Latin Square Design

A Latin square is a grid or matrix containing the same number of rows and columns (k, say). The cell entries consist of a sequence of k symbols (for instance, the integers from 1 to k) inserted in such a way that each symbol occurs only once in each row and once in each column of the grid. By way of an example, Table 1 shows a Latin square that contains the numbers from 1 to 5. This entry describes the classification of Latin squares, their origins in agricultural experiments, and their applications in the social and behavioral sciences.

Table 1 is an example of a standard form, in that the numbers in the first row and the numbers in the first column are in their natural order. Other, nonstandard Latin squares can be constructed by interchanging different rows in the table, by interchanging different columns in the table, or both. In research practice, it is typically recommended that a standard form of the relevant size should be drawn at random from published tables of Latin squares and that its rows and its columns should be interchanged at random using published tables of random sequences.

In 1925, Ronald A. Fisher, a British statistician, proposed that Latin squares could be used to arrange plots in agricultural experiments so as to [Page 949]control for differences in soil fertility. For instance, Table 1 might represent the arrangement of 25 plots available for an experiment, and five different treatments would be applied corresponding to the numbers in the cells in the table. An analysis of variance carried out on some criterion variable would then identify the variation among the rows and the variation among the columns, leaving an unbiased estimate of the effect of the treatments, controlling for any differences among the rows and columns.

Fisher’s writings proved very influential in the social and behavioral sciences, especially in the years after World War II. Benjamin J. Winer, Donald R. Brown, and Kenneth M. Michels identified four main applications of Latin squares in such research: to control nuisance variables, to counterbalance order effects in repeated measures designs, to confound treatment conditions with group main effects, and as balanced fractional replications from a complete factorial design. The second of these applications is probably the most common, although researchers may well not take this feature into account in their analysis of their results. In some fields, the use of Latin square designs has declined, but they remain a potentially important experimental technique.

See also Analysis of Variance; Repeated Measures Analysis of Variance; Repeated Measures Designs