#### Date of Award

Summer 2002

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics and Statistics

#### Program/Concentration

Computational and Applied Mathematics

#### Committee Director

John P. Morgan

#### Committee Member

N. Rao Chaganty

#### Committee Member

Dayanand N. Naik

#### Committee Member

John Stufken

#### Abstract

One of the fundamental principles of experimental design is the separation of heterogeneous experimental units into subsets of more homogeneous units or *blocks* in order to isolate identifiable, unwanted, but unavoidable, variation in measurements made from the units. Given *v* treatments to compare, and having available *b* blocks of *k* experimental units each, the thoughtful statistician asks, “What is the optimal allocation of the treatments to the units?” This is the basic block design problem. Let *n _{ij}* be the number of times treatment

*i*is used in block

*j*and let

*N*be the

*v*x

*b*matrix

*N*= (

*n*). There is now a considerable body of optimality theory for block design settings where binarity (all

_{ij}*n*∈ {0, 1}), and symmetry or near-symmetry of the concurrence matrix

_{ij}*NN*, are simultaneously achievable. Typically the same classes of designs are found to be best using any of the standard optimality criteria. Among these are the balanced incomplete block designs (BIBDs), many species of two-class partially balanced incomplete block designs, and regular graph designs.

^{T}However, there are triples (*v, b, k*) in which binarity *precludes* near-symmetry. For these combinatorially problematic settings, recent explorations have resulted in new optimality results and insight into the combinatorial issues involved. Of particular interest are the *irregular BIRD settings*, that is, triples (*v, b, k*) where the necessary conditions for a BIBD are fulfilled but no such design exists. A thorough study of the smallest such setting, (15,21,5), has produced some surprising optimal designs which will be presented in the first chapter of this document.

An incomplete block design is said to be *resolvable* if the blocks can be partitioned into classes, or *replicates* such that each treatment appears in exactly one block of each replicate. Resolvable designs are indispensable in many industrial and agricultural experiments, especially when the entire experiment can not be completed at one time or when there is a risk that the experiment may be prematurely terminated. In chapters two and three we will investigate the classes of resolvable designs having five or fewer replications and two blocks of possibly unequal size per replicate. Theory for identifying the best designs with respect to important optimality criteria will be developed, and with the optimality theory in hand, optimal designs will be identified and constructions provided. We will conclude with a comment on the robustness of resolvable designs to the loss of a replicate.

#### DOI

10.25777/69nj-7r65

#### ISBN

9780493882932

#### Recommended Citation

Reck, Brian H..
"Nearly Balanced and Resolvable Block Designs"
(2002). Doctor of Philosophy (PhD), dissertation, Mathematics and Statistics, Old Dominion University, DOI: 10.25777/69nj-7r65

https://digitalcommons.odu.edu/mathstat_etds/55