Balanced incomplete block is a reduced version of the balanced complete block design. The design is incomplete if each of the n blocks tests only c of the a levels of treatment A, where c < a. It is balanced provided that each treatment level is tested the same number of times, given by r = nc/a, and each pair of treatment levels appears in the same number of blocks, given by λ = nc(c-1)/[a(a-1)]. Likewise, a two-factor blocked design is incomplete if c < ab combinations of levels of the two treatments A and B, and it is balanced if integer values pertain for both r = nc/(ab) and λ = nc(c-1)/[ab(ab-1)]. In the example on these web pages, a = 4, n = 6, c = 2, so r = 3 and λ = 1. The four levels of factor A could represent a single treatment with four levels, or the four combinations of levels of two treatments each with two levels.

 

Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.

http://southampton.likn.co/~cpd/anovas/datasets/