-5 layout). Use a computer program such
as LatinSquare.exe to ensure that the allocation
of treatment levels is obtained at random from the full set of all possible
layouts. The power of the design depends on the* a *levels
of A, which defines the treatment degrees
of freedom,* p* = *a* - 1, the error degrees of freedom,* q*
= (*a* - 1)(*a* - 2), and the* *replication,
*n* = *a*.

These web pages show an example analysis of a 4-by-4 Latin square for a treatment factor A with blocking factors B and C. This is followed by further examples of the following common forms of replicated Latin square:

- Replicate independent observations in each cell;
- Replicate identical squares;
- Replicate squares independently randomized (allows testing of the treatment-by-square interaction);
- Independent squares in a balanced set of all possible combinations of factors A, B and C (allows tests of the two-way treatment-by-block interactions);
- Stacked squares with rows nested in square (allows testing of the treatment-by-square interaction; often used in crossover designs, with each square representing a replicate trial);
- Completely cross-factored treatments in orthogonal blocks (allows testing of treatment interactions).

For Model-2 analyses, the
replication increases *q* and *n*, and hence power to detect the
treatment effect(s). For Model-1 analyses,
replication that allows estimation of block by treatment interactions will
reduce *q* and power unless the interaction effects are small enough to
permit *post hoc* pooling of error terms.

Click here to download a computer program (LatinSquare.exe) that allocates treatment levels at random.

Doncaster, C. P. & Davey, A. J. H. (2007) *Analysis of Variance and Covariance: How to
Choose and Construct Models for the Life Sciences*. Cambridge: Cambridge
University Press.

http://www.southampton.ac.uk/~cpd/anovas/datasets/