Block Designs: Advances in Theory and Use

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

A problem common to many disparate fields of scientific and industrial enquiry is that of comparing a set of experimental conditions, or `treatments', when faced with heterogeneity in the units of material on which the experiment is to be performed. The technique of blocking specifically addresses this problem. Typically using values of some identifiable nuisance factor, the experimental units are partitioned into homogeneous subsets called blocks. Inferences can then be based on the more precise comparisons of measurements from the same block. The design problem is to determine which treat- ments are assigned to which units in which blocks, said assignment being driven by the desire to maximize the quality of information that will ultimately be produced. The problem becomes more complicated as more nuisance factors, with various inter- relationships, are introduced or identified. This project addresses the design problem for a variety of commonly encountered experimental settings, with two broad goals: (i) to extend the known theory for determining optimal designs for a wider range of settings than is now known or available; and (ii) to produce a comprehensive catalog of designs to be incorporated into a larger, web-based resource that will include a vari- ety of downloadable combinatorial designs useful for statisticians, mathematicians, and other scientists in academia and industry. By collecting optimal designs in a single, easily accessible resource, the use of good designs can be increased and the practice of experimental science consequently sharpened. When comparing v treatments, the most commonly encountered situation across a range of scientific endeavors is that of a single blocking variable partitioning bk experi- mental units into b blocks of k units each. This project will examine optimality problems for these simple block designs, and the attendant combinatorial issues, augmenting the- ory by computation where needed, to produce designs for a practical range of parameter combinations (v; b; k), including settings where equireplication is not possible. The most frequently used designs with two blocking factors are row-column designs, in which two blocking factors can be visualized as row and columns in a rectangular array, and re- solvable designs, in which a second blocking factor partitions a simple block design into subsets of blocks each consisting of a single replicate. This project will also address problems in optimality and construction of designs in both these classes, with special emphasis on plans most demanded in practice: those with few replicates.