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  "body": "\n## Definition\n\nLet $k \u003e p \\ge 1$, and code the levels of $k$ two-level factors as $\\pm 1$.\nThen the runs of the full $2^k$ [[factorial experiment|EHJ3R589ZX]] are the points of $\\set{-1, 1}^k$.\nFor a subset $w \\subseteq \\set{1, \\dots, k}$, called a *word*, write $x^w = \\prod_{j \\in w} x_j$ for $x \\in \\set{-1, 1}^k$.\nIdentify words with vectors of $\\F_2^k$, with the symmetric difference as addition.\nA *regular $2^{k-p}$ fractional factorial design* is a set of runs\n$$F = \\set{x \\in \\set{-1, 1}^k : x^{w} = \\epsilon_w \\text{ for all } w \\in H},$$\nwhere:\n\n1. $H$ is a [[subgroup|0X1JXT47GA]] of $\\F_2^k$ of order $2^p$, the *defining relation*, with no word of size $1$;\n2. the signs $\\epsilon_w \\in \\set{-1, 1}$ satisfy $\\epsilon_{\\emptyset} = 1$ and $\\epsilon_{w \\triangle w'} = \\epsilon_w \\epsilon_{w'}$, so that $F$ is not empty.\n\nThen $F$ has $2^{k-p}$ runs.\nTwo effects with words $u$ and $v$ are *aliased* if $u \\triangle v \\in H$.\nThe *resolution* of the design is the smallest size of a nonempty word of $H$.\n\n## Remarks\n\nOn $F$, the contrast columns $x^u$ and $x^v$ of aliased effects agree up to the sign $\\epsilon_{u \\triangle v}$, so their effects cannot be separated.\nHere $\\F_2$ is the [[field with two elements|0ZXZ82FTX2]].\n\nExample: $k = 3$, $p = 1$, $H = \\set{\\emptyset, \\set{1,2,3}}$ with $\\epsilon = +1$, that is, the generator $C = AB$ and the defining relation $I = ABC$.\nThe runs are $(-1,-1,+1)$, $(+1,-1,-1)$, $(-1,+1,-1)$, $(+1,+1,+1)$, four of the eight.\nThe main effect $A$, with word $\\set{1}$, is aliased with $BC$, since $\\set{1} \\triangle \\set{1,2,3} = \\set{2,3}$.\nThe design has resolution III.\n\nNon-example: the four runs with $x_1 = -1$ form the set $\\set{x : x^{\\set{1}} = -1}$, which uses the word $\\set{1}$ of size $1$, so clause 1 fails.\nIn these runs factor $A$ never changes, so its main effect cannot be estimated at all.\n",
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