21.4.3.6. Three-Level Orthogonal Design
Three-Level orthogonal design is a fractional factorial of \({{3}^{k}}\). For \(p<k\), the \({{3}^{k-p}}\) fractional design first generates \({{3}^{k-p}}\) full factorials for \(k-p\) factors. Then, the remaining \(p\) columns are determined from the pre-generated \(k-p\) columns. To do these, a defining relation is introduced as \(I=A{{B}^{{{\alpha }_{2}}}}{{C}^{{{\alpha }_{3}}}}...{{\left( k-p \right)}^{{{\alpha }_{k-p}}}}\). The component of the \({{k}_{th}}\) column is determined as
\({{x}_{k}}=\sum\limits_{i=1}^{k-1}{{{\beta }_{i}}{{x}_{i}}}\left( \bmod 3 \right)\) where, \({{\beta }_{i}}=\left( 3-{{\alpha }_{k}} \right){{\alpha }_{i}}\text{ }\left( \bmod 3 \right)\text{ for }1\le i\le k-1\).
As an example, we generate \({{3}^{4-2}}\) fractional factorial design with \(I=A{{B}^{2}}C\) and \(I=BCD\). From the rules, the 3rd and 4th columns are \({{x}_{3}}=2{{x}_{1}}+{{x}_{2}}\text{ }\left( \text{mod3} \right)\) and \({{x}_{4}}=2{{x}_{2}}+2{{x}_{3}}\text{ }\left( \text{mod3} \right)\). These results are listed side by side in Table 21.7.
A |
B |
C |
D |
|
1 |
0 |
0 |
0 |
0 |
2 |
1 |
0 |
2 |
1 |
3 |
2 |
0 |
1 |
2 |
4 |
0 |
1 |
1 |
1 |
5 |
1 |
1 |
0 |
2 |
6 |
2 |
1 |
2 |
0 |
7 |
0 |
2 |
2 |
2 |
8 |
1 |
2 |
1 |
0 |
9 |
2 |
2 |
0 |
1 |
The above factorial design of \({{3}^{4-2}}\) is only one of the \({{3}^{2}}\) fractions of \({{3}^{4}}\) full factorial designs. Among those \({{3}^{2}}\) fractions, only the cases that are orthogonal between columns are the three-level orthogonal array for \(k=4\). Hence, the defining relation for \(p\) factors is very important to maintain the orthogonal characteristics in the columns.
We support the automatic generator for the three-level orthogonal array design. If one define the number of factors \(\left( k \right)\), the RecurDyn/AutoDesign automatically generates the \({{3}^{k-p}}\) fractional factorials, where \(p\) is internally determined as possible as minimize the trials.
Trials |
Number of Factors |
Remarks |
9 |
2 - 4 |
\(p=2\) |
18 |
5 - 8 |
Addelman’s design (1961) |
27 |
9 - 13 |
\(p=6~10\) |
54 |
14 - 25 |
Addelman’s design (1961) |
81 |
26 - 40 |
\(p=22~36\) |
243 |
41 - 121 |
\(p=36~116\) |
729 |
122 - 364 |
\(p=116~358\) |
2187 |
365 - 1050 |
\(p=358~1043\) |
Reference
Peter W.M. John, 1998, Statistical Design and Analysis of Experiments, SIAM, Philadelphia.
Douglas C Montgomery, 2000, Design and Analysis of Experiments, John Wiley & Sons, New York.