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  "body": "\n## Definition\n\nLet $T$ be a finite set of treatments, let $U$ be a set of $n$ experimental units, and let $r_t \\ge 1$ be integers with $\\sum_{t \\in T} r_t = n$.\nLet $\\mathcal{D}$ be the set of [[experimental designs|2TE42ZWJ3E]] $d \\colon U \\to T$ with $\\abs{d^{-1}(t)} = r_t$ for every $t \\in T$.\n*Complete randomization* is the choice of a design $D$ at random from $\\mathcal{D}$ with the uniform [[distribution|0QB913WM5E]], that is,\n$$P(D = d) = \\frac{1}{\\abs{\\mathcal{D}}} = \\frac{\\prod_{t \\in T} r_t!}{n!} \\quad \\text{for every } d \\in \\mathcal{D}.$$\nMore generally, a *randomized design* is a design chosen at random from a set of designs by a known probability distribution.\nIn a restricted randomization the set is smaller, for example the designs that give every treatment once in each block.\n\n## Remarks\n\nExample: $T = \\set{A, B}$, $U = \\set{1, 2, 3, 4}$ and $r_A = r_B = 2$.\nThen $\\abs{\\mathcal{D}} = 4!/(2!\\,2!) = 6$, and each of the six splits, such as $\\set{1, 2}$ to $A$ and $\\set{3, 4}$ to $B$, has probability $1/6$.\nEach unit receives $A$ with probability $1/2$.\n\nNon-example: giving $A$ to the first two units to arrive and $B$ to the next two is a fixed rule, not a chance mechanism.\nIf early arrivals differ from late ones, the comparison of $A$ and $B$ is biased.\n\nFix the observed responses and compute a test statistic for every design in $\\mathcal{D}$.\nUnder the hypothesis of no treatment effect, these values give the exact null distribution of a randomization test.\n",
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