Taguchi’s Orthogonal Arrays Are Classical Designs of Experiments

Taguchi’s catalog of orthogonal arrays is based on the mathematical theory of factorial designs and difference sets developed by R. C. Bose and his associates. These arrays evolved as extensions of factorial designs and latin squares. This paper (1) describes the structure and constructions of Taguchi’s orthogonal arrays, (2) illustrates their fractional factorial nature, and (3) points out that Taguchi’s catalog can be expanded to include orthogonal arrays developed since 1960.


Introduction
Today many engineers are using Taguchi's catalog of orthogonal arrays [1] to plan industrial experiments. But Taguchi provides either no information or insufficient information on the methods that were used to construct these arrays. Moreover, Taguchi displays orthogonal arrays in forms that are different from the way these arrays are usually displayed in the statistical literature. It is, therefore, difficult to discern the links between Taguchi's arrays and their counterparts published elsewhere. Recent advertisements and testimonials of the efficacy of experiments based on Taguchi's orthogonal arrays increase the confusion by giving an impression that these arrays are something other than fractional factorials and classical plans of experiments. This paper describes the structure and constructions of Taguchi's orthogonal arrays, illustrates their fractional factorial nature, and points out that his catalog can be expanded to include orthogonal arrays developed since 1960.
The next section of this paper provides the background of orthogonal arrays and introduces the concept that an orthogonal array can be displayed in one of many equivalent forms. This concept is subsequently used to exhibit the equivalence of certain well-known fractional factorial plans and Taguchi's orthogonal arrays. Taguchi's catalog contains 20 arrays. However, only 18 of these arrays are orthogonal arrays. These 18 orthogonal arrays are the focus of this paper, and they have been classified into eight groups defined in such a way that the orthogonal arrays in each group can be constructed by a common method. The subsequent eight sections are devoted to these eight specific groups. In these sections, first the constructions of Taguchi's orthogonal arrays are described and then these arrays are related to fractional factorials and other well-known orthogonal arrays. In order to appreciate the factorial nature of Taguchi's orthogonal arrays, it is necessary to understand the constructions of these arrays. The last section of this paper identifies several useful orthogonal arrays that are not in Taguchi's catalog because they were developed after 1960.

The Background of Orthogonal Arrays
An orthogonal array (more specifically a fixedelement orthogonal array) of 5 elements, denoted by OA^is"'), is an Nxm matrix whose columns have the property that in every pair of columns each of the possible ordered pairs of elements appears the same number of times. The symbols used for the elements of an orthogonal array are arbitrary. This paper uses the symbols (0,1,2,..., s -1) to denote the s elements. Tables 1 and 2 display OA4(2^) and OA8(2'') respectively. Note that in every pair of columns of table 1 each of the 4 ordered pairs (0,0), (0,1), (1,0), and (1,1) appears exactly once. Similarly, every pair of columns in table 2 contains each of the four pairs (0,0), (0,1), (1,0), and (1,1) exactly twice. Taguchi refers to OAw(.s'") by the notation LN{S"'). The letter L in this notation stands for latin square, and it indicates that orthogonal arrays are generalized latin squares. Taguchi uses the symbols (1,2,...,.?) to denote the elements of an orthogonal array. The authors have, however, used the symbols (0,1,..., 5-1) in this paper because these symbols are natural in light of the methods of constructing these arrays.    0  0  0  0  0  2  0  0  0  1  1  1   3  0  1  1  0  1  1  4  0  1  1  1  0  0   5  1  0  1  1  0  1  6  1  0  1  0  1  0   7  1  1  0  1  1  0  S  1  1  0  0  0  1 Orthogonal arrays can be viewed as plans of multifactor experiments where the columns correspond to the factors, the entries in the columns correspond to the test levels of the factors and the rows correspond to the test runs. More specifically, the A^ rows of an OAN{S "') can be viewed as a subset of the possible s"' test runs of a complete factorial plan in m factors each having s test levels. Thus, an OAN{S"') can be viewed as a N/s'" fraction of a complete s'" factorial plan. For example, the four rows of the OA4(2^) that are displayed in Table 1, can be viewed as a 4/2^ = 1/2 fraction of a complete 2^ factorial plan.
A sub-matrix formed by deleting some columns of an orthogonal array is also an orthogonal array. Thus, by deleting certain columns of a given orthogonal array, it is possible to generate many different plans of multifactor experiments.
A fractional factorial plan that enables uncorrelated estimation of every factorial effect included in the underlying linear model assuming that all other effects are zero is called an orthogonal plan. Fractional factorial plans based on orthogonal arrays irrespective of the degree of fractionation are necessarily orthogonal plans. This is the primary reason for the popularity of fractional factorials based on orthogonal arrays.
Some of the most popular arrays in Taguchi's catalog are mixed-element (level) orthogonal arrays. A mixed-element orthogonal array, denoted by OA//(s"'x;"), is a matrix of N rows and m+n columns in which the first m columns have s elements each, the next n columns have t elements each, and in every pair of columns each of the possible ordered pairs of elements appears a constant number of times. The constant, however, depends on the pair of columns selected. Table 3 displays two orthogonal arrays: OAi8(6'x3'^) and OAi8(2'x3''). Note that every pair of columns in Table 3 contains each of the possible ordered pairs a constant number of times. A mixed-level orthogonal array can be viewed as a fractionated multilevel factorial plan.
It follows from the definition of an orthogonal array that an orthogonal array remains an orthogonal array when its (1) rows are permuted or (2) columns are permuted or (3) the elements within a column are permuted. When orthogonal arrays are viewed as plans of multifactor experiments, the row permutation corresponds to reordering of test runs, the column permutation corresponds to relabeling of factors, and the permutation of elements within a column corresponds to relabeling of factor levels. Most experimenters realize that the labels of factors.  1  2  3  4  5  6  7  8   1  0  0  0  0  0  0  0  0  0  2  0  0  0  1  1  1  1  1  1  3  0  0  0  2  2  2  2  2  2   4  1  0  1  0  0  1  1  2  2  5  1  0  1  1  1  2  2  0  0  6  1  0  1  2  2  0  0  1  the labels of factor levels and the order of test runs are arbitrary. Indeed, the order of test runs is usually randomized. Therefore, two orthogonal arrays are defined to be equivalent if one can be obtained from the other via the following permutations: (1) the rows are permuted, (2) the columns are permuted, and (3) the elements (symbols) within a column are permuted (for example, in a threeelement column, the elements (0,1, and 2) can be replaced with any one of their permutations: (0,2, and.^1); (1,0, and 2); (1,2 and 0); (2,0, and 1); or (2,1, and 0) respectively). Taguchi's format for an orthogonal array has the property that the entries in the left most columns change less frequently than the entries in the right most columns. Therefore, when these arrays are used to plan multifactor experiments, the cost of running the experiment can sometimes be reduced by judiciously associating with the left most columns those factors that are most expensive or most difficult to vary.
Taguchi's catalog contains twenty arrays. However, only eighteen of these twenty arrays are orthogonal arrays. The remaining two arrays, denoted by L'9 (2^') and L'27 (3"), are not orthogonal arrays and they are not discussed in this paper. The eighteen orthogonal arrays are classified into eight groups based on the common method of construction. The next eight sections are devoted to these eight groups.

Two-Element Orthogonal Arrays of 2' Rows for r = 2,3,4,5, and 6
The fractional factorial nature of two-element (level) orthogonal arrays follows from the way these arrays are constructed. So this section first describes a simple method of constructing these arrays, then illustrates their fractional factorial nature. A complete two-element orthogonal array with 2' rows has 2''-l columns and it is constructed column by column in three steps.
Step 1: Write in the r column numbers columns specified by 1,2,4,8,..., r-' a complete factorial plan in r factors each having two test levels represented by 0 and 1 respectively. In order to match Taguchi's display format, write this plan in such a way that the entries of the left most columns change less frequently than the entries of the right most columns. The entries of these r columns are used to calculate and define the entries of the remaining columns. Therefore, these r columns are referred to as the basic columns and marked as xi, X2,..; Xr, respectively.
Step 2: These basic columns are used to generate the other columns. The generator of a particular column is a rule of the form This method of construction and analogous methods of constructing three-, four-, and fiveelement orthogonal arrays are based on the mathematical theory of fractional factorials developed by Bose [2].  .
Repeat Repeat entries are 1 The following example illustrates these steps. Example: Construction of an OA8(2^) Here N = 8 = 2\ so r = 3.
Step 1: Write the r = 3 basic columns.
Step 2: List the generators (see rows 1 to 7 of table 4).

X1+X3
Step 3: Complete the array using the generators identified in step 2.
Column No. 12 3 4 5 6 7 Row No. The fractional factorial nature of Taguchi's twoelement orthogonal arrays stems from the fact that the entries of the r columns identified by column numbers 1,2,4,8,...,2'"' form a complete factorial plan, and the remaining columns correspond to the interaction effects. The generators of these columns have a one-to-one correspondence with the main effects and the interaction effects written in Yates' [3] standard order. The r basic columns correspond to the main effects and the remaining columns correspond to the interaction effects, A two-element (two-level) orthogonal array with 2' rows reduces to a fractional factorial plan when more than r factors are associated with the columns of the array and the remaining columns are deleted. In particular, when all 2''-l columns are associated with an equal number of factors, a twolevel orthogonal array OAAf(2"') where N -2' represents a Nil'" = (1/2)'""'' fraction of a complete 2'" factorial plan. For example, an OA8(2') represents a (1/2)'-^ = (1/2)" = l/16th fraction of a complete 2'' factorial plan. That is, an OA8(2') represents a 2'"'' plan in factorial notation.
The test levels of 2*"'' type fractional factorial plans are usually represented [4] by the symbols -and +. Such plans are often constructed by writing a complete factorial plan in the required number of test runs and appending additional columns obtained by multiplying certain columns of the complete factorial plan. This method and the Bose method of constructing two-level orthogonal arrays described here are similar, but since (-x-= -f, -x-f = -, and + y. + = +) while (0-f-0 = 0, 0-H = 1, and 1-1-1 = 0 in modulo 2 arithmetic), the two methods yield different fractions of the same type. However, one fraction can be obtained from another of the same type by switching the test levels, and permuting the rows and columns. For example, Taguchi's OA8(2^) can be obtained from Box, Hunter, and Hunter's 2'"'* plan [4] (shown as table 12.5 on page 391 of their book) by switching -and + in columns 4, 5 and 6; permuting the columns in the order 3,2,6,1,5,4, and 7; and relabeling -as 0 and + as 1.

4, Two-Element Orthogonal
Array OAi2(2") Table 5 displays Taguchi's OAj2(2") in the 0 and 1 notation, and table 6 displays the classic Plackett and Burman [5] plan of 12 runs in the 0 and 1 notation rather than the usual -and 4-notation. Since table 5 can be obtained  from table 6 through the following permutations, Taguchi's OAi2(2") and the Plackett and Burman plan of 12 runs are equivalent.

Three-Element Orthogonal Arrays of 3*^ Rows for r = 2, 3, and 4
A complete three-element orthogonal array with 3' rows has (3''-l)/(3-l) columns and it is constructed in three steps: Step 1: Write in the r columns specified by column numbers 1,2,5,14,...,(3''"'-1)/ (3 -1) +1 a complete factorial plan in r factors each having three test levels represented by 0,1, and 2, respectively. In order to match Taguchi's display format, write this plan in such a way that the entries of the left-most columns change less frequently than do the entries of the right-most columns. Mark these columns asA:i,;t2 Xr, respectively. Step 3: Compute the entries of the remaining columns by using the entries of the r basic columns and the appropriate generators. All calculations are done in modulo 3 arithmetic (that is, an integer larger than or equal to three is replaced with its remainder after division by three).  The following example illustrates these steps. Example: Construction of an OA<; (3'') Here N = 9 = 3\ so r = 2.
Column No, Generator Step 3: Complete the array using the generators identified in step 2. The fractional factorial nature of Taguchi's three-element orthogonal arrays stems from the fact that the entries of the r basic columns identified by column numbers, l,2,5,14,...,(3''"'-l)/(3-l) + l form a complete factorial plan and the other columns correspond to the interaction effects. Since each column contains three distinct elements, two degrees of freedom are associated with each column. Since pairwise interaction effects carry (3 -1) X (3 -1) = 4 degrees of freedom, two columns correspond to each pairwise interaction effect. An interaction effect involving k factors carries (3-1)* = 1^ degrees of freedom. Therefore, 2*"' columns correspond to each interaction effect involving k factors for k = 2,3,4,... . A three-element (level) orthogonal array with 3' rows reduces to a fractional factorial plan when more than /• factors are associated with the columns of the array and the remaining columns are deleted. In particular, when all (3''-l)/(3-l) columns are associated with an equal number of factors, a three-level orthogonal array OAAr (3"') where N = 3' and m = (3''-l)/(3-l) represents a N/3"' = (1/3)""-'■ fraction of a complete 3'" factorial plan. For example, an OA9(3'') represents a (1/3)^ fraction of a complete 3* factorial plan. That is, an OA9(3'') represents a 3*~' plan in factorial notation.

Four-Element Orthogonal Arrays of 4'^ Rows for r = 2 and 3
The method of constructing four-element orthogonal arrays is similar to the method for threeelement arrays. An important difference, however, is that the calculations required to generate the columns are not performed in modulo 4 arithmetic. Instead, special addition and multiplication tables, displayed here as tables 9 and 10 are used. These addition and multiplication tables are based on the "finite arithmetic of the Galois Field Theory" that underlies this method of construction. According to this theory, the calculations required to generate an orthogonal array of s elements are done in modulo s arithmetic when s is a prime number, as is the case with 2, 3, and 5. When j is a power of a prime number such as 4 (which is the square of prime number 2), finite arithmetic of a Galois Field of s elements is used. A four-element orthogonal array with 4' rows and (4''-l)/(4-l) columns is constructed in three steps.
Step 2: As before the generators of the remaining columns are of the form Step 3: Compute the entries of the remaining columns by using the entries of the r basic columns and the appropriate generators. All calculations are done using finite additions and multiplications defined in tables 9 and 10. Table 8. Coefficients of the generators of four-element orthogonal arrays of 4'' rows for /■ = 2, 3,...
The following example illustrates these steps. Example: Construction of an OAi6(4^) Here A^ = 16 = 4^ so r = 2.
Step 1: Write the r = 2 basic columns Generator

X\ x^
Step 2: List the generators (see rows 1 to 5 of table 8).
" This addition table is also the diffefence table for a Galois field of four elements.  The entries of the r basic columns identified by column numbers l,2,6,...,(4'^"'-l)/(4-l) + l form a complete factorial plan and the other columns correspond to the interaction effects. Since each column contains four distinct elements, three degrees of freedom are associated with each column. An interaction effect involving k factors carries (4 -1)* = 3* degrees of freedom. Therefore, 3*"' columns correspond to each interaction effect involving k factors. In particular, three columns correspond to each pairwise interaction effect.

Five-Element Orthogonal Array
OA2s (5*) Taguchi's OA25(5'') is constructed through the same general approach that is used to construct 2-, 3-, and 4-element arrays. The first two columns form a complete factorial plan in two factors each having five test levels represented by 0,1,2,3, and 4. The other columns are generated from these two columns using the following generators where xi and X2 represent the entries of the first two columns. These orthogonal arrays are constructed by the method of Bose and Bush [6]. This method involves four concepts: difference matrices, Kronecker sums, saturated orthogonal arrays, and column replacement. 1) A difference matrix of s elements 0,1,...,(5 -1), denoted by DM{S), is an MxM matrix whose columns have the property that the differences in finite arithmetic between any two columns is a column in which each of the s elements occurs equally often. Table 11 displays the difference matrix D3(3). When .s is a prime number such as 3 or 5, the finite arithmetic used in defining the difference matrix is modulo s arithmetic, and when s is a power of a prime number such as 4 (which is the square of prime number 2), the finite arithmetic is the arithmetic of a Galois Field of s elements. Table 9 defines finite addition for 5=4. The finite difference table for s =4 is the same as the finite addition table.   3) An orthogonal array OAN(s"'xt") is said to be saturated when A^-1 = m(5-l)+/i(f-1). Since an s element column has s -1 degrees of freedom and a t element column has r -1 degrees of freedom, m +n columns of OAN{s"'xt") have m(s -l) + n(t -1) degrees of freedom. Since (heN rows of 0AAI(5"'X/") yield N (independent) data values, the total number of effects that can be estimated after allowing for the grand mean of the N data values isN -1. Therefore when a saturated orthogonal array is used as an experimental plan, the total number of effects that can be estimated is equal to the total degrees of freedom of the columns (factors). When all m +n columns are associated with factors, a saturated orthogonal array can be viewed as a saturated main effect fractional factorial plan. 4) An orthogonal array remains an orthogonal array when one of its columns is replaced with an orthogonal array whose rows have a one-to-one correspondence with the elements of the replaced column. For example, suppose a is a four-element column of an orthogonal array A, and suppose B is an orthogonal array whose rows have a one-to-one correspondence with the elements of column a where Then a matrix obtained from the orthogonal array^4 by replacing the column a with the orthogonal array B is an orthogonal array.
A difference matrix remains a difference matrix when (1) its rows are permuted or (2) its columns are permuted or (3) an integer is added (in finite arithmetic) to any column of the matrix. Because of finite arithmetic, the addition of an integer to a column results in a permutation of the elements of that column. Thus each of these three operations results in a permutation of the elements of the matrix. The difference matrices D6(3) and Ds(4) used by Taguchi table 17 by permuting both the rows and the columns of table 17  in the following order: 1,2,3,4,5 (4) No.
Although the original Japanese version of Taguchi's catalog of orthogonal arrays was developed before 1960, it continues to be very useful. These arrays can be modified to generate many types of multifactor experiments [9] and many other orthogonal arrays can be derived from Taguchi's catalog through established mathematical procedures. Nevertheless, the catalog can now be expanded to include arrays developed after 1960. For example, Taguchi's catalog can be expanded to include OA24(4'x2'"), OA4o(4'x2'*) and OA4s(4^x2'*') first developed by Dey and Ramakrishna [10] and Chacko, Dey, and Ramakrishna [11], and then re-constructed through a unified procedure by Cheng [12]. It is the authors' intent to develop an expanded and revised version of Taguchi's catalog of orthogonal arrays. A companion paper [13] limited to the fixed-element orthogonal arrays appeared in the Journal of Quality Technology.