Row‐column factorial designs with multiple levels

Rahim, Fahim; Cavenagh, Nicholas J.

doi:10.1002/jcd.21799

Cited by 5 publications

(3 citation statements)

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“…The property of being an array of type I 4 (9,9,3,2) eliminates confounding between each of the main effects. We refer the reader to [13] and [7] for a literature review on the application of row-column factorial designs to statistical experimental design. In this paper two arrays are equivalent under any: (a) reordering of rows; (b) reordering of columns; (c) reordering of levels (applied globally); and (d) reordering of the entries in each vector (with the same reordering applied globally).…”

Section: Introductionmentioning

confidence: 99%

“…Necessary and sufficient conditions for a row-column factorial design of strength 1 are given in [13], generalizing [7] and [16].…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1. ( [13]) Let m ≤ n. There exists I k (m, n, q, 1) (that is, an m × n row-column factorial design q k of strength 1) if and only if: i. q|m and q|n;…”

Section: Introductionmentioning

confidence: 99%

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Row-column factorial designs with strength at least $2$

Rahim¹,

Cavenagh²

2022

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The q k (full) factorial design with replication λ is the multi-set consisting of λ occurrences of each element of each q-ary vector of length k; we denote this by λ×[q] k . An m×n row-column factorial design q k of strength t is an arrangement of the elements of λ × [q] k into an m × n array (which we say is of type I k (m, n, q, t)) such that for each row (column), the set of vectors therein are the rows of an orthogonal array of size k, degree n (respectively, m), q levels and strength t. Such arrays are used in experimental design. In this context, for a row-column factorial design of strength t, all subsets of interactions of size at most t can be estimated without confounding by the row and column blocking factors.In this manuscript we study row-column factorial designs with strength t ≥ 2. Our results for strength t = 2 are as follows. For any prime power q and assuming 2 ≤ M ≤ N , we show that there exists an array of type I k (q M , q N , q, 2) if and only if k ≤ M + N , k ≤ (q M − 1)/(q − 1) and (k, M, q) = (3, 2, 2). We find necessary and sufficient conditions for the existence of I k (4m, n, 2, 2) whenever m is odd, assuming the existence of a Hadamard matrix of order 4m. For t = 3 we focus on the binary case. Assuming M ≤ N , there exists an array of type I k (2 M , 2 N , 2, 3) if and only if M ≥ 5, k ≤ M + N and k ≤ 2 M −1 . Most of our constructions use linear algebra, often in application to existing orthogonal arrays and Hadamard matrices.

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