Greco-Latin Squares

Byers, J.A. 1993. Randomization algorithms in BASIC for
    experimental design. Computers in Biology and
    Medicine 23:167-176.
excerpt from pages 173-174:
The program will generate a Greco-Latin square in which every Latin letter occurs once in each row and once in each column, and each Greek letter (represented here by numbers instead) occurs once for each row and once for each column. In addition, each Latin letter and Greek letter (number) occur only once together in the square. Thus, for a Greco-Latin square of 5x5 there are 25 combinations of Latin letters and numbers which can be placed in 25 positions. This means that there are 25! (about 10 to the 25th power) permutations out of which some Greco-Latin squares must be found.
Kempthorne [9] says a Greco-Latin square of side 12 exists, although he gives no solution or reference. He does say that Greco-Latin squares of even numbered sides have not been enumerated except as he shows for a side of 4 (and possibly for a side of 12). It can be seen that in Latin squares of sides N = 3,4,5 or more the number of 45 diagonals made from two or more quadrants number N+j where j is 0,1,2,3..., so that squares of odd numbered sides have an even j while even numbered sides have an odd j. The four-sided (j = 1) square is a special case, while Greco-Latin squares with even numbered sides of 6 or more (j is odd and 3 or more) do not seem to exist.
The program first finds a general solution for a Greco-Latin square with an odd number of letters per side of 3 or more. For example for a 5x5 square, this is done by beginning the first row with ABCDE. The last letter of the row, E, starts on the next row down and arrangement of letters is subsequently ordered (EABCD). This pattern is shifted for the third (DEABC) and all rows resulting in a Latin square. The same procedure is done with numbers but in reverse (right to left) so that the first row is 54321 and the second row is 43215. The two Latin squares of letters and numbers are superimposed and result in a Greco-Latin square. The algorithm does not work for squares with sides with an even number of rows/columns.
By switching two of the columns one obtains another Greco-Latin square. Similarly two rows can be switched to get two different squares. This principle is then used to switch at random various rows and columns to get many different Greco-Latin squares. It was found that the number of random switches of rows or columns needed to be at least as large as the number of rows/columns in order to effect a significant randomization of the initial pattern. With a 4x4 square it is possible to make six different swaps of the rows for a total of seven different squares. Also six (N=6) different swaps of the columns allow a grand total of
1 + [SUM (j=1 to N)of(j-1)] = 37
different Greco-Latin squares including the original one. For a 5x5 there are 101 squares, for a 7x7 there are 442 squares. The general algorithm does not work for even numbered sides, but a Greco-Latin solution exists for a side of four as shown earlier [10]. This unique solution is used with the row and column randomization algorithm to obtain the 37 possible Greco-Latin squares.
A Greco-Latin square could be used if one wanted to test the effects of 7 pheromone blends each at 7 dosages in 7 areas of the forest on each of 7 days. The Greco-Latin square would allow a more powerful analysis since all treatments and dosages were tested at each position and on each day, although each position or day did not have identical treatments/dosages. Several texts discuss the advantages and drawbacks of using Latin and Greco-Latin squares as well as the statistical analysis of variance [1, 3, 4, 6, 8, 9].
selected references:
1. V. L. Anderson and R. A. McLean, Design of Experiments. Marcel Dekker, Inc., New York (1974). 3. W. G. Cochran and G. M. Cox, Experimental Designs. Wiley, New York (1957). 4. B. E. Cooper, Statistics for Experimentalists. Pergamon Press, London (1969). 6. M. N. Das and N. C. Giri. Design and Analysis of Experiments. Wiley, New Delhi (1979). 8. W. T. Federer, Experimental Design. MacMillan Co., New York (1955). 9. O. Kempthorne, The Design and Analysis of Experiments. Wiley, New York (1952). 10. J. A. Byers, Basic algorithms for random sampling and treatment randomization. Comput. Biol. Med. 21, 69 (1991). Other related references:
Byers, J.A. 1991. BASIC algorithms for random sampling and treatment randomization. Computers in Biology and Medicine 21:69-77. Byers, J.A. 1996. Random selection algorithms for spatial and temporal sampling. Computers in Biology and Medicine 26:41-52.
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