A BASIC program for the generation of Latin squares

Abstract

I~ experimental research, it is oftendesirableto expose subjects to more than one stimulus or experimentalconditionto provide for the greaterpowerinherent in repeated measures designs, the economical use of available subjects, or the within-subjects test thatconceptual hypotheses may specifically require. A potential problemin thesedesignsis thepossibility of carryovereffects or the influence of a particular order of presentation. One popular counterbalancing strategy involves the selection of k orders from all possible orders by construction of a Latinsquare, with subjects assigned randomly to one of the k orders. The advantage of the Latinsquareis thata relatively small number of orders is required (i.e., k) and that the orders are selected in such a way as to guarantee the existence of ce~in features. Specifically, each stimulus appears once m every ordinal position, and the set selected is cho~en randomly from among all possible Latin squares of size k. Furthermore, specific statistical analyses have been developed for Latinsquaredesigns, whichallowfor the testingof certain order effects (see, e.g., Kirk, 1968; Myers, 1972; Winer, 1971). Cons~ction of a Latinsquarerequires four basic steps: (1) selection of a standard square of size k, (2) random reordering of the columns of the standard square, ~3) random reordering of the rows of the square result109 from steps I and 2, and (4) random shufflingof the numerical indicators (e.g., Myers, 1972). The resulting rows (or columns) of the final square constitute the k orders, to which each subject in the study is randomly assigned. . The ~jor drawback.to usingthe Latinsquareapproach IS that It can be very tIme-consuming to implement. As k increases, the number of standard squares increases dramatically, and one square must be chosen at random as the startingpoint for the reorderingof rows, columns, and indicators (eachstandard squaregivesrise to a unique subset of all possible Latin squares). For example, although there isonly I standard squarefor k=2 andk=3, and only 4 standard squares for k=4, there are 56 standard squares for k=5, 9,408 standard squares for k=6,