Department of Industrial Engineering The University of Iowa

References:

1. Sohn, Han-Suk, Dennis Bricker, J. Richard Simon, and Yi-Chih Hsieh, "Optimal Sequences of Trials for Balancing Practice and Repetition Effects", Behavior Research Methods, Instruments, and Computers , Volume 29(1997), No. 4, pp. 574-581.
   Abstract: This paper describes procedures for generating trial sequences to balance out practice effects and intertrial repetition effects in experiments consisting of repeated trials. In the sequences presented, each stimulus appears an equal number of times, is preceded equally often by itself and by each other stimulus and is distributed in a "balanced" manner throughout the block of trials. Two criteria for balance are employed. One criterion aims to equalize the average position of each stimulus in the sequence. The second criterion maintains, as much as possible, a uniform interval between appearances of each stimulus in the sequence. For each criterion, optimal or near-optimal sequences are presented for experiments involving from three to nine different stimulus conditions. Suggestions are included for extending (e.g., doubling or tripling) the length of the sequences. Click here to download a .pdf document containing this paper.
2. Emerson, Phillip L. and Randall D. Tobias, "Computer program for quasi-random stimulus sequences with equal transition frequencies", Behavior Research Methods, Instruments, and Computers, vol. 27 (1995), no. 1, pp. 88-98.
   Abstract: C language routines are presented for the generation of randomized stimulus sequences constructed from multiple presentations of m stimuli satisfying sequentyial constraints with respect to the frequencies of the occurrence of n-gram subsequences. Applications are suggested for sequential experiments in which main effects for the present stimulus and the stimuli in the preceding (n-1)-length substring can be tested, as well as the interactions among stimuli at the various positions in the substrings.

Below are listings of the sequences presented in reference #1 above. In the future, additional sequences for N>9 will be added, as well as sequences which have been randomly generated by the algorithm reported in reference #2.

Sequences are selected according to three criteria:

• A. Balance of the integers within the sequence with respect to their positions. That is, for each integer, the sum of its positions within the sequence is summed, and the deviation of that sum from the average position (0.5N[1+N^2])is determined. This criterion judges the quality of a sequence by the maximum value of these deviations, i.e., smaller deviations are preferred. Thus, good sequences do not have an integer appearing predominantly near the beginning or the end of the sequence.
• B. Uniformity of the distribution of the integers within the sequence. That is, the gaps between consecutive appearances of each integer are identified, and their deviation from the average of these gaps (namely N) are determined. This criterion judges the quality of a sequence by the maximum value of these deviations, i.e., smaller deviations are preferred.
• C. This criterion is the same as criterion B, except that the gaps between consecutive appearances of each integer include the gap between the final appearance and the first appearance in the repeated sequence.

(N=3*)
1 2 3 1 3 3 2 2 1
 (N=4*)
1 2 3 4 4 1 3 3 2 4 2 2 1 1 4 3
 (N=5*)
1 2 3 4 5 4 3 3 5 2 2 1 5 5 1 1 4 4 2 4 1 3 2 5 3
 (N=6*)
1 2 3 4 5 6 6 5 5 4 3 3 2 2 1 6 3 1 4 6 4 1 1 5 2 6 2 4 4 2 5 1 3 5 3 6
 (N=7)
1 2 3 4 5 6 7 7 6 6 5 5 4 4 3 3 2 2 1 1 3 1 7 4 7 3 6 2 7 2 5 1 5 7 5 2 4 6 1 6 4 1 4 2 6 3 5 3 7
 (N=8)
1 2 3 4 5 6 7 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 3 1 4 2 4 1 8 2 8 3 8 5 8 1 6 2 7 5 7 3 7 4 6 8 6 3 5 2 6 1 7 2 5 1 5 3 6 4 8 4 7
 (N=9)
1 2 3 4 5 6 7 8 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 4 2 1 1 3 1 9 1 5 9 5 3 9 3 5 2 9 2 6 4 6 2 7 4 7 1 7 3 8 6 8 1 8 4 8 5 7 9 6 9 7 5 8 2 8 3 6 1 4 9 4 1 6 3 7 2

(N=3) d = [-1, 2, -1]
(N=4) d = [0, 0, 0, 0]
(N=5) d = [0, 0, 0, 0, 0]
(N=6) d = [0, 0, 0, 0, 0, 0]
(N=7) d = [0, 0, 0, 1, -1, 0, 0]
(N=8) d = [-13, -14, -12, 0, 14, 15, 12, -2]
(N=9) d = [3, -18, -20, -3, -15, 21, 17, 22, -7]

For example, in the case N=3, the integer "1" appears in positions 1,4, and 9, which yields the sum 14. The target for this sum is 0.5(3)(1+3^2)=15, and so d[1]=-1.

(N=3) g = [1, 2, 3]
(N=4) g = [4, 2, 3, 3]
(N=5) g = [3, 6, 7, 0, 4]
(N=6) g = [4, 12, 4, 5, 0, 5]
(N=7) g = [6, 17, 7, 2, 3, 1, 6]
(N=8) g = [8, 15, 12, 6, 5, 3, 0, 7]
(N=9) g = [15, 18, 14, 3, 6, 2, 2, 4, 8]

For example, in the case N=3, the gaps for the integer "1" are 5 and 3, for the integer "2" they are 2 and 1, and for the integer "3" they are 4 and 1. The target gap is N=3, and so the magnitudes of the deviations are 2, 0, 1, 2, 1, and 2. Thus g[0]=1, i.e. there is 1 deviation of magnitude 0, while g[1]=2, i.e., there are 2 deviations of magnitude 1, etc. Note that in each sequence above, the maximum magnitude of deviations of gaps from their target values is N-1, e.g., in the case N=9 there are 8 deviations of magnitude 8.

(N=3*)
1 2 2 3 1 3 3 2 1
 (N=4*)
1 2 3 4 4 1 3 2 2 4 3 3 1 4 2 1
 (N=5)
1 2 3 4 5 1 3 5 2 2 4 4 1 5 3 3 2 1 4 2 5 5 4 3 1
 (N=6)
1 2 3 4 5 6 1 3 2 4 6 5 1 4 3 6 6 2 2 5 3 1 5 4 2 1 6 3 3 5 5 2 6 4 4 1
 (N=7)
1 2 3 4 4 5 5 2 6 6 7 7 3 1 5 4 6 2 7 1 3 5 6 4 2 1 7 5 3 6 1 4 7 2 5 1 6 3 3 2 2 4 3 7 6 5 7 4 1
 (N=8)
1 2 3 4 5 6 7 8 1 3 2 4 6 5 7 1 8 3 6 2 5 4 7 3 1 6 8 2 7 5 3 8 4 1 7 6 3 5 2 8 8 7 4 2 2 1 5 8 6 6 4 4 3 3 7 7 2 6 1 4 8 5 5 1
 (N=9)
1 2 3 4 5 6 7 8 9 1 3 2 4 6 5 7 9 8 1 4 3 6 2 5 9 7 1 8 4 2 6 3 5 1 9 4 7 2 8 6 1 5 3 9 2 7 4 8 5 2 1 6 9 3 7 5 8 7 7 3 8 2 2 9 5 5 4 1 7 6 6 4 4 9 9 6 8 8 3 3 1
 The Gap Cardinality Vectors of the sequences C
(N=3) g' = [1, 3, 4, 1]
(N=4) g' = [3, 4, 3, 6]
(N=5) g' = [2, 5, 10, 3, 5]
(N=6) g' = [6, 7, 10, 3, 4, 6, 6]
(N=7) g' = [7, 7, 15, 7, 5, 4, 0, 11]
(N=8) g' = [8, 12, 17, 10, 6, 3, 5, 0, 10, 1]
(N=9) g' = [9, 17, 17, 8, 4, 4, 2, 2, 1, 10, 1, 2, 3]

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Last modified: 12 November 1997