Abstract
The main focus of this article is to develop some infinite series to generate circular balanced repeated measurements designs in periods of three different sizes, using method of cyclic shifts (Rule II). Some construction procedures are also developed for the cases where series are not developed. Catalogues of these designs are also presented. These designs have diversified application with varied period sizes, e.g., medicine, pharmacology, animal sciences and psychology. These designs are mainly used to balance out the residuals effects.
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References
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Acknowledgements
Authors are thankful to the reviewers for their valuable correction and suggestions. Authors on 3rd and 4th order are highly grateful also to the Higher Education Commission Pakistan Research and Development Division National Research Program for Universities for providing grant for this research. Grant No: 6294/Punjab/NRPU/R&D/HEC/2016.
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Appendices
Appendix 1
Minimal CBRMDs constructed through series 4.3.17 for v ≤ 70
v | p1 | p2 | p3 | Sets of shifts |
|---|---|---|---|---|
17 | 5 | 4 | 3 | [2,3,4,6] + [13,9,11,10] + [7,14,15] + [8]t |
27 | 5 | 4 | 3 | [2,3,4,16] + [6,7,8,9] + [12,10,14,5] + [25,19,20,23] + [21,24,18] + [13]t |
37 | 5 | 4 | 3 | [2,3,4,26] + [6,7,10,14] + [12,15,13,21] + [17,20,19,22] + [23,24,8,25] + [27,31,29,5] + [33,34,9] + [18]t |
47 | 5 | 4 | 3 | [2,3,4,36] + [7,8,9,16] + [12,13,15,41] + [17,18,19,28] + [22, 45,24,26] + [20,29,30,32] + [33,34,35,5] + [38,39,40,42] + [43,44,37] + [23]t |
57 | 5 | 4 | 3 | [11,3,4,46] + [7,8,9,26] + [55,13,53,15] + [17,18,19,42] + [22,20,24,25] + [12,1,2,31] + [33,34,35,36] + [38,39,40,14] + [43,52,45,5] + [27,49,50,51] + [29,54,41] + [28]t |
67 | 5 | 4 | 3 | [2,3,4,52] + [7,8,9,36] + [12,13,14,16] + [17,18,19,63] + [22,24,23,42] + [27,28,29,49] + [32,34,58,10] + [38,39,43,41] + [61,40,45,56] + [20,30,50,51] + [57,54,55,1] + [35,26,25,59] + [48,46,44] + [33]t |
77 | 5 | 4 | 3 | [16,3,5,65] + [4,56,40,46] + [9,13,17,26] + [14,18,19,33] + [22,23,24,62] + [27,28,29,53] + [32,20,34,35] + [37,75,39,41] + [43,44,45,54] + [48,49,50,51] + [47,10,55,64] + [58,59,69,61] + [1,8,7,66] + [2,60,25,74] + [73,71,12] + [38]t |
87 | 5 | 4 | 3 | [2,3,5,75] + [7,8,9,56] + [12,13,15,36] + [78,14,19,45] + [22,23,25,81] + [27,28,30,61] + [32,33,41,35] + [37,38,39,40] + [17,47,42,68] + [50,49,63,52] + [53,54,55,11] + [58,59,60,24] + [34,64,65,66] + [46,69,4,72] + [73,74,76,70] + [83,79,80,82] + [77,48,71] + [43]t |
97 | 5 | 4 | 3 | [2,3,4,86] + [7,8,9,66] + [12,13,14,46] + [17,18,19,26] + [22,23,31,95] + [27,28,29,88] + [32,33,34,69] + [37,38,39,42] + [40,43,94,70] + [47,50,49,51] + [71,54,55,56] + [58,59,53,61] + [63,64,89,10] + [68,35,45,73] + [85,74,77,76] + [78,79,80,82] + [83,44,75,5] + [30,93,90,15] + [87,84,25] + [48]t |
21 | 7 | 4 | 3 | [2,3,4,5,19,6] + [17,11,13,12,14,15] + [9,8,7] + [10]t |
35 | 7 | 4 | 3 | [2,3,4,5,6,13] + [9,10,11,12,7,20] + [16,19,18,14,32,22] + [24,26,25,8,27,29] + [23,21,28] + [17]t |
49 | 7 | 4 | 3 | [2,3,4,5,6,27] + [9,10,11,12,13,33] + [16,18,17,19,20,39] + [35,23,26,7,28,29] + [31,32,47,25,34,41] + [38,21,46,36,42,43] + [45,40,37] + [24]t |
63 | 7 | 4 | 3 | [2,3,4,5,6,41] + [9,10,11,12,13,61] + [14,17,18,19,20,21] + [23,24,25,26,27,39] + [30,33,32,34,35,36] + [38,28,40,7,42,56] + [45,46,47,29,49,50] + [52,60,54,55,43,57] + [59,53,16] + [31]t |
77 | 7 | 4 | 3 | [2,3,4,5,6,55] + [9,10,11,12,13,22] + [16,17,18,19,20,47] + [23,24,25,26,27,29] + [30,31,32,33,34,40] + [37,35,70,41,42,43] + [45,46,21,48,49,51]+ [52,53,54,58,56,57] + [59,60,61,62,63,68] + [66,67,64,8,39,71] +[73,14,69] + [38]t |
91 | 7 | 4 | 3 | [2,3,4,5,6,69] + [7,10,11,12,15,27] + [16,17,18,21,20,75] + [23,24,25,26,14,46] + [30,32,31,36,34,78] + [37,38,39,40,41,42] + [44,79,47,48,49,50] + [52,53,54,56,85,87] + [59,60,51,62,61,9] + [55,57,67,64,70,71] + [73,74,88,76,77,80] + [81,35,82,83,84,86] + [68,19,28] + [45]t |
25 | 9 | 4 | 3 | [2,3,4, 6,5,7,11,9] + [22,13,14,15,16,17,18,19] + [21,8,23] + [12]t |
43 | 9 | 4 | 3 | [41,5,1,3,6,7,11,12] + [8,13,9,14,15,16,23,18] + [22,38,20,24,26,25,30,28] + [27,31,32,33,34,35,36,37] + [19,4,2] + [21]t |
61 | 9 | 4 | 3 | [2,3,4,5,6,7,8,24] + [25,12, 14,13,15,56,17,18] + [20,22,21,23,9,11,26,29] + [27,31,32,40,34,35,36,37] + [59,33,41,42,43,44,45,55] + [38,49,50,51,53,52,54,16] + [47,48,39] + [30]t |
79 | 9 | 4 | 3 | [2,3,4,5,6,7,8,42] + [11,12,13,14,15,16,17,18] + [20,21,22,23,24,25,26,54] + [29,30,31,32,33,34,35,60] + [38, 41,56,40,43,44,45,46] + [74,49,51,50,52,53,27,65] + [57,59,58,36,61,62,63,55] + [66,67,68,69,70,75,72,73] + [71,76,77] + [39]t |
97 | 9 | 4 | 3 | [2,3,4,5,6,7,8,60] + [11,12,13,80,15,16,17,18] + [31,21,22,23,24,25,26,27] + [40,30,20,32,33,34,35,36] + [38,39,29,41,42,43,70,45] + [47,50,51,49,52,53,54,78] + [57,59,58,9,61,63,93,94] + [91,67,68,69,44,71,92,75] + [73,77,76,19,79,14,81,83] + [84,85,86,87,88,55,65,66] + [62,64,90] + [48]t |
15 | 7 | 5 | 3 | [2,3,5, 8,4,9] + [10,1,12,13] + [7]t |
29 | 7 | 5 | 3 | [2,3,4,5,6,7] + [9,16,11, 13,12,15] + [17,18,19,20,21,22] + [24,25,26,27] + [14]t |
43 | 7 | 5 | 3 | [2,3,4,5,6,7] + [16,10,11,12,13,14] + [9,37,18,22,19,20] + [34,25,27,26,28,29] + [24,31,32,33,17,35] + [40,36,39,30] + [21]t |
57 | 7 | 5 | 3 | [2,3,4,5,6,7] + [9,10,11,12,13,14] + [16,17,18,19,20,21] + [23,30,25,26,27,15] +[51,52,33,34,50,36] + [38,39,40,49,42,35] + [45,46,47,48,41,55]+ [32,53,44,8] + [28]t |
71 | 7 | 5 | 3 | [2,3,4,68,13,49] + [9,10,11,12,6,14] + [16,17,18,34,19,21] + [23,24,60,26,27,28] + [29,31,32,33,20,66] + [38,39,40,41,42,43] + [45,46,47,51,67,50] + [52,53,61,55,56,25] + [59,57,54,63,65,64] + [36,7,30,5] + [35]t |
85 | 7 | 5 | 3 | [2,3,60,5,6,7] + [9,10,11,12,13,21] + [16,17,18,19,20,63] + [83,24,25,26,27,45] + [30,31,32,61,34,35] + [37,38,39,41,40,43] + [52,46,47,48,49,50] + [82,53,54,55,56,69] + [59,4,33,62,36,64] + [66,81,68,80,70,72] + [73,74,75,76,77,58] + [57,23,28,65] + [42]t |
99 | 7 | 5 | 3 | [2,3,4,5,6,7] + [30,10,11,12,13,14] + [16,17,18,19,20,21] + [23,24,25,27,26,28] + [97,31,32,33,34,38] + [37,35,39,40,41,42] +[44,61,46,48,47,50] + [52,53,54,55,70,57] + [59,36,45,67,63,64] +[66,62,73,69,56,1] + [68,74,75,76,95,80] + [78,81,82,90,84,94] +[87,88,89,83,92,72] + [15,91,91,86] + [49]t |
17 | 9 | 5 | 3 | [2,4,3,5,7,6,9,11] + [12,13,14,15] + [8]t |
35 | 9 | 5 | 3 | [2,3,4,5,6,7,8,9] + [11,20,13, 15,16,14,18,19] + [21,23,22,1,25,26,27,28] +[30,12,32,33] + [17]t |
53 | 9 | 5 | 3 | [2,10,4,5,6,7,8,9] + [11,12,13,14,15,16,17,18] + [41,51,22,24,23,25,28,27] +[45,31,32,33,34,35,36,37] + [39, 42,3,50,20,44,30,46] + [48,49,43,21] + [26]t |
71 | 9 | 5 | 3 | [2,3,4,5,6,7,8,9] + [12,11,13,14,15,16,17,18] + [20,21,22,23,10,43,25,27] + [49,30,31,34,32,33,36,37] + [39,40,41,42,1,44,45,46] + [48,69,50,51,38,53,54,60] + [57,58,59,55,61,62,63,28] + [66,56,64,29] + [35]t |
89 | 9 | 5 | 3 | [2,3,4,5,6,7,8,9] + [11,12,13,14,15,16,17,18] + [20,21,22,23,24,25,26,27] + [29,30,31,32,33,34,35,36] + [38,39,40,42,41,43,46,45] + [48,49,50,51,52,53,54,55] + [57,58,59,60,61,62,63,64] + [66,67,68,69,70,71,72,73] + [75,76,77,78,79,80,81,82] + [84,85,86,87] + [44]t |
23 | 7 | 6 | 3 | [2,4,3,5,6,7] + [9,10,19,13,14,15] + [1,18,12,20,21] + [11]t |
37 | 7 | 6 | 3 | [2,3,4,5,6,7] + [10,1,11,15,13,14] + [17,16,,22,20,21] + [24,25,26,27,31,12] + [28,32,33,34,23] + [18]t |
51 | 7 | 6 | 3 | [2,3,4,5,6,7] + [9, 11,10,12,13,14] + [16,17,18,19,20,21] + [24,44,26,27,28,29] + [49,32,33,34,35,37] + [47,48,40,41,45,43] + [42,46,38,15,8] + [25]t |
65 | 7 | 6 | 3 | [2,3,4,5,6,7] + [9,10,11,12,13,14] + [16,17,18,19,22,21] + [23,24,25,26,27,28] + [58,31,33,34,35,36] + [38,20,40,8,42,43] + [45,46,47,48,49,41] + [52,53,54,55,56,63] + [50,60,61,62,57] + [32]t |
65 | 7 | 6 | 3 | [2,3,4,5,6,7] + [9,10,11,12,13,14] + [16,17,18,19,22,21] + [23,24,25,26,27,28] + [58,31,33,34,35,36] + [38,20,40,8,42,43] + [45,46,47,48,49,41]+ [52,53,54,55,56,63] + [50,60,61,62,57] + [32]t |
79 | 7 | 6 | 3 | [2,3,4,5,6,8] + [9,10,11,12,13,16] + [15,17,18,19,20,22] + [23,24,29,26,27,28] + [44,31,32,33,34,35] + [38,3740,42,41,43] + [14,46,47,48,51,76] + [52,53,54,55,56,57] + [36,60,61,62,49,64] + [66,67,68,69,74,59] + [73,70,75,1,21] + [39]t |
93 | 7 | 6 | 3 | [2,3,4,5,6,7] + [9,10,11,12,13,14] + [16,17,18,19,20,21] +[32,24,25,26,27,28] + [30,31,84,33,34,35] + [37,38,39,43,41,42] +[44,45,50,47, 49,48] + [90,53,58,55,56,57] + [59,60,61,62,63,66] +[64,67,68,69,70,71] + [8,74,75,76,77,78] + [80,81,82,83,15,40] +[87,88,54,52,1] + [46]t |
27 | 9 | 6 | 3 | [2,10,5,4 6,7,8,9] + [20,12,11,14,16,17,18,19] + [21,22,23,24,25] + [13]t |
45 | 9 | 6 | 3 | [2,3,4,5,6,7,8,10] + [11,19, 14,13,15, 17,16,18] + [23,34,21,24,25,26,27,28] +[,31,30,32,33,42,38,36,37] + [39,40,41,20,1] + [22]t |
63 | 9 | 6 | 3 | [2,3,4,5,6,7,8,9] + [11,12, 15,13,14,16,17,1] + [20,21,22,28,24,61,26,27] +[29,44,30, 33,32,34,35,36] + [39,40,41,42,43,23,45,46] +[58,59,50,51,52,60,54,55] + [47,48,49,38,10] + [31]t |
81 | 9 | 6 | 3 | [2,3,4,5,6,7,8,9] + [11,12,13,14,15,16,17,18] + [20,21,22,23,24,25,26,27] +[30,29,31,32,33,34,35,68] + [42,39,38,43,45,69,41,46] +[48,49,50,51,73,53,54,55] + [57,58,59,60,61,62,63,64] +[66,67,10,1, 70,71,74,56] + [75,19,77,78,79] + [40]t |
99 | 9 | 6 | 3 | [2,3,4,5,6,7,8,9] + [11,12,13,14,15,16,18,17] + [30,21,23,22,24,25,26,27] + [29,20,31,32,33,34,35,36] + [38,39,40,41,42,43,67,45] + [48,4750,,52,53,51,55,88] + [57,65,59,60,61,62,63,64] + [79,28,68,69,70,71,72,73] + [94,93,77,78,66,10,91,82] + [84,85,86,87,1,89,90,81] + [76,75,74,19,56] + [49]t |
19 | 9 | 7 | 3 | [11,2,6,4,5,10,7,8] + [12,13,14,16,15,17] + [9]t |
37 | 9 | 7 | 3 | [2,3,4,5,6,7,8,9] + [11,12,14,13,15, 17,16,19] + [21,23,22,24,25,34,10,1] + [30,31,32,33,26,35] + [18]t |
55 | 9 | 7 | 3 | [2,3,4,5,6,7,8,9] + [12,11,13,14,15,16,17,18] + [20,29,22,23,24,26,25,28] + [30,31,32,34,33,38,36,37] + [39,40,41,43,42,44,45,49] + [48,1,50,51,52,21] + [27]t |
73 | 9 | 7 | 3 | [2,3,4,5,6,7,8,9] + [29,12,13,14,15,16,17,18] + [20,30,24,23,22,26,25,27] + [65,39,31,32,33, 35,34,37] + [21,40,41,43,42,44,45,46] + [67,68,59,60,61,62,69,64] + [55,57,58,63,70,71] + [36]t |
91 | 9 | 7 | 3 | [2,3,4,5,6,7,8,9] + [11,12,13,14,15,16,17,18] + [,2120,22,23,24,25,26,27] + [40,30,31,32,33,34,35,36] + [38,39,29,41,42,43,44,47] + [85,49,50,51,52,53,54,55] + [79, 59,58,60,61,62,63,10] + [66,67,68, 70,69,71,72,73] + [75,76,77,78, 80,57,81,19] + [84,48,86,65,56,28] + [45]t |
29 | 9 | 8 | 3 | [2,3,4,5,6,7,8,9] + [11, 13,21,15,16,17,18,19] + [1,22,23,24,25,26,27] + [14]t |
47 | 9 | 8 | 3 | [2,3,4,5,6,7,8,10] + [11,12,13,14,15,16,17,18] + [20,21,40,24,25,26,27,28] + [9,31, 33,32,34,35,36,37] + [39,30,41,42,43,44,45] + [23]t |
65 | 9 | 8 | 3 | [2,3,4,5,6,7,8,9] + [11,12, 14,13,15,16,19,18] + [1,21,28,23,24,25,26,27] + [30,31,29,33,34,36,35,37] + [39,40,47,42,43,44,45,46] + [48,49,56,51,54,53,52,22] + [57,58,59,60,61,62,41] + [32]t |
83 | 9 | 8 | 3 | [2,3,4,5,6,7,8,9] + [11,12,13,14,15,16,17,18] + [20,21,22,23,24,25,26,27] + [29,30,31,32,33,34,35,36] + [39,74,40, 43,42,44,45,46] + [81,49,50,51,52,53,54,55] + [57,80,60,71,61,62,63,64] + [67,66,28,69,70,59,77,73] + [75,76,72,78,79,19,10] + [41]t |
Appendix 2
Minimal CBRMDs constructed through series 4.2 for v ≤ 70
v | p 1 | p 2 | p 3 | Sets of shifts |
|---|---|---|---|---|
23 | 7 | 5 | 4 | [11,4,3,5,6,7] + [9,2,19,13,14,15] + [17,18,12,20] + [1,10]t |
37 | 7 | 5 | 4 | [2,3,4,5,6,7] + [8,11,35,12,13,14] + [16,18,19,25,21,22] + [10,24,32,27,28,29] + [31,20,33,34] + [1,17]t |
51 | 7 | 5 | 4 | [17,3,4,5,6,7] + [9,11,26,12,13,14] + [16,18,34,19,20,21] + [23,2,10,27,28,29] + [30,32,33,47,35,36] + [38,49,40,41,42,46] + [45,43,25,48] + [1,24]t |
65 | 7 | 5 | 4 | [2,32,4,5,6,7] + [53,10,11,12,13,14] + [63,17,18,19,20,33] + [23,34,25,26,27,28] + [30,3,21,24,41,36] + [62,39,55,35,42,43] + [61,60,59,48,49,56]+ [52,9,54,40,50,57] + [47,46,45,38] + [1,31]t |
79 | 7 | 5 | 4 | [45,3,4,5,6,7] + [77,10,11,12,13,14] + [16,17,18,15,47,21] + [75,25,24,26,27,28] + [30,31,32,33,34,35] + [37,36,69,41,42,43] + [2,46,66,48,49,50] + [52,53,59,55,56,57] + [76,73,61,62,67,64] + [9,68,70,40,60,71] + [63,74,23,54] + [1,38]t |
93 | 7 | 5 | 4 | [59,3,4,5,6,7] + [17,10,11,12,13,14] + [16,68,18,19,20,21] + [23,24,25,26,27,28] + [77,29,32,33,34,35] + [91,72,39,40,41,42] + [44,79,47,48,49,50] + [90,88,54,55,56,57] + [87,58,61,62,63,64] + [66,67,76,69,73,71] + [70,74,75,84,30,81] + [80,82,78,52,89,85] + [2,83,53,9] + [1,45]t |
27 | 9 | 5 | 4 | [25,3,5,4,6,7,8,9] + [24,13,14,15,16,17,18,19] + [21,22,23,10] + [1,12]t |
45 | 9 | 5 | 4 | [36,3,4,5,6,7,8,9] + [43,12,22,14,15,16,17,18] + [20,13,23,24,30,26,27,28] + [25,32,33,31,34,35,42,37] + [38,40,41,2] + [1,21]t |
63 | 9 | 5 | 4 | [2,11,4,5,6,7,8,9] + [61,12,14,13,15,16,17,18] + [19,21,55,23,24,25,26,27] + [29,31,32,33,34,40,36,37] + [60,59,48,42,43,44,45,46] + [41,49,50,57,52,53,54,22] + [51,58,35,39] + [1,30]t |
81 | 9 | 5 | 4 | [28,3,4,5,6,7,8,9] + [36,12,14,13,15,16,17,18] + [70,21,22,23,25,24,26,27] + [30,29,31,32,33,34,35,59] + [38,54,41,42,43,44,45,46] + [48,49,76,51,52,53,40,55] + [20,58,11,60,77,62,63,64] + [79,67,78,69,57,71,72,73] + [75,50,61,68] + [1,39]t |
99 | 9 | 5 | 4 | [46,3,4,5,6,7,8,9] + [72,12,13,14,15,16,17,18] + [97,21,23,22,24,25,26,27] + [20,30,31,38,33,34,35,36] + [91,65,41,40,42,43,44,45] + [70,49,50,51,52,53,54,55] + [94,58,59,88,61,62,63,64] + [2,67,76,69,47,71,11,73] + [95,96,83,78,77,92,81,89] + [85,84,86,87,60,82,90,32] + [66,68,75,57] + [1,48]t |
17 | 7 | 6 | 4 | [13,4,3,5,6,8] + [10,11,2,12,14] + [1,7]t |
31 | 7 | 6 | 4 | [27,4,3,5,6,7] + [9,10,12,11,13,15] + [29,18,19,16,21,22] + [24,23,2,26,28] + [1,14]t |
45 | 7 | 6 | 4 | [11,3,4,5,6,7] + [9,23,2,12,13,14] + [16,17,18,19,30,22] + [43,42,26,27,28,29] + [41,40,39,34,35,38] + [36,33,32,20,31] + [1,21]t |
59 | 7 | 6 | 4 | [25,3,4,5,6,7] + [41,10,11,12,13,14] + [57,17,18,19,20,21] + [36,24,2,26,27,29] + [38,32,33,35,34,23] + [31,39,40,51,42,43] + [45,46,54,48,49,50] + [52,53,47,55,9] + [1,28]t |
73 | 7 | 6 | 4 | [39,3,4,5,6,7] + [69,10,11,12,13,14] + [27,17,18,19,20,21] + [23,24,26,25,61,28] + [31,71,32,33,48,36] + [70,40,2,47,42,43] + [45,46,41,68,49,60] + [52,53,54,30,56,57] + [59,38,9,62,63,64] + [66,67,34,16,50] + [1,35]t |
87 | 7 | 6 | 4 | [53,3,4,5,6,7] + [9,10,11,12,13,14] + [16,15,18,19,61,21] + [2,35,25,26,27,28] + [68,31,32,33,34,24] + [37,38,39,40,41,43] + [45,46,44,48,49,58] + [52,23,51,55,56,57] + [59,47,70,62,63,64] + [85,67,84,69,60,79] + [73,66,75,76,77,78] + [80,81,82,83,30] + [1,42]t |
19 | 9 | 6 | 4 | [17,3,6,4,5,7,9,10] + [12,13,14,16,15] + [1,8]t |
37 | 9 | 6 | 4 | [20,3,4,5,6,7,8,9] + [11,12,14,13,15,16,18,19] + [35,23,22,24,25,2,27,28] + [29,31,32,33,3] + [1,17]t |
55 | 9 | 6 | 4 | [2,3,4,5,6,7,8,9] + [12,38,13,14,15,17,16,18] + [39,53,22,23,24,25,27,28] + [21,31,32,33,35,34,36,37] + [30,40,41,42,43,44,45,46] + [48,49,50,51,52] + [1,26]t |
73 | 9 | 6 | 4 | [20,3,4,5,6,7,8,9] + [11,21,13,14,15,16,17,18] + [29,12,23,22,24,25,26,27] + [2,45,31,32,33,34,36,37] + [39,40,41,42,43,44,70,66] + [71,49,50,51,65,53,54,55] + [57,58,59,60,61,30,63,64] +[46,67,68,69,62] + [1,35]t |
91 | 9 | 6 | 4 | [38,3,4,5,6,7,8,9] + [56,12,13,14,15,16,17,18] + [21,74,22,23,24,25,26,27] + [89,30,31,32,33,34,35,36] + [20,39,37,41,42,43,45,46] + [48,49,50,51,61,53,54,5] + [57,59,58,60,52,62,63,64] +[88,87,86,69,70,71,72,76] + [85,73,77,78,79,80,81,84] + [82,75,68,67,66] + [1,44]t |
29 | 9 | 7 | 4 | [2,3,4,5,6,7,8,9] + [11,10,14,15,16,17,18,19] + [21,22,23,24,25,26] + [1,13]t |
47 | 9 | 7 | 4 | [40,3,4,5,6,7,8,9] + [11,12,13,14,15,16,17,21] + [30,18,23,24,25,26,27,28] +[45,31,33,32,34,35,36,37] + [38,2,41,42,43,44] + [1,22]t |
65 | 9 | 7 | 4 | [12,3,4,5,6,7,8,9] + [11,2,13,14,15,16,17,18] + [63,21,19,23,24,25,26,27] + [29,46,32,33,34,35,36,37] + [39,53,41,42,43,44,45,30]+ [48,20,50,51,52,62,54,55] + [57,58,59,60,61,40] + [1,31]t |
83 | 9 | 7 | 4 | [30,3,4,5,6,7,8,9] + [11,41,13,14,15,16,17,18] + [59,12,22,23,24,25,26,27] + [29,61,31,32,33,34,35,36] + [38,39,66,,42,43,44,45,46] + [48,49,74,51,52,53,54,55] + [57,50,58,69,2,62,63,64] + [20,65,68,60,70,71,72,73] + [21,76,77,78,79,80] + [1,40]t |
21 | 9 | 8 | 4 | [2,3,5,4,7,6,8,14] + [12,13,10,15,16,17,18] + [1,9]t |
39 | 9 | 8 | 4 | [24,3,4,5,6,7,8,9] + [11,12,13,14,30,16,17,19] + [21,23,22,15,37,26,27,28] + [2,31,32,33,34,35,36] + [1,18]t |
57 | 9 | 8 | 4 | [2,3,4,5,6,7,8,9] + [11,13,45,14,15,16,17,18] + [20,21,22,23,30,25,26,28] + [24,10,32,33,34,36,35,37] + [55,40,41,42,43,44,31,46] + [48,49,47,51,52,53,54] + [1,27]t |
74 | 9 | 8 | 4 | [22,3,4,5,6,7,8,9] + [24,12,13,14,15,16,17,18] + [20,21,41,23,11,25,26,27] + [57,73,31,32,33,34,35,37] + [39,40,2,42,43,44,45,68] + [48,49,50,51,52,53,30,55] + [72,71,69,60,61,67,63,64] + [66,62,46,59,70,58,54] + [1,36]t |
93 | 9 | 8 | 4 | [40,3,4,5,6,7,8,9] + [60,12,13,14,15,16,17,18] + [80,21,22,23,24,25,26,27] + [29,30,31,11,33,34,35,36] + [38,39,2,66,42,43,44,46] + [47,41,50,51,52,53,54,55] + [88,58,59,32,61,62,63,64] + [49,91,68,69,77,71,72,73] + [75,76,70,78,79,20,81,82] + [84,85,86,87,56,89,90] + [1,45]t |
Appendix 3
Minimal CBRMDs constructed through series 4.3 for v ≤ 100
v | p 1 | p 2 | p 3 | Sets of shifts |
|---|---|---|---|---|
25 | 7 | 6 | 5 | [15,3,4,5,6,7] + [22,11,18,13,14,2] + [17,10,19,20,21] + [1,12,23]t |
39 | 7 | 6 | 5 | [2,3,4,5,6,7] + [10,9,36,17,13,14] + [16,18,32,20,21,22] + [24,25,26,27,28,29] + [12,30,33,34,35] + [1,19,37]t |
53 | 7 | 6 | 5 | [19,3,4,5,6,7] + [9,30,11,12,13,14] + [16,17,18,42,20,21] + [50,25,3927,28,29] + [49,32,33,34,35,40] + [38,24,36,41,31,43] + [45,46,44,48,2] + [1,26,51]t |
67 | 7 | 6 | 5 | [2,34,4,5,6,7] + [9,58,11,12,13,14] + [64,17,18,19,37,21] + [39,24,25,26,27,28] + [31,41,32,3,35,36] + [38,23,40,30,46,43] + [63,62,61,48,52,59] + [49,53,50,55,56,57] + [54,60,47,42,45] + [1,65,33]t |
81 | 7 | 6 | 5 | [47,3,4,5,6,7] + [78,17,11,12,13,14] + [16,44,18,19,20,21] + [77,24,25,26,27,28] + [43,31,32,29,34,35] + [37,39,38,41,45,30] + [42,46,23,48,49,61] + [67,53,54,55,56,57] + [9,60,76,62,63,64] + [65,59,68,75,70,71] + [73,74,69,50,2] + [1,40,79]t |
95 | 7 | 6 | 5 | [61,3,4,5,6,7] + [9,20,11,12,13,14] + [16,17,19,18,75,21] + [23,24,25,32,27,28] + [87,31,26,33,34,35] + [37,38,39,40,41,42] + [89,46,43,48,49,50] + [52,44,54,55,56,57] + [59,60,2,62,63,64] + [53,67,69,68,70,71] + [73,74,10,76,77,78] + [80,81,65,83,84,85] + [30,88,92,90,91] + [1,47,93]t |
29 | 9 | 6 | 5 | [2,3,4,5,6,7,8,9] + [11,13,10,15,16,17,18,19] + [20,22,23,24,25] + [1,14,27]t |
47 | 9 | 6 | 5 | [40,3,4,5,6,7,8,9] + [11,12,13,14,15,16,17,18] + [31,21,19,24,25,26,27,28] + [30,44,32,33,34,35,39,37] + [36,2,41,42,43] + [1,23,45]t |
65 | 9 | 6 | 5 | [12,3,4,5,6,7,8,9] + [11,2,13,14,15,16,17,18] + [62,21,19,23,24,25,26,27] + [28,31,48,33,34,35,36,37] + [30,40,41,42,43,44,45,52] + [60,49,50,51,20,53,54,55] + [57,58,59,39,61] + [1,32,63]t |
83 | 9 | 6 | 5 | [30,3,4,5,6,7,8,9] + [40,12,13,14,15,16,17,18] + [50,21,22,23,24,25,26,27] + [29,61,31,32,33,34,35,36] + [38,39,66,42,43,44,45,46] + [48,49,74,51,52,53,54,55] + [57,59,58,60,2,62,63,64] + [75,65,68,69,70,71,72,73] + [20,76,77,78,79] + [1,41,81]t |
21 | 9 | 7 | 5 | [2,3,4,5,7,6,8,9] + [12,14,13,15,11,17] + [1,10,19]t |
39 | 9 | 7 | 5 | [24,3,4,5,6,7,8,9] + [27,13,12,14,15,16,18,17] + [21,2,22,23,25,26,11,28] + [30,31,29,33,34,35] + [1,19,37]t |
57 | 9 | 7 | 5 | [2,3,4,5,6,7,8,9] + [46,13,10,14,15,16,17,18] + [20,21,22,23,24,25,26,34] + [40,31,32,33,54,35,36,37] + [39,50,52,42,43,44,45,30] + [48,49,11,51,41,53] + [1,28,55]t |
75 | 9 | 7 | 5 | [22,3,4,5,6,7,8,9] + [11,12,26,14,15,16,17,18] + [20,21,41,23,24,25,13,27] + [72,59,31,32,33,34,36,35] + [61,2,40,42,43,44,45,46] + [48,49,50,51,52,53,30,55] + [57,58,54,60,29,68,63,64] + [66,67,62,69,70,71] + [1,37,73]t |
93 | 9 | 7 | 5 | [40,3,4,5,6,7,8,9] + [60,12,13,14,15,16,17,18] + [80,21,22,23,24,25,26,27] + [90,30,41,32,33,34,35,36] + [38,39,2,31,79,43,45,44] + [48,49,42,51,52,53,54,55] + [57,58,59,11,20,62,63,64] + [65,67,68,69,85,71,72,73] + [75,77,76,78,29,61,81,84,83] + [82,70,86,87,88,89] + [1,46,91]t |
31 | 9 | 8 | 5 | [2,4,3,6,5,7,8,9] + [11,12,13,14,26,17,18,19] + [21,22,23,25,24,10,27] + [1,15,29]t |
49 | 9 | 8 | 5 | [44,3,4,5,6,7,8,9] + [20,12,13,15,14,16,17,18] + [39,21,22,28,27,25,26,23] + [2,41,32,33,34,35,36,37] + [11,40,31,42,43,30,45] + [1,24,47]t |
67 | 9 | 8 | 5 | [14,3,4,5,6,7,8,9] + [11,21,13,2,15,16,17,19] + [20,64,22,23,24,30,26,27] + [62,31,25,32,35,34,36,37] + [49,39,41,43,42,44,45,46] + [48,40,50,51,52,53,54,55] + [58,57,56,60,61,29,63] + [1,33,65]t |
85 | 9 | 8 | 5 | [32,3,4,5,6,7,8,9] + [11,45,13,14,15,16,17,18] + [21,56,22,23,24,25,26,27] + [29,30,31,71,33,34,35,36] + [38,40,39,41,44,43,82,46] + [81,80,58,51,52,53,54,55] + [12,50,59,68,61,62,63,64] + [66,67,60,69,70,49,72,78] + [75,76,77,73,79,2,48] + [1,42,83]t |
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Rasheed, M., Ismail, M., Shehzad, F. et al. Minimal Repeated Measurements Designs Balanced for Residual Effects in Circular Periods of Three Different Sizes. J Stat Theory Pract 13, 31 (2019). https://doi.org/10.1007/s42519-018-0030-7
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DOI: https://doi.org/10.1007/s42519-018-0030-7
Keywords
- Repeated measurements designs
- Residual effects
- Circular balanced repeated measurement designs
- Minimal designs