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Math Details¶
This page gives what a Taguchi design is built from: the ten arrays the product carries, how many factors and levels each one supports, which of them are stored as arrays and which are generated from a two level factorial, and how factors are assigned to array columns.
Notation¶
| Term | Description |
|---|---|
| \(n\) | the number of runs, the \(L\) number of the array |
| \(k\) | the number of factors placed in the array |
| \(R\) | the number of replicates |
The ten arrays¶
The product carries ten arrays. A mixed array holds factors at two levels and factors at three levels in the same design, in two contiguous groups of columns; the others hold one level count throughout.
| Array | Runs | Levels | Factors | Mixed |
|---|---|---|---|---|
| L4 | 4 | 2 | 2 to 3 | no |
| L8 | 8 | 2 | 2 to 7 | no |
| L9 | 9 | 3 | 2 to 4 | no |
| L12 | 12 | 2 | 2 to 11 | no |
| L16 | 16 | 2 | 2 to 15 | no |
| L18 | 18 | one at 2, up to seven at 3 | 2 to 8 | yes |
| L27 | 27 | 3 | 2 to 13 | no |
| L32 | 32 | 2 | 2 to 31 | no |
| L36 | 36 | up to eleven at 2, up to twelve at 3 | 3 to 23 | yes |
| L36 | 36 | up to three at 2, exactly thirteen at 3 | 14 to 16 | yes |
| L54 | 54 | one at 2, three to twenty five at 3 | 4 to 26 | yes |
Two entries in that table need reading carefully.
There are two different L36 arrays. They have the same run count and are not interchangeable. The first splits its columns as eleven two level and twelve three level; the second as three two level and thirteen three level. Their factor ranges barely overlap, so in practice the factor counts asked for select between them.
The second L36 requires exactly thirteen three level factors, not up to thirteen. Its three level part has a minimum equal to its maximum, so its total factor count runs from 14 to 16: thirteen three level factors, plus one, two or three at two levels.
For each mixed array the product also records the column index at which the three level part begins, which is what divides the array's columns into the two groups: column 1 for L18 and L54, column 11 for the first L36, and column 3 for the second.
Stored arrays and generated arrays¶
Four of the ten are generated rather than stored. L4, L8, L16 and L32 are the arrays whose run counts are powers of two, and each is produced from a two level full factorial on \(\log_2 n\) factors: the factorial's own columns and the columns of its interactions together make up the array's columns. So an L8 is a \(2^3\) full factorial with its three main effect columns and its four interaction columns, seven in all, which is the array's stated factor limit.
Because those four are generated this way, the product also holds a description of what each of their columns is: a main effect, or the particular interaction whose column it is. The other six arrays are stored as arrays and carry no such description.
The remaining six, L9, L12, L18, L27 and the two L36 arrays together with L54, are stored directly.
Assigning factors to columns¶
An array has a fixed number of columns and the design may use fewer. Which columns the factors occupy is a choice, and the product supplies a default.
L9, L27 and L12 take the leading columns in order: factor 1 to column 1, factor 2 to column 2, and so on. The three level arrays use that rule always, and L12 is treated the same way despite being a two level array.
On a mixed array the two groups are filled independently, the two level factors from the start of the two level group and the three level factors from the start of the three level group. So on the first L36 the two level factors take columns 1 onward and the three level factors take columns 12 onward, column 12 being where that array's three level part begins.
On the four generated arrays the default is built in two stages, because those arrays' columns are main effects and interactions of the underlying factorial rather than interchangeable slots.
The main effect columns are taken first. They sit at the positions
which is columns 1, 2, 4, 8 and 16, and there are \(\log_2 n\) of them: two for L4, three for L8, four for L16 and five for L32.
If the design needs exactly one factor more than that, the last column of the array is taken, which is the interaction of all the underlying factors. Otherwise interaction columns are appended by rules that depend on the array and the factor count:
| Array | Factors | Columns appended after the main effects |
|---|---|---|
| L4 | up to 3 | none needed |
| L8 | 5 or more | the ordered list, columns 1, 2, 3 and onward |
| L16 | 6, 7 or 8 | two, three or four three factor interaction columns |
| L16 | more than 8 | the ordered list |
| L32 | 7 to 10 | two to five four factor interaction columns |
| L32 | 11 to 15 | six to ten three factor interaction columns |
| L32 | 16 | the first ten three factor interactions and one five factor interaction |
| L32 | 17 or more | the ordered list |
The dialog can override the default and place any factor in any column of the appropriate group.
Replicates¶
Replicating joins whole copies of the array end to end, so replicate 2 repeats the same \(n\) rows in the same order, and the design holds \(n \cdot R\) rows. There are no center points in this family.
Aliasing¶
The aliasing report is built by comparing columns, as Aliasing and Resolution describes. Two features of these arrays interact with it.
The alias search has a lower ceiling on the largest arrays. Products of factor columns are formed up to seven factors when the design has fifteen factors or fewer, and only up to three factors when it has more than fifteen. Four arrays can carry more than fifteen factors and so can fall on the second side of that line: L32 above fifteen, both L36 arrays above fifteen, and L54 above fifteen. For those the deepest pairing the report can find is a main effect against a three factor interaction. The second L36 reaches sixteen factors at most, so only its largest case is affected.
On a generated array the column assignment determines the aliases. A factor placed in a column that is an interaction of the underlying factorial is aliased with that interaction by construction, which is why the column choice on L8, L16 and L32 is a design decision and not a presentation one.
Shared Math Details used here¶
This design uses shared formulas defined once in Shared Math Details. See those pages for the exact definitions.
| Shared concept | Used here for | Reference |
|---|---|---|
| Aliasing and resolution | the column comparison the aliasing report is built from, and its size ceiling | Aliasing and Resolution |
| Design coding | turning a coded level into the value written on the design sheet | Design Coding |
See Also¶
- Taguchi Designs
- Two Level Factorial Designs Math Details, the engine L4, L8, L16 and L32 are generated from
- Plackett Burman Math Details, two level designs at run counts that are multiples of four
References¶
- Taguchi, G. (1987). System of Experimental Design: Engineering Methods to Optimize Quality and Minimize Costs. UNIPUB/Kraus International Publications.
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.