Balanced Incomplete Block Designs

Abstract

Balanced incomplete block designs (BIBD) are experimental designs that can be described by a labeled, fully filled rectangular array or by a larger equivalent rectangular display with some empty cells. A cell in the rectangle identifies one treatment in a given block. The number of different treatments for a given block (individual) is less than the total number of treatments; this precludes having each treatment appear once in every block. Hence, the design is an incomplete block design. Without some constraints on the number of occurrences of certain treatments in different rows, the average performance in different parts of the design can be biased. This bias yields different averages for two treatments, for example, because of differences in the cells in which those treatments were employed. To eliminate the bias, a special form of balance is employed in which every possible pair of treatments appears in an equal number of blocks. Accordingly, there also is balance in the total number of times each treatment occurs. Analysis methods are discussed for general linear fixed, mixed, and Bayesian models. I discuss a special case of a BIBD design, Youden squares, that permits but does not require the assessment of one more independent variable than other BIBDs. In the behavioral sciences, the rows of Youden squares can be different blocks, typically different people or animals, and the columns can be different observation occasions that permit the assessment of so-called period effects.