Randomized block design

The defining constraint is that every treatment appears inside every block, and the randomization is carried out separately within each one. Blocks are built first, from a variable measured before any treatment is given, so that units inside a block are alike on it. What happens in one block has no bearing on what happens in another.

Forty plots run from full sun to deep shade and two fertilizers are to be compared. Sort the plots into 4 blocks of 10 by sunlight level, then inside each block randomly choose 5 plots for each fertilizer. A block of 10 splits $\binom{10}{5} = 252$ ways, so the design has $252^4 = 4{,}032{,}758{,}016$ equally likely assignments. A completely randomized design on the same 40 plots would have $\binom{40}{20} = 137{,}846{,}528{,}820$, about 34 times as many. The assignments blocking deletes are exactly the ones where a fertilizer drew more than its share of the sunny plots.

"The sunny plots got fertilizer A and the shaded plots got fertilizer B, so we blocked by sunlight" describes a different study. There, sunlight and fertilizer change together with no way to separate them, which is the confounding that blocking exists to prevent. A block is not a treatment group and blocks are not tested against each other: the comparison is made inside a block and then combined across blocks.

A block therefore has to be large enough to hold every treatment, so with three treatments a block of two is impossible. When there are exactly two treatments and each block holds two units, the design has its own name, matched pairs.

Topic 1.13 lists the randomized block design as one of three named designs, the others being the completely randomized design and matched pairs. The choice between the first two is worked through in randomized block design vs completely randomized design.