## Introduction

A strength 2 orthogonal array (OA) of size n with*k*

*s*-level columns (

_{i}*i*= 1,...,

*k*), denoted by

*L*

_{n}(

*s*

_{1}...

*s*) is an

_{k}*n*x

*k*matrix in which all possible combinations of levels in any two columns appear the same number of times (Rao 1947). There is an OA library of over 200 OAs maintained by Prof. N. J. A. Sloane. This library has been recently updated by Dr. W. F. Kuhfeld of SAS at his OA site. This site contains all OAs listed in the Appendix of Kuhfeld & Tobias (2005) as well as new ones contributed by other authors. A simple introduction to OA can be found in most textbook on design of experiments (e.g. Chapter 7 of Wu & Hamada, 2000). A more comprehensive reference of OA is Hedayat, et. al. (1999).

In a near-OA
*L' _{n}*(

*s*

_{1}...

*s*), to reduce the run size, the orthogonality of some pairs of columns is necessarily sacrificed. The concept of near-OA (see Taguchi 1959; Wang and Wu 1992; Nguyen 1996b; Wu & Hamada 2000; Ma et. al. 2000 and Xu 2002) provides a genuine answer to situations when OAs are not available. An array is called a saturated design when ∑(

_{k}*s*-1)=

_{i}*n*-1 (e.g. a Hadamard matrix) and is called a supersaturated design when ∑(

*s*-1)>

_{i}*n*-1. The 2-level supersaturated designs were discussed in Booth & Cox (1962), Lin (1993), Nguyen (1996a), Tang & Wu (1997), Cheng (1997) and Wu & Hamada (2000) Section 8.6. The mixed level supersaturated designs were discussed in Fang, et. al. (2003, 2004). Additional references on this subject can be found in the references of these two papers.

NOA is a Gendex program for constructing mixed level OA, near-OAs and
supersaturated designs. NOA is extremely useful when you want to augment OAs or
near-OAs with additional columns. Several new OAs can be obtained by adding
additional columns to the Sloane OA library. The optimality criterion used in NOA,
called the *average* criterion is the minimization of
*E*(*d*^{2})=∑∑*d*^{2}_{ij}
/* _{k}*C

_{2}(where

*i*and

*j*are different columns of the array and

*k*is the total number of columns of this array) discussed Ma et. al (2000), Lu et al. (2003) and Fang, et. al. (2003, 2004) (note that E(

*d*

^{2}) is

*D*

^{2}in the first reference and

*E*(

*f*

_{NOD}) in the last two references). The following matrix shows the frequency distribution of the levels of the columns of an

*L'*

_{84}(3

^{1}6

^{1})

5 4 5 5 4 5 4 5 5 4 5 5 5 5 4 5 5 4

which say the combination (0,0) appears 5 times and the combination (0,1) appears 4 times, etc. The expected frequency for each level combination is

*n*/(*s*_{1}*s*_{2})=84/(3x6)=4.6667.
The differences between the observed and the expected frequencies of each cell
are in the following matrix:

0.3333 -0.6667 0.3333 0.3333 -0.6667 0.3333 -0.6667 0.3333 0.3333 -0.6667 0.33333 0.3333 0.3333 0.3333 -0.6667 0.3333 0.3333 -0.6667

Here *d*^{2}_{12} (=4) is defined as the squared
*Euclidean* distance between the observed and expected frequencies (or
the sum of squares of the values in the difference matrix). Details of the NOA
algorithm which uses this optimality criterion is discussed Nguyen
& Liu (2007).

## Using NOA

Let's assume all Gendex class files are in the directory c:\gendex and
suppose you want to construct an
*L*_{12}(3^{1}2^{4}) (Example 1). At the working directory,
type the following command at the command prompt (case is important):

java -cp c:\gendex NOA

The NOA GUI will pop up. Enter the the number of factors at the appropriate factor levels .

Now click **START**, two windows will pop up. The first one asks you
to choose the number of runs: choose 12 and click OK. The second one asks you whether you want to force
you want the first 3-level column to be orthogonal to the remaining columns: click No.

NOA will start running and after try 2, the plan of the constructed array for this try
pops up in the NOA output window (as *E(d ^{2})* reaches 0) and then
NOA stops:

**START**button has been changed to the

**RESET**one. If you click this

**RESET**button, the output will disappear and you can use NOA for a new design problem. Also note that the default random seed is the one obtained from the system clock and the default number of tries is 1000. You can change these default values if you wish to.

Now, suppose you want to construct an
*L*_{12}(3^{1}2^{9}) (Example 11) by adding five 2-level
columns to an array *L*_{12}(3^{1}2^{4})
obtained in the previous NOA session and stored in a file called
*base.txt:*

0 0 0 0 1 2 0 0 1 0 0 1 1 1 0 2 1 1 0 0 1 1 0 0 1 0 1 0 0 0 1 1 1 1 1 1 0 1 0 0 2 0 1 0 1 1 0 0 1 0 2 1 0 1 1 0 0 1 1 1 |

Enter *base.txt* at the **File** text field and enter 5
at the 2-level text field of the NOA GUI and click **START
** and **Yes** to the question on whether you want to force
the first 3-level column to be orthogonal to the remaining columns. In less than a second.
the NOA output window will become (Example 11):

## Output

The result of the best try is displayed in the NOA output window and is also saved
in the file NOA.htm in the working directory. This file can be read by a browser such as
IE or *Google Chrome*. Information for this try includes:

- Try number;
- The number of iterations;
*f*=∑*d*^{2}(and_{ij}*g*if you answer**Yes**for to the question on whether you want to force the first*x*columns to be orthogonal to the remaining columns).*g*=0 indicates that these columns are indeed orthogonal to the remaining columns;*E*(*d*^{2}) and its lower bound. NOA bound improves the one of Fang, et. al. (2003, 2004) when the constructed design is a near-OA. Also, unlike the latter, the former does not make the restriction that*n*should be divisible by*s*(_{i}*i*=1,...*,k*). Details of the calculation of this lower bound appears in Nguyen & Liu (2008). Note that for 2-level column arrays, the popular criterion*E*(*s*^{2})=4*E*(*d*^{2}) (Eq. (14) of Fang, et. al. (2003)).*d*_{max}and the frequency of*d*=_{ij}*d*_{max};*N*(the number of_{p}*d*>0 or the number of pairs of nonorthogonal columns in the array)._{ij}*A*_{2}(measurement of the overall aliasing or nonorthogonality between all possible pairs of columns).*D*-efficiency;- The array;
- The
*d*^{2}matrix; - The Cramer's V matrix: Cramer's V is the most popular of the chi-square-based measures of association between two nominal variables.
- The number of protected columns if any;
- The time in seconds NOA used to construct the above design.

**Note**: The calculations of items 6-8 are described in Section 2.2 of Xu
(2002). Items 10 and 11 are not printed when the constructed array is an OA (i.e. when *f*=0).

## Examples

- An
*L*_{12}(3^{1}2^{4}) (http://designcomputing.net/gendex/noa/m1.html). - An
*L*_{18}(2^{1}3^{7}) (http://designcomputing.net/gendex/noa/m2.html). - An
*L*_{20}(5^{1}2^{8}) (http://designcomputing.net/gendex/noa/m3.html). - An
*L*_{25}(5^{6}) (http://designcomputing.net/gendex/noa/m4.html). - An
*L'*_{6}(3^{1}2^{3}) (http://designcomputing.net/gendex/noa/n1.html). - An
*L*'_{10}(5^{1}2^{5}) (http://designcomputing.net/gendex/noa/n2.html). - An
*L*'_{12}(4^{1}3^{4}) (http://designcomputing.net/gendex/noa/n3.html). - An
*L*'_{12}(2^{3}3^{4}) (http://designcomputing.net/gendex/noa/n4.html). - An
*L*'_{12}(6^{1}2^{5}) (http://designcomputing.net/gendex/noa/n5.html). - An
*L*'_{12}(6^{1}2^{6}) (http://designcomputing.net/gendex/noa/n6.html). - An
*L*'_{12}(3^{1}2^{9}) (http://designcomputing.net/gendex/noa/n7.html). - An
*L*'_{12}(2^{1}3^{5}) (http://designcomputing.net/gendex/noa/n8.html). - An
*L*'_{12}(3^{2}2^{7}) (http://designcomputing.net/gendex/noa/n9.html). - An
*L*'_{12}(3^{2}2^{7}) (http://designcomputing.net/gendex/noa/n9bis.html). - An
*L*'_{12}(2^{5}3^{3}) (http://designcomputing.net/gendex/noa/n10.html). - An
*L*'_{15}(5^{1}3^{5}) (http://designcomputing.net/gendex/noa/n11.html). - An
*L*'_{18}(2^{1}3^{8}) (http://designcomputing.net/gendex/noa/n12.html). - An
*L*'_{18}(3^{7}2^{3}) (http://designcomputing.net/gendex/noa/n13.html). - An
*L*'_{18}(9^{1}2^{8}) (http://designcomputing.net/gendex/noa/n14.html). - An
*L*'_{20}(5^{1}2^{15}) (http://designcomputing.net/gendex/noa/n15.html). - An
*L*'_{24}(8^{1}3^{8}) (http://designcomputing.net/gendex/noa/n16.html). - An
*L*'_{24}(3^{1}2^{21}) (http://designcomputing.net/gendex/noa/n17.html). - An
*L*'_{24}(6^{1}2^{15}) (http://designcomputing.net/gendex/noa/n18.html). - An
*L*'_{24}(6^{1}2^{18}) (http://designcomputing.net/gendex/noa/n19.html). - An
*S*_{16}(2^{15}) (http://designcomputing.net/gendex/noa/s1.html). - An
*S*_{11}(2^{11}) (http://designcomputing.net/gendex/noa/s2.html). - An
*S*_{8}(2^{14}) (http://designcomputing.net/gendex/noa/s3.html). - An
*S*_{12}(2^{18}) (http://designcomputing.net/gendex/noa/s4.html). - An
*S*_{12}(6^{5}4^{4}) (http://designcomputing.net/gendex/noa/s5.html). - An
*S*_{20}(5^{5}4^{1}) (http://designcomputing.net/gendex/noa/s6.html). - An
*S*_{16}(2^{16}8^{8}) (http://designcomputing.net/gendex/noa/s7.html). - An
*S*_{20}(2^{19}4^{1}5^{5}) (http://designcomputing.net/gendex/noa/s8.html).

**Notes**:

- Examples 1-4: mixed level OAs. Certain new OAs can be constructed by adding new columns to an existing
array. As an additional example, the
*L*_{60}(2^{15}6^{1}10^{1}) can be constructed by adding sequentially five 2-level columns to the*L*_{60}(2^{10}6^{1}10^{1}) in the Sloane OA library. Similarly, the*L*_{84}(2^{14}6^{1}14^{1}) can be constructed by adding sequentially six 2-level columns to the*L*_{84}(2^{8}6^{1}14^{1}) in the Sloane OA library. - Examples 5-24: mixed level near-OAs. These examples correspond
to those in Table 2 of
Nguyen & Liu (2008). Some of the arrays in these examples were
contructed by adding additional columns to existing OAs. As an additional example, the
*L*'_{100}(10^{4}2^{4}3^{2}), for example, was constructed by adding sequentially two 3-level column to the OA*L*_{100}(10^{4}2^{4}). This new OA was in turn obtained by adding sequentially four 2-level columns to the*L*_{100}(10^{4}) in the Sloane OA library. Examples 13 and 14 provide two solutions to the*L*'_{12}(3^{2}2^{7}). In the second solution (example 14), the first two 3-level columns are orthogonal to the remaining columns. - Examples 25-28: 2-level saturated and supersaturated designs.
- Examples 29-32: mixed level supersaturated designs. Like the designs in Table 2 of
Koukouvinos & Mantas (2005), the ones in the
last two examples were constructed by adding additional columns to the
OAs
*L*_{16}(2^{15}) and*L*_{20}(2^{19}) respectively.

## References

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Cheng, C.S. (1997) *E*(*s*^{2})-optimal supersaturated
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Fang, K-T. , Lin, D. K. J. & Liu, M-Q. (2003) Optimal mixed-level
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