Methods and Applications of Statistics in Clinical Trials, Volume 1: Concepts, Principles, Trials, and Designs

Chapter 53

Partially Balanced Designs

Rosemary A. Bailey

53.1 Introduction

Balanced incomplete block designs (BIBDs) have many desirable qualities. They are easy to analyze; many families of them are easily constructed; the loss of information on treatment estimates (in the sense of increase in variance) due to blocking is as small as it can be for the given block size; and it is easy to compare them with other incomplete block designs, because usually the BIBDs are superior in every respect. However, for many combinations of block size, number of treatments and number of replications, there is no balanced incomplete block design. Bose and Nair [5] introduced partially balanced incomplete block designs in 1939 for use in such situations, hoping to retain many of the desirable properties of BIBDs. They succeeded in their aim to a certain extent: some of the new class of designs are very good, while others are undoubtedly useless for practical experimentation.

Unfortunately, any discussion of partial balance must necessarily be more technical than that of BIBDs, because there is an extra layer of complication. Moreover, many of the ideas have been developed and clarified by pure mathematicians, without ever being reexpressed in statistical language. Thus some sections of this article are unavoidably technical. It is hoped that the division into sections with titles will enable the nonmathematical reader to learn something from this article.

During the development of the subject there ...