# Orthogonal arrays

Post-publication activity

Orthogonal Arrays represent a versatile class of combinational arrangements useful for conducting experiments to determine the optimum mix of a number of factors in a product to maximize the yield, and in the construction of a variety of designs for agricultural, medical and other experiments. An article in Forbes Magazine (March 11, 1996, pp.114-118) refers to Orthogonal Arrays as a New Mantra in a variety of industrial establishments in USA.

## Definition

Consider a set $$S$$ of $$s$$ symbols which may be taken as $$1,2,\cdots,s\ ,$$ and a $$t \times n$$ matrix $$T$$ of elements from $$S\ .$$ A certain arrangement of symbols in $$T\ ,$$ called hypercube of strength $$d\ ,$$ was introduced in Rao (1943), which was further developed in a series of papers, Rao (1946a, 1946b, 1947). A full combinatorial description of hypercubes of strength $$d$$ with some mathematical properties was given in Rao (1949), which has been later termed as orthogonal array (OA) by Bush (1950). Slight variations of OA leading to combinational arrangements, known as OA:I and OA:II were given in Rao (1961). The definitions of different kinds of OA’s are as follows.

1. A $$t \times n$$ matrix $$T$$ is called OA of strength $$d\ ,$$ constraints $$t$$ and index $$\lambda\ ,$$ and is represented by $$OA(n,t,s,d)\ ,$$ if in every $$d$$ rows, the $$n$$ columns contain each of the $$s^d$$ ordered combinations of $$s$$ elements taken $$d$$ at a time allowing repetitions $$\lambda$$ times.
2. $$T$$ is called OA:I of strength $$d\ ,$$ constraints $$t$$ and index $$\lambda\ ,$$ represented by $$OA(n,t,s,d) \ :$$ I if in every set of $$d$$ rows, the $$n$$ columns contain each of the $$s!/(s-d)!$$ ordered combinations of $$s$$ elements taken $$d$$ at a time without repetitions, $$\lambda$$ times.
3. $$T$$ is called OA:II of strength $$d\ ,$$ constraints $$t$$ and index $$\lambda$$ and represented by $$OA(n,t,s,d)\ :$$ II if in every $$d$$ rows, the n columns contain each of $$s!/d!$$ combinations of $$s$$ elements taken $$d$$ at a time ignoring order and without repetitions, $$\lambda$$ times.

For example, when $$s=3$$ with the elements represented by $$1,2,3$$ different orthogonal arrays of strength $$2$$ are as follows.

$$OA(9,4,3,2)$$ $$OA:I(6,3,3,2)$$ $$OA:II(3,3,3,2)$$
$$\begin{array}{ccccccccccc} 1&1&1&~&2&2&2&~&3&3&3\\ 1&2&3&~&1&2&3&~&1&2&3\\ 1&2&3&~&2&3&1&~&3&1&2\\ 1&2&3&~&3&1&2&~&2&3&1 \end{array}$$ $$\begin{array}{ccccccc} 1&2&3&~&1&2&3\\ 2&3&1&~&3&1&2\\ 3&1&2&~&2&3&1 \end{array}$$ $$\begin{array}{ccc} 1&2&3\\ 2&3&1\\ 3&1&2 \end{array}$$

The book on OA’s by Hedayat, Sloane, and Stufken (1999) is an excellent introduction to OA’s describing new areas of research in combinatorial mathematics they generated and numerous practical applications in industrial and medical experiments. Some of the current developments in OA’s are described in a special issue of the Journal of Planning and Inference (vol.56, 1996) edited by Singhi. The concept of OA’s was extended to $$t \times n$$ matrices where the elements in the $$i$$-th row come from the set $$1,2,\cdots,s_i\ ,$$ and the appropriate number of combinations in any chosen set of $$d$$ rows is considered as described in Rao (1947).

## History

When I joined the Indian Statistical Institute in 1941 to study statistics, I was surprised to find that combinatorics, based on the concepts of Galois fields and finite projective geometrics, was the main field of research of many faculty members under the leadership of R.C.Bose. I found the subject very fascinating, and within a few months I acquired the necessary knowledge of combinatorics and started doing research and publishing papers from 1942 in collaboration with K.R.Nair, an associate of R.C.Bose. I also started working on the construction of Galois fields and using the results for generating incomplete block designs and constructing confounded factorial designs during the years 1943 - 1945. I introduced what are called hypercubes of strength $$d$$ in my thesis for M.A. degree of Calcutta University, Rao (1943). In two subsequent papers, Rao (1946a, 1946b), the concepts were further developed . A general definition of orthogonal arrays and their applications are given in Rao (1947, 1949). The contents of the last two papers were first submitted to Biometrika as a consolidated paper as they provide a generalization of the paper by Placket and Burman (1946) published in Biometrika. But it was rejected as too mathematical by the editor of Biometrika at that time. I prepared two papers, one giving the applications of OA’s and another on combinatorial properties of OA and sent them to two different journals, Journal of the Royal Statistical Society and Proceedings of the Edinburgh Mathematical Society. Both received favorable comments from referees as highly original and published without any revision. These two papers provided the foundation for subsequent publications by various authors on combinatorics based on OA’s and the applications of OA in design of experiments.

## Applications

The book by Hedayat, Sloane, and Stufken contain a number of practical applications of OA’s. Of special significance is the use of OA’s in industrial experimentation by Taguchi (1986, 1987) to determine the optimum mix of factors to maximize yield and the effect of noise factors such as environmental conditions on production. The array OA:I was used by Bose, Shrikande and Parkar (1960) in disproving Euler’s conjecture on orthogonal Latin squares. Morgan and Chakravarti (1988) used OA:II in constructing optimal incomplete block designs when correlations between neighboring plots in an experimental block are considered. Majumdar and Martin (2004) give a variety of applications of OA:I and OA:II to produce useful designs for experiments where the responses are correlated. Due to their rich structure, OA:I and OA:II have the potential to produce efficient designs for use in a variety of applications. Reference may be made to Kunnet and Stufken (2002), Reghavarao and Zhou (1996), and Afsarinejad, and Hedayat (2002).