Using graphs to find the best block designs

A statistician designing an experiment wants to get as much information as possible from the data gathered. Often this means the most precise estimate possible (that is, an estimate with minimum possible variance) of the unknown parameters. If there are several parameters, this can be interpreted in many ways: do we want to minimize the average variance, or the maximum variance, or the volume of a confidence region for the parameters? In the case of block designs, these optimality criteria can be calculated from the concurrence graph of the design, and in many cases from its Laplacian eigenvalues. The Levi graph can also be used. The various criteria turn out to be closely connected with other properties of the graph as a network, such as number of spanning trees, isoperimetric number, and the sum of the resistances between pairs of vertices when the graph is regarded as an electrical network. In this chapter, we discuss the notions of optimality for incomplete-block designs, explain the graph-theoretic connections, and prove some old and new results about optimality.

[1]  Sreenivasan Ravi,et al.  Association Schemes: Designed Experiments, Algebra and Combinatorics , 2006 .

[2]  N.R. Malik,et al.  Graph theory with applications to engineering and computer science , 1975, Proceedings of the IEEE.

[3]  A. Kshirsagar.  A Note on Incomplete Block Designs , 1958 .

[4]  D. S. Tracy,et al.  Generalized Inverse Matrices: With Applications to Statistics , 1971 .

[5]  Byron Jones,et al.  Exchange and Interchange Procedures to Search for Optimal Designs , 1980 .

[6]  J. Kiefer.  Construction and optimality of generalized Youden designs II , 1975 .

[7]  K. D. Tocher.  The Design and Analysis of Block Experiments , 1952 .

[8]  Radoslav Harman.  Minimal efficiency of designs under the class of orthogonally invariant information criteria , 2004 .

[9]  J. Dodziuk.  Difference equations, isoperimetric inequality and transience of certain random walks , 1984 .

[10]  Raya Khanin,et al.  Near‐optimal designs for dual channel microarray studies , 2005 .

[11]  T. Tjur.  Block Designs and Electrical Networks , 1991 .

[12]  R. A. Bailey,et al.  Optimality of Some Two-Associate-Class Partially Balanced Incomplete-Block Designs , 1991 .

[13]  Peter J. Cameron,et al.  Strongly regular graphs , 2003 .

[14]  Shaun M. Fallat,et al.  Maximizing Algebraic Connectivity Over Unicyclic Graphs , 2003 .

[15]  R. A. Bailey.  Designs for two‐colour microarray experiments , 2007 .

[16]  R. A. Bailey.  Association Schemes: Designed Experiments, Algebra and Combinatorics , 2004 .

[17]  Rainer Schwabe,et al.  Optimum Designs for Multi-Factor Models , 1996 .

[18]  Ravindra B. Bapat,et al.  Optimal block designs with minimal number of observations , 1991 .

[19]  D. Preece.  Cyclic and Computer Generated Designs, 2nd Edition (Monographs on Statistics and Applied Probability 38) , by J. A. John & E. R. Williams, xii + 255 pp. London: Chapman & Hall (1995). £30.00 (hardback). ISBN 0 412 57580 9. , 1996, The Journal of Agricultural Science.

[20]  D-optimal designs with minimal and nearly minimal number of units , 1996 .

[21]  Peter J. Cameron,et al.  Combinatorics of optimal designs , 2008 .

[22]  Nripes Kumar Mandal,et al.  Uncertain Resources and Optimal Designs : Problems and Perspectives , 1990 .

[23]  Michael William Newman,et al.  The Laplacian spectrum of graphs , 2001 .

[24]  L. Shapiro.  An Electrical Lemma , 1987 .

[25]  E. M. Elsayed,et al.  DIFFERENCE EQUATIONS , 2001 .

[26]  Ching-Shui Cheng,et al.  Optimality of Certain Asymmetrical Experimental Designs , 1978 .

[27]  Noga Alon,et al.  lambda1, Isoperimetric inequalities for graphs, and superconcentrators , 1985, J. Comb. Theory, Ser. B.

[28]  Olaf Krafft,et al.  A-optimal connected block designs with nearly minimal number of observations , 1997 .

[29]  John P. Morgan,et al.  The completely symmetric designs with blocksize three , 2002 .

[30]  R. A. Bailey.  Variance and concurrence in block designs and distance in the corresponding graphs , 2009 .

[31]  J. John,et al.  Cyclic and Computer Generated Designs , 1995 .

[32]  Ching-Shui Cheng,et al.  Maximizing the total number of spanning trees in a graph: Two related problems in graph theory and optimum design theory , 1981, J. Comb. Theory B.

[33]  N. Abreu.  Old and new results on algebraic connectivity of graphs , 2007 .

[34]  F. W. Levi,et al.  Finite geometrical systems , 1942 .

[35]  Alan D. Sokal.  The multivariate Tutte polynomial (alias Potts model) for graphs and matroids , 2005, Surveys in Combinatorics.

[36]  N. Gaffke,et al.  D-optimal block designs with at most six varieties , 1982 .

[37]  N. Alon,et al.  il , , lsoperimetric Inequalities for Graphs , and Superconcentrators , 1985 .

[38]  G. Churchill,et al.  Experimental design for gene expression microarrays. , 2001, Biostatistics.

[39]  F. Yates.  A new method of arranging variety trials involving a large number of varieties , 1936, The Journal of Agricultural Science.

[40]  Norman Balabanian,et al.  Electrical Network Theory , 1969 .

[41]  J. John,et al.  Conjectures for Optimal Block Designs , 1982 .

[42]  G. Kirchhoff.  Ueber die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Vertheilung galvanischer Ströme geführt wird , 1847 .

[43]  R. A. Bailey,et al.  Design of comparative experiments , 2008 .

[44]  J. A. John,et al.  Optimal Incomplete Block Designs , 1977 .

[45]  A. Hora,et al.  Distance-Regular Graphs , 2007 .

[46]  N. Urquhart.  Generalized Inverse Matrices (with Applications to Statistics) , 1973 .

[47]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[48]  Edgar Brunner,et al.  Robustness considerations in selecting efficient two-color microarray designs , 2009, Bioinform..

[49]  Ching-Shui Cheng,et al.  Do nearly balanced multigraphs have more spanning trees? , 1985, J. Graph Theory.

[50]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[51]  K. R. Shah,et al.  Theory of Optimal Designs , 1989 .