Depicted here:

• Four naturally-occuring groups (i.e., you did not randomly select participants from a larger population nor did you randomly assign them to the four groups).

• Four different experimental treatments, or variations on the same treatment, to be compared.

• Four different occasions, which allows each group to receive all four treatments, but in a different order for each group.

To continue with our diabetic patient example, you might find yourself using such an approach if you have four different educational approaches, four different (equivalent) clinics to try them out in, and you expect different effects from different orders of presentation. (This might be stretching our example a bit, but it is conceivable that you could do this if the time between presentations was long enough, maybe six months or so.)

Yes, I realize you could just randomly assign each of the four educational approaches to each of the four clinics. If one approach then proved superior, you would have some evidence that it was better. But there would be potential competing explanations due to the possible influence of extraneous factors. You would need to replicate to increase confidence in your results (unless they were overwhelming, of course). Think of this design as a systematic means of arranging your replications.

The Latin-square design is typically analyzed with the analysis of variance (ANOVA), as long as there are no major problems with missing data, as might occur if you had participants absent from one or more groups at one or more times.