Orthogonal array

An OA(18, 7, 3, 2) on the right side. Any pair of columns (extracted to the left side) contains each ordered pair twice as its row. Here to play the SVG animation varying the pair of columns.

For t ≤ k, an orthogonal array of type (N, k, v, t) – an OA(N, k, v, t) for short – is an N × k array whose entries are chosen from a set X with v points (a v-set) such that in every subset of t columns of the array, every t-tuple of points of X is repeated the same number of times. The number of repeats is usually denoted λ.

In many applications these parameters are given the following names:

N is the number of experimental runs,

k is the number of factors,

v is the number of levels,

t is the strength, and

λ is the index.

The definition of strength leads to the parameter relation

N = λv^t.

An orthogonal array is simple if it does not contain any repeated rows. (Subarrays of t columns may have repeated rows, as in the OA(18, 7, 3, 2) example pictured in this section.)

An orthogonal array is linear if X is a finite field F_q of order q (q a prime power) and the rows of the array form a subspace of the vector space (F_q)^k.[2] The right-hand example in the introduction is linear over the field F₂. Every linear orthogonal array is simple.

In a mixed-level orthogonal array, the symbols in the columns may be chosen from different sets having different numbers of points, as in the following example:[3]

0	0	0	0	0
1	1	1	1	0
0	0	1	1	1
1	1	0	0	1
0	1	0	1	2
1	0	1	0	2
0	1	1	0	3
1	0	0	1	3

This array has strength 2:

• Any pair of the first four columns contains each of the ordered pairs (0, 0), (0, 1), (1, 0) and (1, 1) two times.
• Columns 4 and 5 – or column 5 with any one of the other columns – contains each ordered pair (i, j) once, where i = 0 or 1 and j = 0, 1, 2, or 3.

It may thus be denoted may be denoted OA(8, 5, 2⁴4¹, 2), as is discussed below. The expression 2⁴4¹ indicates that four factors have 2 levels and one has 4 levels.

As in this example, there is no single ``index" or repetition number λ in a mixed-level orthogonal array of strength t: Each subarray of t columns can have a different λ.

The terms symmetric and asymmetric are sometimes used for fixed-level and mixed-level. Here symmetry refers to the property that all factors have the same number of levels, not to the "shape" of the array: a symmetric orthogonal array is almost never a symmetric matrix.

The notation OA(N, k, v, t) is sometimes contracted so that one may, for example, write simply OA(k, v),[4] as long as the text makes clear the unstated parameter values. In the other direction, it may be expanded for mixed-level arrays. Here one would write OA(N, k, v₁···v_k, t), where column i has v_i levels. This notation is usually shortened when values v are repeated, so that one writes OA(8, 5, 2⁴4¹, 2) for the example at the end of the last section, rather than OA(8, 5, 2·2·2·2·4, 2). In similar fashion, one may shorten OA(N, k, v, t) to OA(N, v^k, t) for fixed-level arrays.

This OA notation does not explicitly include the index λ, but λ can be recovered from the other parameters via the relation N = λv^t. This is effective when the parameters all have specific numerical values, but less so when a class of orthogonal arrays is intended. For example, when indicating the class of arrays having strength t = 2 and index λ=1, the notation OA(N, k, v, 2) is insufficient to determine λ by itself. This is typically remedied by writing OA(v², k, v, 2) instead. While notations that explicitly include the parameter λ do not have this problem, they cannot easily be extended to denote mixed-level arrays.

Some authors define an OA(N, k, v, t) as being k × N rather than N × k. In such cases the strength of the array is defined in terms of a subset of t rows rather than columns.

Except for the prefix OA, the notation OA(N, k, v, t) is the same as that introduced by Rao.[5] While this notation is very common, it not universal. Hedayat, Sloane and Stufken[6] recommend it as standard, but list eight alternatives found in the literature, and there are others.[8]

An example of an OA(16, 5, 4, 2); a strength 2, 4-level design of index 1 with 16 runs:

An example of an OA(27, 5, 3, 2) (written as its transpose for ease of viewing):[9]

This example has index λ = 3.

An OA(n², 3, n, 2) is equivalent to a Latin square of order n. For k ≤ n+1, an OA(n², k, n, 2) is equivalent to a set of k − 2 mutually orthogonal Latin squares of order n. Such index one, strength 2 orthogonal arrays are also known as Hyper-Graeco-Latin square designs in the statistical literature.

Let A be a strength 2, index 1 orthogonal array on an n-set of elements, identified with the set of natural numbers {1,...,n}. Choose and fix, in order, two columns of A, called the indexing columns. Because the strength is 2 and the index is 1, all ordered pairs (i, j) with 1 ≤ i, j ≤ n appear exactly once in the rows of the indexing columns. Here i and j will in turn index the rows and columns of a n×n square. Take any other column of A and fill the (i, j) cell of this square with the entry that is in this column of A and in the row of A whose indexing columns contain (i, j). The resulting square is a Latin square of order n. For example, consider this OA(9, 4, 3, 2):

By choosing columns 3 and 4 (in that order) as the indexing columns, the first column produces the Latin square

while the second column produces the Latin square

These two squares, moreover, are mutually orthogonal. In general, the Latin squares produced in this way from an orthogonal array will be orthogonal Latin squares, so the k − 2 columns other than the indexing columns will produce a set of k − 2 mutually orthogonal Latin squares.

This construction is completely reversible and so strength 2, index 1 orthogonal arrays can be constructed from sets of mutually orthogonal Latin squares.[10]

Orthogonal arrays provide a uniform way to describe these diverse objects which are of interest in the statistical design of experiments.

The concepts of Latin squares and mutually orthogonal Latin squares were generalized to Latin cubes and hypercubes, and orthogonal Latin cubes and hypercubes by Kishen (1942).[14] Rao (1946) generalized these results to arrays of strength t. The present notion of orthogonal array as a generalization of these ideas, due to legendary scientist C. R. Rao, appears in Rao (1947),[15] with his generalization to mixed-level arrays appearing in 1973.[16]

Rao initially used the term "array" with no modifier, and defined it to mean simply a subset of all treatment combinations – a simple array. The possibility of non-simple arrays arose naturally when making treatment combinations the rows of a matrix. Hedayat, Sloane and Stufken[17] credit K. Bush[18] with the term "orthogonal array".

• Combinatorial design
• Latin hypercube sampling
• Graeco-Latin squares

• Box, G. E. P.; Hunter, W. G.; Hunter, J. S. (1978). Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building. John Wiley and Sons. ISBN 9780471093152.
• Bush, K. A. (1950). Orthogonal arrays (PhD). University of North Carolina.
• Dénes, J.; Keedwell, A. D. (1974), Latin Squares and Their Applications, New York-London: Academic Press, ISBN 0-12-209350-X, MR 0351850
• Hedayat, A.S.; Sloane, N.J.A.; Stufken, J. (1999), Orthogonal Arrays, Theory and Applications, New York: Springer
• Kishen, K. (1942), "On latin and hyper-graeco cubes and hypercubes", Current Science, 11: 98–99
• Kishen, K. (1950), "On the construction of latin and hyper-graeco-latin cubes and hypercubes", J. Indian Soc. Agric. Statistics, 2: 20–48
• Raghavarao, Damaraju (1988). Constructions and Combinatorial Problems in Design of Experiments (corrected reprint of the 1971 Wiley ed.). New York: Dover.
• Raghavarao, Damaraju and Padgett, L.V. (2005). Block Designs: Analysis, Combinatorics and Applications. World Scientific.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Rao, C.R. (1946), "Hypercubes of strength ''d'' leading to confounded designs in factorial experiments", Bulletin of the Calcutta Mathematical Society, 38: 67–78
• Rao, C.R. (1947), "Factorial experiments derivable from combinatorial arrangements of arrays", Supplement to the Journal of the Royal Statistical Society, 9 (1): 128–139, doi:10.2307/2983576, JSTOR 2983576
• Rao, C. R. (1973). "Some combinatorial problems of arrays and applications to design of experiments". In Srivastava, Jagdish N. (ed.). A Survey of Combinatorial Theory. North Holland. ISBN 0-7204-22620.
• Stinson, Douglas R. (2003), Combinatorial Designs: Constructions and Analysis, New York: Springer, ISBN 0-387-95487-2
• Street, Anne Penfold & Street, Deborah J. (1987). Combinatorics of Experimental Design. Oxford U. P. [Clarendon]. ISBN 0-19-853256-3.
• Taguchi, Genichi (1986). Orthogonal Arrays and Linear Graphs. Dearborn, MI: American Supplier Institute.

• Hyper-Graeco-Latin square designs
• A SAS example using PROC FACTEX
• Kuhfeld, Warren F. "Orthogonal Arrays". SAS Institute Inc. SAS provides a catalog of over 117,000 orthogonal arrays.

 This article incorporates public domain material from the National Institute of Standards and Technology.