= Counterbalancing for immediate sequential effects =
'''ALGORITHM (Williams)'''

 1. Write down the N conditions in some order, say {1, 2, 3, ...} and its N-1 cyclic rotations
 1. Apply the interleaving permutation {1, 2, N, 3, N-1,4, N-2, ...} to each of these sequences
 1. 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 :

 {{{
123456
234561
345612
456123
561234
612345
}}}

we get the sequentially counterbalanced 6x6 design:

 {{{
126354
231465
342516
453621
564132
615243.
}}}

'''''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 :

 {{{
1234567
2345671
3456712
4567123
5671234
6712345
7123456
}}}

we get the intermediate matrix:

 {{{
1273645
2314756
3425167
4536271
5647312
6751423
7162534.
}}}

Appending the mirror image of the intermediate matrix we get the sequentially counterbalanced 14x7 design:

 {{{
1273645
2314756
3425167
4536271
5647312
6751423
7162534
5463721
6574132
7615243
1726354
2137465
3241576
4352617.
}}}

'''Here is some MATLAB code to perform this:'''

 {{{
function design = williams(n)
% Ian Nimmo-Smith (MRC CBU) April 2003
cyclic = toeplitz([1,(n:(-1):2)],[1:n]);
cyclic = cyclic([1,(n:(-1):2)],:);
baseperm = [1];
half = floor(n/2);
if n == 2*half    % even
    for j = 1:(half-1)
        baseperm = [baseperm, j+1, n-j+1];
    end
    baseperm = [baseperm, half+1];
else              % odd
    for j = 1:half
        baseperm = [baseperm, j+1, n-j+1];
    end
end
design = cyclic(:,baseperm);
if n ~= 2*half
    design = [design;design(:,(n:(-1):1))];
end      
}}}

'''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) [[http://www.isid.ac.in/~ashish/workshop/mausumiw3.pdf|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. [[http://designcomputing.net/gendex/pdf/federer.pdf|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: [[http://www.maths.qmul.ac.uk/postgraduate/anncook/ollis(ac).pdf|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]
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