Combinatorics of optimal designs

Abstract

To a combinatorialist, a design is usually a 2-design or balanced incompleteblock design. However, 2-designs do not necessarily exist in all cases where a statistician might wish to use one to design an experiment. As a result, statisticians need to consider structures much more general than the combinatorialist's designs, and to decide which one is "best" in a given situation. This leads to the theory of optimal design. There are several concepts of optimality, and no general consensus about which one to use in any particular situation.For block designs with fixed block size k, all these optimality criteria are determined by a graph, the concurrence graph of the design, and more specifically, by the eigenvalues of the Laplacian matrix of the graph. It turns out that the optimality criteria most used by statisticians correspond to properties of this graph which are interesting in other contexts: D-optimality involves maximizing the number of spanning trees; A-optimality involves minimizing the sum of resistances between all pairs of terminals (when the graph is regarded as an electrical circuit, with each edge being a one-ohm resistor); and E-optimality involves maximizing the smallest eigenvalue of the Laplacian (the corresponding graphs are likely to have good expansion and random walk properties). If you are familiar with these properties, you may expect that related "nice" properties such as regularity and large girth (or even symmetry) may tend to hold; some of our examples may come as a surprise! The aim of this paper is to point out that the optimal design point of view unifies various topics in graph theory and design theory, and suggests some interesting open problems to which combinatorialists of all kinds might turn their expertise. We describe in some detail both the statistical background and the mathematics of various topics such as Laplace eigenvalues of graphs. PreliminariesThis first section of the paper sets the scene. We look briefly at the different ways in which a combinatorialist and a statistician view a simple block design such as the Fano plane, and also introduce a running example which would not be recognised as a design under the standard combinatorial definition but is in fact the best design for a statistician in certain circumstances.Section 2 looks at the way that information about treatment differences is recovered from the results of an experiment conducted using a block design. We look briefly at what makes a good design, and show that all criteria for this can be expressed in terms of the concurrence graph of the design. Having thus focussed our attention on graphs, we give a brief survey of the Laplacian matrix of a graph and its eigenvalues, and mention the interesting question of which graphs can be concurrence graphs of block designs with given block size.Section 3 covers the three most important kinds of optimality, described by the letters A (average), D (determinant) and E (extreme), and how they look when expressed in terms of the concurrence matrix.In section ...

Relations among partitions

“…For further explanation of how combinatorial design problems arise from statistically designed experiments, see [22,33,177,202].…”

Graph-theoretic design and analysis of key predistribution schemes

“…These ideas have been formalised and studied in the theory of optimal designs for statistical experiments and networks using the three measures of A-, D-and E-optimality. We refer the reader to [3,4,5,22] for more details. Currently, little is known about these optimality criteria for designs with block size k > 2 or n > 20, so the results may be of limited use to KPS applications.…”

Optimal and efficient semi-Latin squares

“…If R is non-empty we apply a Sturm sequence (see Childs (1995, p.280-285)) to determine the number of (real) zeros of g( Since ∆ is an equireplicate design, the canonical efficiency factors of ∆ not equal to 1, and their multiplicities, are the same as those of the dual block design ∆ * of ∆ (see Bailey and Cameron (2009 There are of course other measures of efficiency than the A-, D-and Emeasures. One such measure, which is rational and can be computed exactly by the most recent version of the DESIGN package, is the MV-measure (see Bailey and Royle (1997); Bailey and Cameron (2009)).…”

“…One such measure, which is rational and can be computed exactly by the most recent version of the DESIGN package, is the MV-measure (see Bailey and Royle (1997); Bailey and Cameron (2009)). …”

“…We say that ∆ is A-optimal in a class C of (v, b, r, k)-designs containing ∆ if A ∆ ≥ A Γ for each Γ ∈ C. D-optimal and E-optimal are defined similarly. We refer the reader to Bailey and Cameron (2009) for the definitions of Schur-optimality and MV-optimality. When we apply the terms canonical efficiency factors, connected, and efficiency measure to a semi-Latin square S, we shall always mean the same as when those terms are applied to the underlying block design of S.…”