Latin Square Design

In designed experiments, Latin Square Designs are used when 1) the factors of interest have more than two levels, 2) multiple treatments can be administered to the same experimental units, and 3) when only main effects are of interest, while 4) interaction effects are assumed to be negligible. For example, it can be used when four drugs are administered to the same group of patients, but in four different orders in four different experimental subsamples to control for the possible carry-over effects (e.g., drug #1 could be more effective if it is preceded by the administration of drug #2, but less effective if it is preceded by #3 or when it is administered as the very first treatment).

The main advantage of Latin Square Designs is that they provide relatively unbiased estimates of main effects in a highly economical manner because Latin Square Design experiments can be performed with far fewer runs compared to a full factorial designs where all possible orders of treatments would have to be represented. For example, in the following valid Latin Square Design with four levels of treatment, each treatment is represented an equal number of times in each serial position, and there is no systematic correlation between any treatments and treatments that precede it or follow it:

The disadvantage of Latin Square Designs is that they do not provide information on interactions between the treatments, and they will provide unbiased estimates of main effects for treatments only if the assumption that the interactions are negligible is valid, which cannot be guaranteed as long as full factorial experiments are not conducted. Still, in practice, Latin Square Designs are often the only realistic option to collect data on the main effects of interest, and researchers usually rely on the domain knowledge to estimate the risk that the assumption about interactions being negligible is invalid.