Orthogonal Latin Square Designs with n = j2
One frequently encountered use of a Latin Square design is to counterbalance the various sequences in which the levels of an independent variable might occur. For example: To counterbalance the potential order effects of j = 4 levels (A, B, C, and D) of a certain stimulus variable, each of j = 4 subjects could have the levels presented in a different sequence, as shown in the following 4x4 table.
Sequence    In this table, each subscripted entry represents a measure of the dependent variable for one particular subject on one particular level of the independent variable. Thus, A1 represents the measure for subject 1 on level A, B3 represents the measure for subject 3 on level B, and so on.
1 2 3 4
subject 1 A1 B1 C1 D1
subject 2 B2 C2 D2 A2
subject 3 C3 D3 A3 B3
subject 4 D4 A4 B4 C4
In all versions of a 4x4 Latin Square, each of the 4 alphabetic designations—A, B, C, D—would appear exactly once in each of the 4 rows and in each of the 4 columns. In the orthogonal version there is the additional stipulation that for each row sequence, as read from left to right, there must be a corresponding column sequence, as read from top to bottom. Thus, in the present example, row 1 (A, B, C, D) is orthogonal with column 1 (A, B, C, D); row 2 (B, C, D, A) is orthogonal with column 2 (B, C, D, A); and so on.
In another version of the orthogonal Latin Square design, there are three independent variables, each with j quantitative or categorical levels. The levels of two of the independent variables delineate j rows and j columns, respectively, and the j levels of the third variable are then distributed among the jxj cells of the matrix in the same fashion as shown above. For example (designating "independent variable" as "IV" and "dependent variable" as "DV"): Suppose IV-X to be four different levels of visual stimulation, IV-Y to be four different levels of auditory stimulation, IV-Z to be four different levels (A, B, C, D) of task difficulty, and DV to be a measure of task performance. As shown in the next table, an orthogonal Latin Square design in this situation could be achieved by measuring task performance for 4x4 = 16 subjects, each under a different combination of the levels of independent variables X, Y, and Z.
In this table, each subscripted entry represents a measure of task performance for one particular subject under one particular combination of the levels of X, Y, and Z. Thus, A1 represents the measure for one subject on level 1 of IV-X, level 1 of IV-Y, and level A of IV-Z; B3 represents the measure for one subject on level 4 of IV-X, level 3 of IV-Y, and level B of IV-Z; and so on.

The analysis of variance within an orthogonal Latin Square results in three F-ratios: one for the row variable, one for the column variable, and one for the third IV whose j levels are distributed orthogonally among the cells of the rows x columns matrix. In the Latin Square computational pages on this site, the third IV, with levels designated as A, B, C, etc., is listed as the "treatment" variable.
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The Latin Square ANOVA designs on this site are confined to the situation where there is only one measure per cell (restricted full rank model).
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