Latin Square Design

Latin square design is a type of experimental design that can be used to control sources of extraneous variation or nuisance factors. Latin squares design is an extension of the randomized complete block design and is employed when a researcher has two sources of extraneous variation in a research study that he or she wishes to control or eliminate. This entry addresses how to set up a study using Latin square design, the advantages and disadvantages of this design approach, and the analysis of data when using Latin square design.

Let us begin with an example to illustrate why one might use a Latin square design in an experimental study. Perhaps you want to test the influence [Page 847]of four different protocols for producing a product at a factory. Specifically, you want to test the average time to produce a product based on protocol. However, you suspect that there may also be an influence of the type of machine (there are four types), as well as the human operator of the machine (there are four human operators), on production time. How do you control for the influence of these two factors? One way to assess or control for the influence of the machine and the machine operator is to set up a Latin square design. By using the Latin square design, you will be able to assess the influence of the protocol on production time, while controlling for the influence of the machine and the machine operator.

With Latin square design, the number of treatment levels must correspond to the number of levels of each of the extraneous factors. To set up a Latin square design, the levels of one factor are represented by rows and the levels of another factor are represented by columns to create a square. Next, each treatment is given at each combination of the levels of the two factors. Returning to the earlier example, in the factory experiment, each operator would follow each of the four protocols on one of the four machines in a Latin square design. It is possible that both the machine and the operator of the machine will influence production time. By employing a Latin square design, the researcher can control for these influences to ascertain the effects of the production protocol.

The Latin square design described herein may be visually understood as follows:

In each row (operator), each protocol (represented by the Latin letters) appears only once, and in each column (machine), each protocol appears only once. Often researchers use randomization to determine the order or sequence of treatment in the rows and the columns, and to which Latin letter the protocols will be assigned to in the study design. Because each treatment may only appear once in each column and once in each row, there are limitations on the possible number of combinations in the table of a Latin square design. Researchers can use statistical software to randomly select the combinations of the rows, columns, and treatments for the study design.

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