Randomized Block Design vs Completely Randomized 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.

A completely randomized design has one pool of experimental units and one randomization: treatments go to all of the units entirely at random, with nothing grouped, sorted or paired first. That last clause is what the word completely is pointing at.

Take 45 seedlings and three light levels. Number the seedlings 1 to 45, generate random integers from 1 to 45 while ignoring repeats, send the first 15 distinct numbers to the low level, the next 15 to the medium level, and the 15 whose numbers never came up to the high level. Each seedling had probability $1/3$ of each treatment, and the design imposes nothing beyond that.

The error is a mislabel: "we randomly assigned the men to the two treatments and the women to the two treatments, so the design is completely randomized." Randomizing inside groups you formed beforehand is a randomized block design. If a variable was used to sort the units before the draw, the design is blocked, whatever the write-up calls it.

Equal group sizes are conventional, not required. A run that puts 22 subjects in one treatment and 18 in the other is still completely randomized. Flipping a fair coin for each of 20 subjects is genuine random assignment, and an exact 10 and 10 split then has probability $\binom{20}{10}/2^{20} = 184756/1048576 \approx 0.176$, so it turns up on about 18 percent of runs. Use a random number generator instead whenever the group sizes have to come out as promised.

A known nuisance variable does not invalidate this design. It stays valid and simply leaves that variation in the noise, which is the variation blocking pulls out instead. Completely randomized, randomized block and matched pairs are the three designs named in topic 1.13 Experimental Design, and a question asking for one of them is not answered by another.