Optimal semi-Latin squares with side six and block size two

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Bailey R. A. and Royle G. 1997Optimal semi-Latin squares with side six and block size twoProc. R. Soc. Lond. A.4531903–1914http://doi.org/10.1098/rspa.1997.0102SectionRestricted accessOptimal semi-Latin squares with side six and block size two R. A. Bailey R. A. Bailey School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK Google Scholar Find this author on PubMed Search for more papers by this author and G. Royle G. Royle Department of Computer Science, University of Western Australia, Nedlands 6009, Western Australia, Australia Google Scholar Find this author on PubMed Search for more papers by this author R. A. Bailey R. A. Bailey School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS, UK Google Scholar Find this author on PubMed Search for more papers by this author and G. Royle G. Royle Department of Computer Science, University of Western Australia, Nedlands 6009, Western Australia, Australia Google Scholar Find this author on PubMed Search for more papers by this author Published:08 September 1997https://doi.org/10.1098/rspa.1997.0102AbstractAn (n x n)/k semi-Latin square is like an n × n Latin square except that there are kletters in each cell. Each of the nkletters occurs once in each row and once in each column. Designs for experiments are assessed according to the statistical concept of efficiency factor. A high efficiency factor corresponds to low variances of within-block estimators. There are four widely used measures of the efficiency factor of a design: for each, any design which maximizes the value of the efficiency factor among a given class of designs is said to be optimal in that class. Previous theory gives optimal semi-Latin squares for various values of k for all values of n except for n=6. In this paper we therefore examine (6 x 6)/2 semi-Latin squares. We restrict attention to those semi-Latin squares whose quotient block designs are regular-graph designs, because a plausible and widely believed conjecture is that optimal regular–graph designs are optimal overall. For each of the four measures of efficiency factor, we find the optimal (6 × 6)/2 semi–Latin square among regular–graph semi–Latin squares of that size. Previous ArticleNext Article VIEW FULL TEXT DOWNLOAD PDF FiguresRelatedReferencesDetailsCited by Uto N and Bailey R (2022) Constructions for regular-graph semi-Latin rectangles with block size two, Journal of Statistical Planning and Inference, 10.1016/j.jspi.2022.02.007, 221, (81-89), Online publication date: 1-Dec-2022. Mba E, Chigbu P and Ukaegbu E (2021) Evaluating Popular Statistical Properties of Incomplete Block Designs: A MATLAB Program Approach, Mathematics, 10.3390/math9111281, 9:11, (1281) Bailey R, Cameron P, Soicher L and Williams E (2020) Substitutes for the Non-existent Square Lattice Designs for 36 Varieties, Journal of Agricultural, Biological and Environmental Statistics, 10.1007/s13253-020-00388-1, 25:4, (487-499), Online publication date: 1-Dec-2020. Uto N and Bailey R (2020) Balanced Semi-Latin Rectangles: Properties, Existence and Constructions for Block Size Two, Journal of Statistical Theory and Practice, 10.1007/s42519-020-00118-3, 14:3, Online publication date: 1-Sep-2020. Bailey R (2017) Semi‐Latin Squares Wiley StatsRef: Statistics Reference Online, 10.1002/9781118445112.stat02644.pub2, (1-8) Morgan J (2014) L atin Squares and Related Experimental Designs Wiley StatsRef: Statistics Reference Online, 10.1002/9781118445112.stat04085 Soicher L (2013) Optimal and efficient semi-Latin squares, Journal of Statistical Planning and Inference, 10.1016/j.jspi.2012.08.010, 143:3, (573-582), Online publication date: 1-Mar-2013. Soicher L (2013) Designs, Groups and Computing Probabilistic Group Theory, Combinatorics, and Computing, 10.1007/978-1-4471-4814-2_3, (83-107), . Soicher L (2012) Uniform Semi-Latin Squares and Their Schur-Optimality, Journal of Combinatorial Designs, 10.1002/jcd.21300, 20:6, (265-277), Online publication date: 1-Jun-2012. Bailey R (2011) Symmetric Factorial Designs in Blocks, Journal of Statistical Theory and Practice, 10.1080/15598608.2011.10412047, 5:1, (13-24), Online publication date: 1-Mar-2011. Morgan J (2007) L atin Squares and Related Experimental Designs Encyclopedia of Statistics in Quality and Reliability, 10.1002/9780470061572.eqr013, Online publication date: 14-Dec-2007. Bailey R (2007) Designs for two-colour microarray experiments, Journal of the Royal Statistical Society: Series C (Applied Statistics), 10.1111/j.1467-9876.2007.00582.x, 56:4, (365-394), Online publication date: 1-Aug-2007. Bailey R and Chigbu P (1997) Enumeration of semi-Latin squares, Discrete Mathematics, 10.1016/S0012-365X(96)00217-8, 167-168, (73-84), Online publication date: 1-Apr-1997. Bailey R (1997) A Howell design admitting A51, Discrete Mathematics, 10.1016/S0012-365X(96)00216-6, 167-168, (65-71), Online publication date: 1-Apr-1997. This Issue08 September 1997Volume 453Issue 1964 Article InformationDOI:https://doi.org/10.1098/rspa.1997.0102Published by:Royal SocietyPrint ISSN:1364-5021Online ISSN:1471-2946History: Published online08/09/1997Published in print08/09/1997 License: Citations and impact