×

zbMATH — the first resource for mathematics

Optimal regular graph designs. (English) Zbl 1384.62280
Summary: 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 \(k \leq 10\) and replication \(r\leq 10\) and \(r(k-1)-(v-1)\lfloor r(k-1)/(v-1)\rfloor \leq 9\).

MSC:
62K10 Statistical block designs
62K05 Optimal statistical designs
05B05 Combinatorial aspects of block designs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Amerine, M.A., Roessler, E.B.: Wines: Their Sensory Evaluation. W. H, Freeman, San Francisco (1979)
[2] Bailey, R.A., Cameron, P.J.: Combinatorics of optimal designs. In: Surveys in Combinatorics 2009, London Mathematical Society Lecture Notes, vol 365, pp 19-73. Cambridge University Press, Cambridge (2009) · Zbl 1182.05010
[3] Bailey, RA, Designs for two-colour microarray experiments, Appl. Stat., 56, 365-394, (2007)
[4] Beckenbach, E.F., Bellman, R.: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer, Berlin (1965)
[5] Best, DJ; Rayner, JC; Allingham, D, A statistical test for ranking data from partially balanced incomplete block designs, J. Sensory Stud., 26, 81-84, (2011)
[6] Bose, RC; Nair, KR, Partially balanced incomplete block designs, Sankhya, 4, 337-372, (1939)
[7] Bose, RC; Shimamoto, T, Classification and analysis of partially balanced incomplete block designs with two associate classes, J. Am. Stat. Soc., 47, 151-184, (1952) · Zbl 0048.11603
[8] Cheng, CS, Optimality of certain asymmetrical experimental designs, Annal Stat., 6, 1239-1261, (1978) · Zbl 0396.62055
[9] Cheng, CS, Maximizing the total number of spanning trees in a graph: two related problems in graph theory and optimum design theory, J. Comb. Theory, 31, 240-248, (1981) · Zbl 0475.05050
[10] Cheng, CS, On the optimality of (M.S)-optimal designs in large systems, Sankhya, 54, 117-125, (1992)
[11] Cheng, CS; Bailey, RA, Optimality of some two-associate-class partially balanced incomplete-block designs, Annal Stat., 19, 1667-1671, (1991) · Zbl 0741.62071
[12] Clatworthy, W.H.: Tables of two-associate-class partially balanced designs, NBS Applied Mathematics Series, vol 63. The United States Department of Commerce Publications, National Bureau of Standards (U.S.) (1973) · Zbl 0443.62064
[13] Cochran, W.G., Cox, G.: Experimental Designs. John Wiley and Sons, New York, NY (1957) · Zbl 0077.13205
[14] Constantine, GM, On the E-optimality of PBIB designs with a small number of blocks, Annal Stat., 10, 1027-1031, (1982) · Zbl 0489.62067
[15] Constantine, GM, On the optimality of block designs, Ann. Inst. Stat. Math., 38, 161-174, (1986) · Zbl 0588.62123
[16] Gacula, M.C., Singh, J., Bi, J., Altan, S.: Statistical Methods in Food and Consumer Research. Academic Press, New York, NY (2009)
[17] John, J.A., Mitchell, T.J.: Optimal Incomplete Block Designs. ORNL/CSD-8 Available from the National Technical Information Service, The United States Department of Commerce, 5285 Port Royal Road, Springfield, VA (1976)
[18] John, J.A., Wolock, F.W., David, H.A.: Cyclic designs, NBS Applied Mathematics Series, vol 62. The United States Department of Commerce Publications, National Bureau of Standards (U.S.) (1972) · Zbl 0918.05062
[19] John, JA, Reduced group divisible paired comparison designs, Ann. Math. Stat., 38, 1887-1893, (1967)
[20] John, J.A., Mitchell, T.J.: Optimal incomplete block designs. J. R. Stat. Soc. 39B, 39-43 (1977) · Zbl 0354.05015
[21] John, JA; Williams, ER, Conjectures for optimal block designs, J. R. Stat. Soc., 44B, 221-225, (1982) · Zbl 0491.62063
[22] Jones, B; Eccleston, JA, Exchange and interchange procedures to search for optimal designs, J. R. Stat. Soc., 42, 238-243, (1980) · Zbl 0443.62064
[23] Kerr, MK; Churchill, GA, Experimental design for gene expression microarrays, Biostatistics, 2, 183-201, (2001) · Zbl 1097.62562
[24] Kiefer, J.: Optimality and Construction of Generalized Youden Designs. A Survey of Statistical Designs and Linear Models, pp. 333-354. North-Holland, Amsterdam (1975)
[25] Kirchhoff, G, Über die auflösung der gleichung, auf welche man bei der untersuchung der linearen verteilung galvanischer ströme geführt wird, Ann. Phys. Chem., 72, 497-508, (1847)
[26] Lockshin, L; Mueller, S; Louviere, J; Francis, L; Osidacz, P, Development of a new method to measure how consumers choose wine, Aust. N. Z. Wine Ind. J., 24, 81-84, (2011)
[27] Meringer, M, Fast generation of regular graphs and construction of cages, J. Graph Theory, 30, 137-146, (1999) · Zbl 0918.05062
[28] Morgan, JP, Optimal incomplete block designs, J. Am. Stat. Assoc., 102, 655-663, (2007) · Zbl 1172.62321
[29] Shah, K.R., Sinha, B.K.: Theory of Optimal Designs, Lecture Notes in Statistics, vol 54. Springer, Berlin (1989) · Zbl 0688.62043
[30] Soicher, L.H.: The DESIGN Package for GAP, Version 1.4. http://designtheory.org/software_design/ (2006)
[31] Takeuchi, K, On the optimality of certain type of PBIB designs, Rep. Stat. Appl. Res. Union Jpn. Sci. Eng., 8, 140-145, (1961) · Zbl 0104.12503
[32] Wit, E; Nobile, A; Khanin, R, Near-optimal designs for dual channel microarray studies, Appl. Stat., 54, 817-830, (2005) · Zbl 05188715
[33] Wolfram Research, I.: Mathematica (2012) · Zbl 0104.12503
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.