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  "body": "\n## Definition\n\nLet $T$ be a finite set of treatments and $U$ a finite set of experimental units.\n*Blocking* is a partition $U = B_1 \\cup \\cdots \\cup B_b$ of the units into disjoint *blocks* of units that are expected to respond alike.\nThe partition is fixed before the treatments are assigned, and treatments are then assigned and compared within blocks.\nA [[design|2TE42ZWJ3E]] $d \\colon U \\to T$ is a *randomized complete block design* if:\n\n1. each block has exactly $\\abs{T}$ units;\n2. each treatment occurs exactly once in each block, that is, $d$ restricted to $B_j$ is a bijection onto $T$ for every $j$;\n3. the bijections on the different blocks are chosen independently and uniformly at random, by [[randomization|734Q6Q51X7]] within blocks.\n\n## Remarks\n\nThe usual model for a randomized complete block design is $y_{ij} = \\mu + \\tau_i + \\beta_j + \\varepsilon_{ij}$, with treatment effects $\\tau_i$, block effects $\\beta_j$ and independent errors.\nDifferences $y_{ij} - y_{i'j}$ within block $j$ do not contain $\\beta_j$, so block effects drop out of treatment comparisons.\n\nExample: three varieties $T = \\set{A, B, C}$ on four fields, each cut into three plots.\nThe fields are the blocks, each of size $3$; in each field the three varieties are put on the three plots in an order chosen at random.\n\nNon-example: if each field receives a single variety on all its plots, clause 2 fails.\nThe variety effect is then confounded with the field effect, and the two cannot be separated.\n\nWhen blocks are smaller than the number of treatments, clause 1 fails and one uses incomplete block designs, such as balanced incomplete block designs.\n",
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