Balanced Incomplete Block Designs

Suppose you have n different fertilizers you want to test. The simplest experiment would be to take a big plot of land, divide it into n blocks, and apply each fertilizer to a block. That’s fine if you have one big enough plot of land, and the land itself doesn’t matter. If the fertilizer supplies a nutrient that is already abundant in a particular soil, you won’t see as much benefit from that fertilizer on that soil. Plots of land will vary in other ways besides soil composition—some may receive more sun than others, some may have better drainage, etc.—that may interact with the effect of the fertilizer. So you’d like to test your fertilizers on several different plots of land. Call the number of plots b. Suppose each of your plots is large enough to test q < n fertilizers, i.e. the plots are not large enough to test every fertilizer on every plot. You would like to design your experiment so that each fertilizer is tested the same number of times λ, and so that every pair of…

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