# The partial aliasing patterns for Taguchi's L12 orthogonal array.

Introduction

In Taguchi's experiment, the use of the [L.sub.12] orthogonal array has been highly recommended and many successful cases have been reported. However, its partial aliasing patterns (PAP) are rather complicated. Taguchi[1] stated that "the interaction components for two given columns are confounded with the remaining nine columns" and that "this array should therefore not be used for experiments requiring interactions". Many books, such as those by Montgomery[2], Phadke[3], Ross[4] and Taguchi et al.[5], do not discuss the PAP for the [L.sub.12] orthogonal array. The purpose of this article is to propose a method for computing the PAP of an interaction for the [L.sub.12] orthogonal array.

The computation of partial aliasing patterns

Although Taguchi[1] provided the [L.sub.12]([2.sup.11]) orthogonal array with symbols 1 and 2, for the convenience of this presentation, Table I gives the [L.sub.12] orthogonal array with symbol 2 being replaced by -1. The first column of all 1s, denoted by I, has been added for analysis; and the remaining columns are denoted by factors A-K. An interactive column is equal to the product of the corresponding columns of factors, row by row. For example, an interactive column (of AB interaction) is listed last in Table I. An inner product (IP) of two (or more) vectors is defined as the sum for all products of elements with the same position in vectors[6]. For example, let a = ([a.sub.1], [a.sub.2], ..., [a.sub.k]), b = ([b.sub.1], [b.sub.2], ..., [b.sub.k]), and c = ([c.sub.1], [c.sub.2], ..., [c.sub.k]); so that the IP of a, b, and c is equal to

(a [absolute value of] b c) = [summation of][a.sub.i][b.sub.i][c.sub.i] where i=1 to k.

The partial alias (PA) of an interaction with a certain factor is defined as the IP of the interactive column and the column of that factor divided by 12. For example, in the last row of Table I, PA(AB,C) = 1/3 indicates that 1/3 of the effect of AB will be confounded with the main effect of C. The PAP of an interaction is defined by its PAs with all 12 columns (including column I). For example, the [TABULAR DATA FOR TABLE I OMITTED] PAP of the AB interaction is shown in the last row of Table I. Note that the sum of all 12 PAs for an interaction is equal to 1 (Sum = 1 in Table I).

For a quick computation, the PA of an interaction with a certain factor is equal to the IP of all corresponding columns of factors divided by 12. For example, PA(C, AB) = IP(ABC)/12 = 1/3. By this definition, it is easily to show that PA(C, AB) = PA(A, BC) = PA(B, AC) = PA(ABC, I) = IP(ABC)/12.

PAP for two-factor interaction

Since 11 factors could be assigned to the [L.sub.12] orthogonal array, there are 45 two-factor interactions of interest. The results of the PAPs for all two-factor interactions are shown in Table II. For convenience, suppose the effect of the AB interaction has three units, then one unit of effect will be confounded with C, D, E, F, G, and H, and -1 unit of effect will be confounded with I, J, and K, respectively. In other words, assume that all higher-order interactions do not exist, so that the effect of column C is equal to C + AB/3, or the effect of column K is equal to K - AB/3. The SAS program for computing the PAP of an interaction is listed in the Appendix.

Table II has several interesting properties:

* The interaction of any two columns is confounded partially with the remaining nine columns. However, the effects of partial aliases are not uniformly distributed, the patterns can be read from Table II. Suppose the effect of a two-factor interaction has three units, then there are six partial aliases with one unit, and three [TABULAR DATA FOR TABLE II OMITTED] partial aliases with -1 unit, and the total effect of all partial aliases remains the same as the original effect, which has three units.

* Since the IP of any two columns is equal to 0 - for example, IP(AB) = IP(JK) = IP (AK) = 0, a two-factor interaction is orthogonal to the column of I.

* Since the sum of all elements in a single column is equal to zero, and [A.sup.2] = I, etc., then PA(A, AB) = IP(BI)/12 = 0. This shows that a two-factor interaction is orthogonal to the column of its components.

* Excepting zero in Table II, a PA is equal to either 1 or -1. This fact shows that the IP for three-factor interaction is equal to either 4 or -4.

PAP for higher factor interactions

In this section, three-factor and higher-order interactions will be studied further. First, we observe that the interaction of ABCDEFGHIJK is complete by being confounded with I, which implies that I = ABCDEFGHIJK. From this fact, it is clear that any interaction has its own complementary interaction. For example, AB = CDEFGHIJK, or ABCD = EFGHIJK, etc. Moreover, if all IPs for three-factor, four-factor and five-factor interactions are known, then the PAP for any order of interaction can easily be obtained. There are 330 four-factor interactions. Table III lists 110 of those whose IPs are equal to -4, and all the others are equal to +4. There are 462 five-factor interactions. Table IV lists 55 of those whose IPs are equal to +8, and 11 whose IPs are equal to -8, and all the others are equal to zero.

Now, the PAP of a three-factor interaction can be read from Table III. For example, IP(ABCA) = IP(BC) = 0; IP(ABCD) = 4; IP(ABCF) = -4; IP(ABCK) = 4. Now, suppose that the effect of ABC interaction has three units, then PA is equal to IP divided by four, and the result of PAP for ABC interaction is shown in the first row of Table V.
```Table III
```

```List of four-factor interactions whose inner products are equal to -4
```

```ABCF    ABHJ    ADEJ    AFHK    BCEI    BEFG    CFGH    DEGI    DHJK
ABCG    ACDF    ADEK    AFIK    BCFK    BEHI    CFGJ    DEHJ    EFGH
ABCJ    ACDI    ADFG    AGHI    BDFK    BFJK    CFHI    DFGH    EFGI
ABDG    ACDK    ADHK    AGHK    BDGH    BFGJ    CFIK    DFGK    EFHK
ABDH    ACEH    ADIJ    AGJK    BDIJ    BFGK    CFJK    DFHJ    EFIJ
ABDI    ACEI    AEFJ    AHIJ    BDEF    BFHI    CGHK    DFIJ    EFJK
ABEF    ACEJ    AEGH    AIJK    BDEI    BFHK    CGIJ    DFIK    EGHJ
ABEH    ACFH    AEIK    BCDH    BDEJ    BFIJ    CGIK    DGHI    EGIK
ABEK    ACGI    AFGI    BCDJ    BDFH    BGHI    CHIJ    DGIJ    EGJK
ABFI    ACJK    AFGJ    BCDK    BDGJ    BGHJ    CHJK    DGJK    EHIJ
ABGK    ADEG    AFHJ    BCEG    BDIK    BGKI    DEFK    DHIK    EHIK
Table IV
```

```List of five-factor interactions
```

```                     IP = +8                          IP = -8
```

```ABCFK      ACDHI      AFGJK      BDEGI      CEFJK      ABCDE
ABCGI      ACDJK      AFHIK      BDEJK      CFGHJ      ABFGH
ABCHJ      ACEFH      AGHIJ      BDFGK      CFGIK      ADEFI
ABDFI      ACECJ      BCDFJ      BEGHJ      CHIJK      ADGIK
ABDGJ      ACEIK      BCDGH      BFGIJ      DEHIK      AEHJK
ABDHK      ADGHK      BCDIK      BFHJK      DFGHI      BCGJK
ABEFJ      ACEFK      BCEFG      BGHIK      DFIJK      BDHIJ
ABEGK      ACEGH      BCEHK      CDEFI      DGHJK      BEFGK
ABEHI      ACEIJ      BCEIJ      CDEGK      EFGHK      CDFGK
ABIJK      ADFHJ      BCFHI      CDEHJ      EFHIJ      CEGHI
ACDFG      AEFGI      BDEFH      CDGIJ      EGIJK      DEFGJ
```

[TABULAR DATA FOR TABLE V OMITTED]

If the four-factor interaction of ABCD is of interest, the PAP can be obtained as follows: from Table II, IP(ABCDA) = IP(BCD), IP(ACD), IP(ABD),and IP(ABC) are all equal to four; from Table IV, IP(ABCDE) = -8, since ABCDF, ABCDG, 0ABCDH, ABCDI, ABCDJ, and ABCDK are not in the list, their IPs are equal to zero. Since the effect of interaction is assumed to have three units, all IPs are divided by four and the result of PAP is shown in the third row of Table V.

If the six-factor interaction of ABCDEF is of interest, the PAP can be obtained as follows: (for example), from Table IV, IP(ABCDEFI) = IP(GHIJK) = 0, IP(ABCDEFA) = IP(BCDEF) = 0, etc; from Table III, IP(ABCDEFG) = IP(HIJK) = 4, etc. All the other IPs can be computed in the same manner, and the result is shown in the seventh row of Table V. In Table V, some other interactions are arbitrarily selected and their PAPs are listed for reference.

Findings and discussions

Six findings of the preceding analysis can be listed:

(1) The IP values for one-factor and two-factor interactions is zero; the value of IP takes on either a +4 or a -4 value for three-factor (Table II) as well as four-factor (Table III) interactions, and takes on +8, 0, or -8 for five-factor interactions (Table IV).

(2) In Tables II and V, all entries in the last column are equal to three, hence the total effect of the PAP for any order of interaction remains the same as the original effect.

(3) A two-factor interaction is orthogonal to the I column, while a higher-order interaction may partially confound with column I.

(4) A two-factor interaction is orthogonal to the column of its components while a higher-order interaction may partially confound with the column of its components. For example, for the AB interaction in Table II, both entries are zero under columns A and B; on the other hand, for the interaction ABCD in Table V, all entries are 1 for columns of A, B, C and D.

(5) The partial alias of two interactions can be computed in the same manner. For example, from Table III, PA(ABC, CDEFJK) = IP(ABDEFJK)/12 = IP(CGHI)/12 = 4/12 = 1/3.

(6) From each IP, we may derive many PAs. For example, from Table II, we have IP(CHK) = 4, then

1/3 = PA(ACHK, A) = PA(BCHK, B) = PA(HK, C) = PA(CDHK, D) = PA(CEHK, E) = PA(CFHK, F) = PA(CGHK, G) = PA(CK, H) = PA(CIHK, I) = PA(CJHK, J) = PA(CH, K) = PA(ACH, AK) = PA(ACK, AH) = PA(AHK, AC) = PA(ABCHK, AB) = PA(AHK, AC) = PA(ACDHK, AD) = PA(ACEHK, AE) = PA(ACFHK, AF) = PA(ACGHK, AG) = PA(ACK, AH) = PA(ACIHK, AI) = PA(ACJI-IK,AJ) = PA(ACH, AK)...

Conclusions

Although Taguchi's [L.sub.12] orthogonal array has complex partial aliasing patterns, it is highly recommended and many successful cases have been reported. In this article, a method based on the inner product (IP) of vectors for computing the PAP of an interaction for the [L.sub.12] orthogonal array was proposed. The partial alias of an interaction with a certain factor is equal to the IP of all corresponding columns of factors divided by 12. For example, PA(ABC, D) = IP(ABCD)/12. The PAP of an interaction is defined as its PA with 12 columns (I plus A - K factors). Table II gives a complete list of the PAPs for all two-factor interactions. Because I = ABCDEFGHIJK, the IP value for one-factor and two-factor interactions is zero, the IP values for three-factor interactions and four-factor interactions are +4 or-4; and for five-factor interactions the IP values are +8, 0, or -8; and we can easily obtain the PAP for any order of interaction. With the same method, we can study the PAPs of Plackett-Burman designs[7] for n = 12, 20, 24, 28, etc.

References

1. Taguchi, G., Introduction to Quality Engineering, Asian Productivity Organization, Tokyo, 1986, p. 182.

2. Montgomery, D.C., Design and Analysis of Experiments, 2nd ed., John Wiley & Sons, New York, NY, 1984.

3. Phadke, M.S., Quality Engineering Using Robust Design, Prentice-Hall, London, 1989.

4. Ross, P.J., Taguchi Techniques for Quality Engineering, McGraw-Hill, New York, NY, 1989.

5. Taguchi, G., Elsayed, E.A. and Hsiang, T.C., Quality Engineering in Production Systems, McGraw-Hill, New York, NY, 1989.

6. Hoffman, K. and Kunze, R., Linear Algebra, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1971.

7. Plackett, R.L. and Burman, J.P., "The design of optimum multifactorial experiments", Biometrika, Vol. 33, 1946, pp. 305-25.

Appendix: The SAS program for computing the partial aliasing pattern of an interaction

data 112; input x1-x12; int=x1*x2; array x{12} x1-x12; array a{12} a1-a12; do i=1 to 12; a{i}=x{i}*int; end; cards;
```1     1     1     1     1     1     1     1     1     1     1     1
1     1     1     1     1     1    -1    -1    -1    -1    -1    -1
1     1     1    -1    -1    -1     1     1     1    -1    -1    -1
1     1    -1     1    -1    -1     1    -1    -1     1     1    -1
1     1    -1    -1     1    -1    -1     1    -1     1    -1     1
1     1    -1    -1    -1     1    -1    -1     1    -1     1     1
1    -1     1    -1    -1     1     1    -1    -1     1    -1     1
1    -1     1    -1     1    -1    -1    -1     1     1     1    -1
1    -1     1     1    -1    -1    -1     1    -1    -1     1     1
1    -1    -1    -1     1     1     1     1    -1    -1     1    -1
1    -1    -1     1    -1     1    -1     1     1     1    -1    -1
1    -1    -1     1     1    -1     1    -1     1    -1    -1     1
```

proc means; var a1-a12; run;

(Hsien-Tang Tsai is Professor of Business Management at the National Sun Yat-sen University, Kaohsiung, Taiwan, ROC.)
COPYRIGHT 1995 Emerald Group Publishing, Ltd.
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