Efficient and optimal designs for correlated observations.

This thesis considers some aspects of the problem of finding efficient and optimal designs when observations are correlated. The two main areas that are examined are nested row-column (NRC) designs and early generation variety trials (EGVTs). In NRC designs, the experimental area is divided into b blocks, and each block is divided into P1 rows and P2 columns (blocks of size P1 x P2). Here, optimal NRC designs, which can be constructed from semi-balanced arrays, are obtained under the assumption that within-block observations are correlated. For a stationary reflection symmetric dependence structure, optimal NRC designs with blocks of size 2 x 2 are obtained for models with fixed block effects, which may also include row and/or column effects. It is shown that the efficiency of binary designs can be very low for some correlation values. Also, optimal NRC designs for blocks of size 3 x 3 and P1 x 2 (P1 ≥ 3 ) are determined. The optimality region for blocks of size P1 x P2 (P1 P2 ≥ 2) under the AR( 1)* AR( 1) process is also specified. It is shown that optimal NRC designs are highly specific to the correlation values. The purpose of EGVTs is to select top performing new crop varieties for further testing. Recently there has been much interest in the spatial analysis of EGVTs, but there has been little work on the design of efficient EGVTs when a spatial analysis is intended. Several intuitively simple criteria to assess the efficiency of designs for EGVTs are examined, and simulation studies suggest that some of these criteria are well associated with probabilities of selecting the highest yielding new varieties. Also, the efficiency and robustness of some systematic designs for EGVTs is investigated over several models and dependence structures. For the examples considered, it is shown that designs in which the plots containing control varieties are at least a knight's move apart are robust

A Census Of Highly Symmetric Combinatorial Designs

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v,k,1) designs, mainly for t = 2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t > 2 most of these characterizations have remained longstanding challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 2 < t < 7 is of particular interest and has been open for about 40 years (cf. [11, p. 147] and [12, p. 273], but presumably dating back to 1965). The present paper continues the author's work [20, 21, 22] of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general results on highly symmetric Steiner t-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics

Design Lines

The two basic equations satisfied by the parameters of a block design define a three-dimensional affine variety $\mathcal{D}$ in $\mathbb{R}^{5}$. A point of $\mathcal{D}$ that is not in some sense trivial lies on four lines lying in $\mathcal{D}$. These lines provide a degree of organization for certain general classes of designs, and the paper is devoted to exploring properties of the lines. Several examples of families of designs that seem naturally to follow the lines are presented.Comment: 16 page

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc