52013-03-08 10:17:26localhostconverted to 1.6 markup42006-07-12 16:43:50devel06.mrc-cbu.cam.ac.uk32006-06-30 22:56:57Scripting Subsystem22006-06-30 22:55:26Scripting Subsystem12006-06-30 21:37:47Scripting Subsystem

ALGORITHM (Williams) Write down the N conditions in some order, say {1, 2, 3, ...} and its N-1 cyclic rotations Apply the interleaving permutation {1, 2, N, 3, N-1,4, N-2, ...} to each of these sequences If N is even, STOP, otherwise append the N sequences obtained by completely reversing the N sequences generated by steps 1 and 2. illustrated for N=6 (even) The interleaving permutation maps {1,2,3,4,5,6} to {1,2,6,3,5,4}. Applying this to the columns of the cyclic matrix : we get the sequentially counterbalanced 6x6 design: Illustrated for N=7 (odd) The interleaving permutation maps {1,2,3,4,5,6,7} to {1,2,7,3,6,4,5}. Applying this to the columns of the cyclic matrix : we get the intermediate matrix: Appending the mirror image of the intermediate matrix we get the sequentially counterbalanced 14x7 design: Here is some MATLAB code to perform this: BIBLIOGRAPHY Archdeacon, D.S., Dinitz, J.H., and Stinson, D.R. (1980). Some new row­complete Latin Squares. Journal of Combinatorial Theory, Ser. A, 29, 395-- 398. Mausumi Bose (Applied Statistics Unit, Indian Statistical Institute Kolkata, India) Crossover Designs: Analysis and Optimality Using the Calculus for Factorial Arrangements, Design Workshop Lecture Notes ISI, Kolkata, 25-29 November 2002, 83-192. Bradley, J. V. (1958). Complete counterbalancing of immediate sequential effects in a Latin square design, Journal of the American Statistical Association, 53, 525-528. Durso, F. T. (1984). A Subroutine for counterbalanced assignment of stimuli to conditions. Behaviour Research Methods, Instruments & Computers, 16(5), 471-472 Federer, Walter T. and Nguyen, Nam-Ky. Incomplete block designs. Volume 2, pp 1039–1042 in Encyclopedia of Environmetrics (ISBN 0471 899976) Edited by Abdel H. El-Shaarawi and Walter W. Piegorsch . John Wiley & Sons, Ltd, Chichester, 2002. Lewis, J. R. (1993). Pairs of Latin squares that produce digram-balanced Greco-Latin designs: A BASIC program. Behaviour Research Methods, Instrument, & Computers, 25(3), 414-415 Ollis, Matt: Terraces and the Oberwolfach Problem Prescott, P. (1999). Construction of sequentially counterbalanced designs formed from two Latin squares. Utilitas Mathematica, 55, 135-52. Prescott, P. (1999). Construction of uniform-balanced cross-over designs for any odd number of treatments. Statistics in Medicine, 18, 265-72. Williams, E. J. (1949). Experimental designs balanced for the estimation of residual effects of treatments. Australian Journal of Scientific Research, 2, 149-168. [Last updated on 27 November, 2003] Return to Statistics FAQ page Return to Statistics main page Return to CBU main page These pages are maintained by Ian Nimmo-Smith and Peter Watson