Row-column factorial designs with multiple levels
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An m× n row-column factorial design is an arrangement of the elements of a factorial design into a rectangular array. Such an array is used in experimental design, where the rows and columns can act as blocking factors. If for each row/column and vector position, each element has the same regularity, then all main effects can be estimated without confounding by the row and column blocking factors. Formally, for any integer q, let [q]={0,1,… ,q-1}. The q^k (full) factorial design with replication α is the multi-set consisting of α occurrences of each element of [q]^k; we denote this by α× [q]^k. A regular m× n row-column factorial design is an arrangement of the the elements of α× [q]^k into an m× n array (which we say is of type I_k(m,n;q)) such that for each row (column) and fixed vector position i∈ [q], each element of [q] occurs n/q times (respectively, m/q times). Let m≤ n. We show that an array of type I_k(m,n;q) exists if and only if (a) q|m and q|n; (b) q^k|mn; (c) (k,q,m,n)≠ (2,6,6,6) and (d) if (k,q,m)=(2,2,2) then 4 divides n. This extends the work of Godolphin (2019), who showed the above is true for the case q=2 when m and n are powers of 2. In the case k=2, the above implies necessary and sufficient conditions for the existence of a pair of mutually orthogonal frequency rectangles (or F-rectangles) whenever each symbol occurs the same number of times in a given row or column.
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