Abstract
A typical problem in optimal design theory is finding an experimental design that is optimal with respect to some criteria in a class of designs. The most popular criteria include the A- and D-criteria. Regular graph designs occur in many optimality results, and if the number of blocks is large enough, an A-optimal (or D-optimal) design is among them (if any exist). To explore the landscape of designs with a large number of blocks, we introduce extensions of regular graph designs. These are constructed by adding the blocks of a balanced incomplete block design repeatedly to the original design. We present the results of an exact computer search for the best regular graph designs and the best extended regular graph designs with up to 20 treatments v, block size \(k \le 10\) and replication r \(\le 10\) and \(r(k-1)-(v-1)\lfloor r(k-1)/(v-1)\rfloor \le 9\).
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Acknowledgements
The author would like to thank Peter J. Cameron and R. A. Bailey for the many conversations on optimal designs and is extremely grateful to J. P. Morgan for bringing the topic to her attention and the helpful discussions about RGDs and the arising computational difficulties. Joachim Kunert has been an invaluable source of information for preference designs. Parts of the computations were performed using the facilities at the London Research Institute, Cancer Research UK.
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Appendices
Appendix 1: Catalogue of A- and D-best RGDs
There follows a table of the A- and D-best RGDs (except for the case \(v=14\), \(r=5\), \(k=2\), where only the D-best RGD is included). These RGDs also produce optimal BIBD-extended RGDs for large y. We list \(\delta =r(k-1)-\lambda (v-1)\) and the smallest \({\tilde{\lambda }}\) such that a 2-\((v,k,{\tilde{\lambda }})\)-design exists. There exists only one irreducible matrix for \(\delta =2\) or \(\delta =v-1\), and we therefore exclude these cases.
Most designs can be found in either Clatworthy (1973) or John et al. (1972). To follow the convention as in John and Mitchell (1977), we write in the reference column P.XY for a design in Clatworthy (1973) with reference number XY and C.XY for a design in John et al. (1972) with reference number XY. If the design is not in either catalogues but can be found in John and Mitchell (1977) we give the reference number as JM.XY, and if it is cyclic, we give the initial blocks. If the design is the complement of a design in one of the catalogue, we add an (C) to the reference number. All other designs can be found in the appendix. However, some of these designs are possible to construct with known methods, such as the ones in John (1967). A full catalogue of all designs as XML files in external representation format is included in the electronic appendix. For more information on external representation see Soicher (2006).
v | k | r | \({\tilde{\lambda }}\) | \(\delta \) | Reference |
|---|---|---|---|---|---|
6 | 2 | 3 + 5y | 1 | 3 | P.SR6,C.A2 |
6 | 2 | 4 + 5y | 1 | 4 | P.R18,C.A3 |
6 | 2 | 8 + 5y | 1 | 3 | P.R24 |
6 | 2 | 9 + 5y | 1 | 4 | P.R27 |
6 | 3 | 4 + 5y | 2 | 3 | (1,2,4)(1,3,5) |
6 | 3 | 7 + 5y | 2 | 4 | P.R46 |
6 | 3 | 9 + 5y | 2 | 3 | P.R52 |
6 | 4 | 6 + 10y | 6 | 3 | P.SR35 |
6 | 4 | 8 + 10y | 6 | 4 | P.R96 |
7 | 2 | 4 + 6y | 1 | 4 | #1 |
7 | 2 | 10 + 6y | 1 | 4 | JM.1 |
7 | 5 | 10 + 15y | 10 | 4 | #2 |
8 | 2 | 3 + 7y | 1 | 3 | C.A7 |
8 | 2 | 4 + 7y | 1 | 4 | P.SR9,C.A8 |
8 | 2 | 5 + 7y | 1 | 5 | JM.3 |
8 | 2 | 6 + 7y | 1 | 6 | P.R29,C.A10 |
8 | 2 | 10 + 7y | 1 | 3 | (1,2)(1,2)(1,3)(1,4)(1,5)(1,5) |
8 | 3 | 3 + 21y | 6 | 6 | P.R54,C.B3 |
8 | 3 | 6 + 21y | 6 | 5 | JM.4 |
8 | 3 | 9 + 21y | 6 | 4 | P.R58 |
8 | 4 | 4 + 7y | 3 | 5 | C.B6 |
8 | 4 | 6 + 7y | 3 | 4 | P.SR38 |
8 | 4 | 8 + 7y | 3 | 3 | (1,2,3,5)(1,2,3,6) |
8 | 4 | 9 + 7y | 3 | 6 | P.R101 |
8 | 5 | 5 + 35y | 20 | 6 | P.R314,C.B3(C) |
8 | 5 | 10 + 35y | 20 | 5 | JM.4(C) |
8 | 6 | 9 + 21y | 15 | 3 | C.A7(C) |
9 | 2 | 4 + 8y | 1 | 4 | C.A11 |
9 | 2 | 6 + 8y | 1 | 6 | P.R34,C.A13 |
9 | 3 | 3 + 4y | 1 | 6 | P.SR23 |
9 | 3 | 6 + 4y | 1 | 4 | JM.8 |
9 | 3 | 7 + 4y | 1 | 6 | P.R62 |
9 | 3 | 10 + 4y | 1 | 4 | JM.10 |
9 | 4 | 4 + 8y | 3 | 4 | C.B12 |
9 | 5 | 5 + 10y | 5 | 4 | C.B12(C) |
9 | 6 | 6 + 8y | 5 | 6 | P.SR65 |
10 | 2 | 3 + 9y | 1 | 3 | P.T2 |
10 | 2 | 4 + 9y | 1 | 4 | C.A16 |
10 | 2 | 5 + 9y | 1 | 5 | P.SR11,C.A17 |
10 | 2 | 6 + 9y | 1 | 6 | JM.12 |
10 | 2 | 7 + 9y | 1 | 7 | JM.13 |
10 | 2 | 8 + 9y | 1 | 8 | P.R36,C.A20 |
10 | 3 | 3 + 9y | 2 | 6 | JM.14 |
10 | 3 | 6 + 9y | 2 | 3 | P.T12 |
10 | 4 | 4 + 6y | 2 | 3 | P.T33 |
10 | 4 | 8 + 6y | 2 | 6 | JM.15 |
10 | 4 | 10 + 6y | 2 | 3 | P.T37 |
10 | 5 | 6 + 9y | 4 | 6 | JM.16 |
10 | 5 | 8 + 9y | 4 | 5 | JM.17 |
10 | 5 | 10 + 9y | 4 | 6 | JM.19 |
10 | 6 | 6 + 9y | 5 | 3 | P.T60 |
10 | 7 | 7 + 21y | 14 | 6 | JM.14(C) |
11 | 2 | 4 + 10y | 1 | 4 | C.22 |
11 | 2 | 6 + 10y | 1 | 6 | #3 |
11 | 2 | 8 + 10y | 1 | 8 | JM.20 |
11 | 3 | 3 + 15y | 3 | 6 | #4 |
11 | 3 | 9 + 15y | 3 | 8 | JM.21 |
11 | 4 | 8 + 20y | 6 | 4 | C.B25(C) |
11 | 8 | 8 + 40y | 28 | 6 | #4(C) |
12 | 2 | 3 + 11y | 1 | 3 | JM.22 |
12 | 2 | 4 + 11y | 1 | 4 | C.A27 |
12 | 2 | 5 + 11y | 1 | 5 | #5 |
12 | 2 | 6 + 11y | 1 | 6 | P.SR13,C.A29 |
12 | 2 | 7 + 11y | 1 | 7 | JM.24 |
12 | 2 | 8 + 11y | 1 | 8 | P.R38,C.A31 |
12 | 2 | 9 + 11y | 1 | 9 | P.R39,C.A32 |
12 | 3 | 3 + 11y | 2 | 6 | #6 |
12 | 3 | 4 + 11y | 2 | 8 | P.SR26 |
12 | 3 | 7 + 11y | 2 | 3 | JM.25 |
12 | 3 | 9 + 11y | 2 | 7 | JM.27 |
12 | 3 | 10 + 11y | 2 | 9 | P.R78,C.B34 |
12 | 4 | 3 + 11y | 3 | 9 | P.SR41 |
12 | 4 | 5 + 11y | 3 | 4 | C.B37 |
12 | 4 | 6 + 11y | 3 | 7 | JM.28 |
12 | 5 | 5 + 55y | 20 | 9 | P.R145,C.B43 |
12 | 6 | 10 + 11y | 5 | 6 | P.SR71 |
12 | 7 | 7 + 77y | 42 | 9 | P.R176, |
C.B43(C) | |||||
12 | 9 | 9 + 33y | 24 | 6 | #6(C) |
13 | 2 | 4 + 12y | 1 | 4 | C.A35 |
13 | 2 | 6 + 12y | 1 | 6 | #7 |
13 | 2 | 8 + 12y | 1 | 8 | #8 |
13 | 3 | 9 + 6y | 1 | 6 | #9 |
7 13 | 5 | 5 + 15y | 5 | 8 | C.B58 |
14 | 2 | 3 + 13y | 1 | 3 | #10 |
14 | 2 | 4 + 13y | 1 | 4 | #11 |
14 | 2 | 5 + 13y | 1 | 5 | #12 |
14 | 3 | 9 + 39y | 6 | 5 | #13 |
14 | 3 | 15 + 39y | 6 | 4 | #14 |
14 | 6 | 6 + 39y | 15 | 4 | C.B72 |
15 | 2 | 4 + 14y | 1 | 4 | #15 |
15 | 3 | 9 + 28y | 1 | 4 | C.B77 |
16 | 2 | 3 + 15y | 1 | 3 | #16 |
16 | 3 | 9 + 13y | 2 | 3 | #17 |
16 | 4 | 6 + 5y | 1 | 3 | P.R118 |
18 | 2 | 3 + 17y | 1 | 3 | #18 |
18 | 3 | 10 + 17y | 2 | 3 | #19 |
18 | 5 | 5 + 85y | 20 | 3 | #20 |
20 | 2 | 3 + 19y | 1 | 3 | #21 |
Appendix 2: Optimal regular graph designs
There follows a list of all designs not found in the used reference catalogues. Blocks are represented as k-tuples of treatments in the rows.
\(\underline{\#1\,v=7,\,k=2,\,r=4+6y}\)
\(\underline{\#2\,v=7,\,k=5,\,r=10+15y}\)
\(\underline{\#3\,v=11,\,k=2,\,r=6+10y}\)
\(\underline{\#4\,v=11,\,k=3,\,r=3+15y}\)
\(\underline{\#5\,v=12,\,k=2,\,r=5+11y}\)
\(\underline{\#6\,v=12,\,k=3,\,r=3+11y}\)
\(\underline{\#7\,v=13,\,k=2,\,r=6+12y}\)
\(\underline{\#8\,v=13,\,k=2,\,r=8+12y}\)
\(\underline{\#9\,v=13,\,k=3,\,r=9+6y}\)
\(\underline{\#10\,v=14,\,k=2,\,r=3+13y}\)
\(\underline{\#11\,v=14,\,k=2,\,r=4+13y}\)
\(\underline{\#12\,v=14,\,k=2,\,r=5+13y}\)
\(\underline{\#13\,v=14,\,k=3,\,r=9+39y}\)
\(\underline{\#14\,v=14,\,k=3,\,r=15+39y}\)
\(\underline{\#15\,v=15,\,k=2,\,r=4+14y}\)
\(\underline{\#16\,v=16,\,k=2,\,r=3+15y}\)
\(\underline{\#17\,v=16,\,k=3,\,r=9+13y}\)
\(\underline{\#18\,v=18,\,k=2,\,r=3+17y}\)
\(\underline{\#19\,v=18,\,k=3,\,r=10+17y}\)
\(\underline{\#20,v=18,\,k=5,\,r=5+85y}\)
\(\underline{\#21\,v=20,\,k=2,\,r=3+19y}\)
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Cakiroglu, S.A. Optimal regular graph designs. Stat Comput 28, 103–112 (2018). https://doi.org/10.1007/s11222-016-9720-8
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DOI: https://doi.org/10.1007/s11222-016-9720-8