Nested Balanced Incomplete Block Designs

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b1;r;k1) are each partitioned into sub-blocks of size k2, and the b2 =b1k1=k2 sub-blocks themselves constitute a BIBD with parameters (v; b2;r;k2), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k1 =2k2= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, they are almost unknown in the combinatorial literature, except in the literature of tournaments, and detailed combinatorial studies of them have been lacking. The present paper therefore reviews and extends mathematical knowledge of NBIBDs. Isomorphism and automorphisms are defined for NBIBDs, and methods of construction are outlined. Some special types of NBIBD are de4ned and illustrated. A first-ever detailed table of NBIBDs with v⩽16, r⩽30 is provided; this table contains many newly discovered NBIBDs. © 2001 Elsevier Science B.V. All rights reserved

Optimal Row-Column Designs for Correlated Errors and Nested Row-Column Designs for Uncorrelated Errors

In this dissertation the design problems are considered in the row-column setting for second order autonormal errors when the treatment effects are estimated by generalized least squares, and in the nested row-column setting for uncorrelated errors when the treatment effects are estimated by ordinary least squares. In the former case, universal optimality conditions are derived separately for designs in the plane and on the torus using more general linear models than those considered elsewhere in the literature. Examples of universally optimum planar designs are given, and a method is developed for the construction of optimum and near optimum designs, that produces several infinite series of universally optimum designs on the torus and near optimum designs in the plane. Efficiencies are calculated for planar versions of the torus designs, which are found to be highly efficient with respect to some commonly used optimality criterion. In the nested row-column setting, several methods of construction of balanced and partially balanced incomplete block designs with nested rows and columns are developed, from which many infinite series of designs are obtained. In particular, 149 balanced incomplete block designs with nested rows and columns are listed (80 appear to be new) for the number of treatments, v \u3c 101, a prime power

Optimality and Construction of Designs with Generalized Group Divisible Structure

This thesis is an investigation of the optimality and construction problems attendant to the assignment of v treatments to experimental units in b blocks of size k, paying special attention to settings for which equal replication of the treatments is not possible. The model is that of one way elimination of heterogeneity, in which the expectation of an observation on treatment i in block j is Ti + βj (treatment effect + block effect), where Ti and βj are unknown constants, 1 ≤ i ≤ v and 1 ≤ j ≤ b. All observations are assumed to be uncorrelated with same variance. The generalized group divisible design with s groups, or GGDD(s), is defined in terms of the elements of the information matrix, instead of in terms of the elements of the concurrence matrix as done by Adhikary (1965) and extended by Jacroux (1982). This definition extends the class of designs to include non-binary members, and allows for broader optimality results. Some sufficient conditions are derived for GGDD(s) to be E- and MV-optimal. It is also shown how augmentation of addition blocks to certain GGDD(s)s produces other nonbinary, unequally replicated E- and MV-optimal block designs. Where nonbinary designs are found, they are generally preferable to binary designs in terms of interpretability, and often in terms of one or more formal optimality criteria as well. The class of generalized nearly balanced incomplete block designs with maximum concurrence range l, or NBBD(l), is defined. This class extends the nearly balance incomplete block designs as defined by Cheng & Wu (1981), and the semi-regular graph designs as defined by Jacroux (1985), to cases where off-diagonal entries of the concurrence matrix differ by at most the positive integer l. Sufficient conditions are derived for a NBBD(2) to be optimal under a given type-I criterion. The conditions are used to establish the A- and D-optimality of an infinite series of NBBD(2)s having unequal numbers of replicates. Also, a result from Jacroux (1985) is used to establish the A-optimality of a new series of NBBD(1)s. Several methods of construction of GGDD(s)s are developed from which many infinite series of designs are derived. Generally these designs satisfy the obtained sufficient conditions for E- and MV-optimality. Finally, in the nested row-column setting, the necessary conditions for existence of 2 x 2 balanced incomplete block designs with nested rows and columns (BIBRCs) are found to be sufficient. It is also shown that, sufficient for a BIBRC with p=q to generally balanced, is that the row and column classifications together form a balanced incomplete block design, as does the block classification. All of the 2 x 2 BIBRCs are constructed to have this property

Recursive Methods for Construction of Balanced N-ary Block Designs

2000 Mathematics Subject Classification: Primary 05B05; secondary 62K10.This paper presents a recursive method for the construction of balanced n-ary block designs. This method is based on the analogy between a balanced incomplete binary block design (B.I .E .B) and the set of all distinct linear sub-varieties of the same dimension extracted from a finite projective geometry. If V1 is the first B.I .E .B resulting from this projective geometry, then by regarding any block of V1 as a projective geometry, we obtain another system of B.I .E .B. Then, by reproducing this operation a finite number of times, we get a family of blocks made up of all obtained B.I .E .B blocks. The family being partially ordered, we can obtain an n-ary design in which the blocks are consisted by the juxtaposition of all binary blocks completely nested. These n-ary designs are balanced and have well defined parameters. Moreover, a particular balanced n-ary class is deduced with an appreciable reduction of the number of blocks

Statistical Algorithms for Optimal Experimental Design with Correlated Observations

This research is in three parts with different although related objectives. The first part developed an efficient, modified simulated annealing algorithm to solve the D-optimal (determinant maximization) design problem for 2-way polynomial regression with correlated observations. Much of the previous work in D-optimal design for regression models with correlated errors focused on polynomial models with a single predictor variable, in large part because of the intractability of an analytic solution. In this research, we present an improved simulated annealing algorithm, providing practical approaches to specifications of the annealing cooling parameters, thresholds and search neighborhoods for the perturbation scheme, which finds approximate D-optimal designs for 2-way polynomial regression for a variety of specific correlation structures with a given correlation coefficient. Results in each correlated-errors case are compared with the best design selected from the class of designs that are known to be D-optimal in the uncorrelated case: annealing results had generally higher D-efficiency than the best comparison design, especially when the correlation parameter was well away from 0. The second research objective, using Balanced Incomplete Block Designs (BIBDs), wasto construct weakly universal optimal block designs for the nearest neighbor correlation structure and multiple block sizes, for the hub correlation structure with any block size, and for circulant correlation with odd block size. We also constructed approximately weakly universal optimal block designs for the block-structured correlation. Lastly, we developed an improved Particle Swarm Optimization(PSO) algorithm with time varying parameters, and solved D-optimal design for linear regression with it. Then based on that improved algorithm, we combined the non-linear regression problem and decision making, and developed a nested PSO algorithm that finds (nearly) optimal experimental designs with each of the pessimistic criterion, index of optimism criterion, and regret criterion for the Michaelis-Menten model and logistic regression model

Decomposition tables for experiments I. A chain of randomizations

One aspect of evaluating the design for an experiment is the discovery of the relationships between subspaces of the data space. Initially we establish the notation and methods for evaluating an experiment with a single randomization. Starting with two structures, or orthogonal decompositions of the data space, we describe how to combine them to form the overall decomposition for a single-randomization experiment that is ``structure balanced.'' The relationships between the two structures are characterized using efficiency factors. The decomposition is encapsulated in a decomposition table. Then, for experiments that involve multiple randomizations forming a chain, we take several structures that pairwise are structure balanced and combine them to establish the form of the orthogonal decomposition for the experiment. In particular, it is proven that the properties of the design for such an experiment are derived in a straightforward manner from those of the individual designs. We show how to formulate an extended decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated.Comment: Published in at http://dx.doi.org/10.1214/09-AOS717 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Decomposition tables for experiments. II. Two--one randomizations

We investigate structure for pairs of randomizations that do not follow each other in a chain. These are unrandomized-inclusive, independent, coincident or double randomizations. This involves taking several structures that satisfy particular relations and combining them to form the appropriate orthogonal decomposition of the data space for the experiment. We show how to establish the decomposition table giving the sources of variation, their relationships and their degrees of freedom, so that competing designs can be evaluated. This leads to recommendations for when the different types of multiple randomization should be used.Comment: Published in at http://dx.doi.org/10.1214/09-AOS785 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Resolvable designs with large blocks

Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix and consequently given in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities and a detailed study of E-optimality is offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization.Comment: Published at http://dx.doi.org/10.1214/009053606000001253 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Construction of some new three associate class partially balanced incomplete block designs in two replicates

Paper presented at the 2nd Strathmore International Mathematics Conference (SIMC 2013), 12 - 16 August 2013, Strathmore University, Nairobi, Kenya.Search for experimental designs which aid in research studies involving large number of treatments with minimal experimental units has been desired overtime. This paper constructs some new series of three associate Partially Balanced Incomplete Block (PBIB) designs having n(n - 2) /4 treatments with three associate classes in two replicates using the concept of triangular association scheme. The design is constructed from an even squared array of n rows and n columns (n _> 8) with its both diagonal entries bearing no treatment entries and that given the location of any treatment in the squared array, the other location of the same treatment in the array is predetermined. The design and association parameters for a general case of an even integer n >_8 are obtained with an illustrated case for n = 8. Efficiencies of the designs within the class of designs are obtained for a general case of even n >_8 with a listing of efficiencies of designs with blocks sizes in the interval [8,22]. The designs constructed have three associate classes and are irreducible to minimum number of associate classes.Search for experimental designs which aid in research studies involving large number of treatments with minimal experimental units has been desired overtime. This paper constructs some new series of three associate Partially Balanced Incomplete Block (PBIB) designs having n(n - 2) /4 treatments with three associate classes in two replicates using the concept of triangular association scheme. The design is constructed from an even squared array of n rows and n columns (n _> 8) with its both diagonal entries bearing no treatment entries and that given the location of any treatment in the squared array, the other location of the same treatment in the array is predetermined. The design and association parameters for a general case of an even integer n >_8 are obtained with an illustrated case for n = 8. Efficiencies of the designs within the class of designs are obtained for a general case of even n >_8 with a listing of efficiencies of designs with blocks sizes in the interval [8,22]. The designs constructed have three associate classes and are irreducible to minimum number of associate classes