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  "body": "\n## Definition\n\nLet $T$ be a finite set of *treatments* and let $U$ be a finite set of $n$ *experimental units*.\nAn *experimental design* for $T$ on $U$ is a map $d \\colon U \\to T$, which assigns treatment $d(u)$ to unit $u$.\nThe number $r_t = \\abs{d^{-1}(t)}$ is the *replication* of treatment $t$.\nAfter the treatments are applied, a response $y_u$ is observed on each unit.\nA [[statistical model|K3AMMSY8DD]] relates $y_u$ to the treatment $d(u)$ and to the unit $u$.\n*Design of experiments* is the choice of $T$, $U$ and $d$, together with the plan of analysis, before the data are collected.\n\n## Remarks\n\nExample: two diets $T = \\set{A, B}$ and four animals $U = \\set{1, 2, 3, 4}$.\nThe map $d(1) = A$, $d(2) = B$, $d(3) = B$, $d(4) = A$ is a design with $r_A = r_B = 2$.\n\nNon-example: in an observational study the units choose their own treatments, for example smokers and non-smokers.\nThe map $d$ is not chosen by the investigator, so this is not a designed experiment.\nDifferences in response may then come from other differences between the groups.\n\nFor experiments with a quantitative input $x$ in a region $\\mathcal{X}$, a design is a choice of the input values $x_1, \\dots, x_n \\in \\mathcal{X}$.\nIn the approximate theory, it is a probability measure on $\\mathcal{X}$ with finite support.\n",
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