Encyclopaedia of DesignTheory: Block Designs

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Topic essays on block designs:

Block designs

Block designs mean different things to statisticians and combinatorialists!

In an agricultural experiment, we may have access to a number of plots on experimental farms in different parts of the country. We expect that fertility, drainage, etc., will vary from one plot to another, but that plots on the same farm will be more alike than plots on different farms. In allocating treatments to plots, we have to take account of these, so that plots receiving the same treatment are spread over the different farms suitably; if this is not done, the effect of the treatments cannot be disentangled efficiently from the effect of the different farms.

Thus, to a statistician, we have a set (of plots) which carries a partition (into blocks), and we wish to impose another partition (into treatments) in an intelligent way.

A combinatorialist takes the treatments as the basic objects, and often refers to them as "points". Assuming that no treatment occurs more than once in a block (a design with this property is called binary), each block can be identified with a subset of the set of all treatments, and the design is a family of subsets (called "blocks") of the set of points.

A block design is called balanced if any two treatments occur together in a constant number of blocks. As a general rule, if a balanced design can be found, then it is best to use such a design. As another general rule, combinatorialists have tended to concentrate their efforts on constructing balanced designs, while statisticians are more concerned with what to do if no balanced design exists.

At present, the Encyclopaedia contains three topic essays on general block designs: the first describes the two viewpoints in more detail, the second defines the above representations and others, while the third describes estimation and variance in block designs.

There are also Encyclopaedia entries on some specific types of block design (t-designs and Steiner triple systems), as well as topic essays on pairwise balanced designs, projective and affine planes, and square 2-designs (also known as symmetric BIBDs).

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Peter J. Cameron
5 July 2006