Chapter I. Introduction to the Tables

1. History. Incomplete block designs have become highly developed as statistical tools for the planning, conduct, analysis, and interpretation of scientific experimentation conducted in laboratories and test installations where a high degree of experimental control is desirable or even necessary to cope with the experimental variability that invariably arises throughout the conduct of the experiment. Historically the statistical use of these designs began with the balanced incomplete block designs introduced by Yates (1936a) and developed by Fisher and Yates (1938) and Bose (1939) and extensively tabled in Fisher and Yates (1963) and Cochran and Cox (1957) along with the lattice designs of Yates (1936b, 1937) and Cochran and Cox (1957) all of which were developed for use in agricultural and biological experiments. Further refinements in statistically based experimental arrangements of the first type were developed by Youden (1937, 1951) and of the second by Harshbarger (1946, 1947) and Nair (1951). Bose and Nair (1939) developed a very general class of incomplete block designs which they called partially balanced incomplete block designs with m associate classes (abbreviated PBIBD(m)). These designs include the balanced incomplete block designs (BIB designs) and the square and rectangular lattice designs as special cases. Generally, they have the advantage of providing an extremely large selection of experimental arrangements from which investigators may choose the ones best suited to their needs. Besides the advantage of control of experimental variation through the blocking technique, they offer (1) the opportunity to select plans of smaller size [fewer repetitions of the treatments, varieties, test conditions, etc.], (2) statistical analysis of the experimental data by relatively simple computational procedures, (3) simplicity of presentation and interpretation of the results from comparative experiments in terms of clear-cut statements about their precision based on modern statistical developments.

In the United States and England during the period of World War II, a strong interest was developed in industry and in government to adapt the developments in probability and small sample theory to the control of quality of production processes of industry. This interest was extended after the war to the introduction of statistical procedures to the experimental problems of the physical and engineering sciences. This effort has been particularly significant in the vast and expensive investigative activity associated with atomic power and aerospace developments. Today there is intense interest in current developments in probability and statistics for possible application to research and development in all branches of science, engineering, health and medical research, business, and in fact, in any area where decisions need be made on the basis of incomplete information. Thus the incomplete block designs have potential for application to many fields of investigation.

By 1952 R. C. Bose and a group of his advanced degree students at the University of North Carolina (including S. S. Shrikhande, W. S. Connor, T. Shimamoto, and W. H. Clatworthy) had sufficiently extended and developed the partially balanced incomplete block designs with two associate classes [PBIBD(2), or even more briefly "D(2)" or "D(2) designs"] that several hundred were now available. With the encouragement of Gertrude Cox, Director of the Institute of Statistics, Consolidated University of North Carolina, and with the support of the Agricultural Experiment Station, North Carolina State College, some 375 of these D(2) designs were classified, catalogued and generally put into convenient form for use by experimenters. A monograph was published by Bose, Clatworthy and Shrikhande (1954) very soon after Bose and Shimamoto (1952) had developed the concept of association schemes, on which the classification of D(2) designs is based. Since 1952 the concept of association schemes has been the subject of intensive research by statisticians interested in the combinatorial problems of the design of experiments. The intimate relations between association schemes and certain kinds of graphs has attracted many mathematicians interested in graph theory to join in their investigation. The resulting improved insight into the construction problems of D(2) designs accompanying the improved knowledge of association schemes, led to the construction of literally hundreds

referred to briefly as "BCS (1954)"].

The extremely widespread and frequent referencing of BCS (1954) by scholars all over the world, the vast development of new designs of D(2) type, and the very high potential for use of these combinatorial arrangements in scientific experimentation led the present writer to the decision to prepare an up-to-date catalog of D(2) designs along with associated, but incomplete, auxiliary information heretofore unpublished (in fact never before attempted for a large collection of designs). These special features are described in sections 4 and 5 of this Chapter.

2. Definition of a partially balanced incomplete block design with two associate classes and the relations between its parameters. Following BCS (1954), an incomplete block design is said to be partially balanced with two associate classes if it satisfies the following requirements.

(i) The experimental material is divided into b blocks of k units each, different treatments being applied to the units in the same block.

(ii) There are v ( > k) treatments each of which occurs in r blocks.

(iii) There can be established a relation of association between any two treatments satisfying the following requirements:

(a) Two treatments are either first associates or second associates.

(b) Each treatment has exactly n ith associates (i = 1, 2).

(c) Given any two treatments which are ith associates, the number of treatments common to the jth associate of the first and the kth associate of the second is p and is independent of the pair of treatments we start with. Also pj = på¿¡(i,j,k−1, 2).

(iv) Two treatments which are ith associates occur together in exactly λ; blocks (i = 1,2).

For a proper partially balanced incomplete block design λ1 # λ2. If λ1 = λ2 or if one of the n; vanishes, the design becomes a balanced incomplete block design.

The numbers v, r, k, b, ni, n2, A1, and A2 are called the parameters of the first kind, whereas the numbers pik (i, j, k=1, 2) are called the parameters of the second kind.

It is to be noted that the association relations between the treatments of a partially balanced design are governed solely by the requirements of paragraph (iii) of the definition and do not depend upon how the treatments are distributed in blocks. The association scheme depends only on the parameters n1, n2, and pk (i, j, k=1, 2).

The following relations between the parameters are known to hold: