This is actually a family of designs, generically referred to as a Latin-square arrangement. Other names for such designs have been rotation experiment, cross-over design, and switch-over design
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 patientexample, you might find yourself usingsuch an approach if you have fourdifferent educational approaches, fourdifferent (equivalent) clinics to try themout in, and you expect different effectsfrom different orders of presentation.(This might be stretching our example abit, but it is conceivable that you could dothis if the time between presentationswas long enough, maybe six months orso.)
Yes, I realize you could justrandomly assign each of the foureducational approaches to eachof the four clinics. If oneapproach then proved superior,you would have some evidencethat it was better. But therewould be potential competingexplanations due to the possibleinfluence of extraneous factors.
You would need to replicate to increaseconfidence in your results (unless theywere overwhelming, of course). Think ofthis design as a systematic means ofarranging your replications. The Latin-square design is typicallyanalyzed with the analysis of variance(ANOVA), as long as there are no majorproblems with missing data, as mightoccur if you had participants absent fromone or more groups at one or more times.
differ from randomized complete block designs in that the experimental units are grouped in blocks in two different ways, that is, by rows and columns. Therefore, two different sources of variation can be isolated.
A requirement of the latin squareis that the number of treatments,rows, and number of replications,columns, must be equal; therefore,the total number of experimentalunits must be a perfect square. Forexample, if there are 4 treatments,there must be 4 replicates, or 4 rowsand 4 columns.
This is a 4x4 latin square which gives atotal of 16 experimental units. Because ofthis restriction, latin square experimentscan become large and unmanageablevery readily. Also, if the number oftreatments is too small, there are too fewdf for error so that the most commonsquares are in the range of 5x5 to 8x8
Since two sources of variation areisolated in rows and columns, themean square for error will be smallerthan for the same data analyzed aseither a completely randomized orrandomized complete block design.However, the df for error will also besmaller as shown in the analysis ofvariance table.
Controls more variation than CR or RCB designs because of 2-way stratification. Results in a smaller mean square for error. Simple analysis of data Analysis is simple even with missing plots.
Number of treatments is limited to the number of replicates which seldom exceeds 10. If have less than 5 treatments, the df for controlling random variation is relatively large and the df for error is small.