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  "body": "\n## Definition\n\nLet $k \\ge 2$ and let $L_1, \\dots, L_k$ be finite sets with $s_i = \\abs{L_i} \\ge 2$, the *levels* of factors $1, \\dots, k$.\nThe treatments are the combinations $t = (t_1, \\dots, t_k) \\in L_1 \\times \\cdots \\times L_k$.\nA *full factorial experiment* is an [[experiment|2TE42ZWJ3E]] whose design applies every one of the $s_1 s_2 \\cdots s_k$ combinations to at least one unit.\nWhen every $s_i = 2$, it is a *$2^k$ factorial experiment*, and the levels are usually coded $-1$ and $+1$.\n\n## Remarks\n\nIn a $2^k$ experiment with one run per combination and responses $y_t$, the *main effect* of factor $i$ is the mean of $y_t$ over $t_i = +1$ minus the mean over $t_i = -1$.\nThe *interaction* of factors $i$ and $j$ is half of the difference between the effect of factor $i$ at $t_j = +1$ and at $t_j = -1$.\nEquivalently, it is the mean of $y_t$ over $t_i t_j = +1$ minus the mean over $t_i t_j = -1$.\n\nExample: a $2^2$ experiment with factors $A$ and $B$ and responses $y_{--} = 10$, $y_{+-} = 14$, $y_{-+} = 12$, $y_{++} = 20$, where the first sign is the level of $A$.\nThe main effect of $A$ is $(14 + 20)/2 - (10 + 12)/2 = 6$, and that of $B$ is $(12 + 20)/2 - (10 + 14)/2 = 4$.\nThe interaction is $\\big( (20 - 12) - (14 - 10) \\big)/2 = 2$, which also equals $(10 + 20)/2 - (14 + 12)/2$.\n\nNon-example: the runs $(-,-)$, $(+,-)$, $(-,+)$ change one factor at a time from a baseline.\nThe combination $(+,+)$ is missing, so this is not a full factorial experiment, and the interaction of $A$ and $B$ cannot be estimated.\n\nThe data of a factorial experiment are usually analysed by analysis of variance, with one sum of squares for each main effect and each interaction.\n",
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