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Math Details¶
This page gives how a Plackett Burman design is constructed: the stored first column each design is grown from, the cyclic shift that produces the rest of the columns, the final row, the two run counts that are stored whole instead, and how many factors each run count can carry.
Notation¶
| Term | Description |
|---|---|
| \(n\) | the number of runs in the design before center points |
| \(L\) | \(n - 1\), the length of the stored first column |
| \(k\) | the number of factors |
| \(c_i\) | entry \(i\) of the stored first column, \(i = 0, 1, \dots, L-1\) |
| \(X_{ij}\) | the coded entry of the design in row \(i\), column \(j\) |
| \(C\) | the number of center points |
The run counts on offer¶
The design is available at nine run counts:
The dialog offers those of them that exceed the number of factors, and no others. The factor limit is
so a 12 run design carries up to 11 factors and a 48 run design up to 47. Asking for \(k \ge n\) is refused. These are all multiples of four and none of them is a power of two, which is the point of the family: it fills in run counts that a two level factorial cannot reach.
Construction from the first column¶
Seven of the nine designs are grown from a single stored column. The product stores one vector of \(L\) entries of \(+1\) and \(-1\) per run count, and the whole design follows from it.
The first factor's column is the stored vector as it stands. Each following column is the previous one shifted down by one position, cyclically, so that the entry that falls off the bottom reappears at the top:
That fills \(L\) rows and as many columns as there are factors. A final row of \(-1\) in every column is then appended, which brings the row count to
For 12 runs the stored column is
Only \(k\) of the \(L\) available columns are generated. A design run with fewer factors than \(n-1\) simply stops after \(k\) shifts, and the columns that would have followed are never formed.
The two designs stored whole¶
28 runs and 40 runs are not grown from a first column. Neither has a single generating vector that produces it by cyclic shifting, so both are stored as complete matrices and written out directly. The result is the same kind of design and carries the same factor limit; only the construction differs.
Center points¶
Center points are appended after the \(n\) design rows, \(C\) of them, with every factor at coded \(0\). They add to the row count directly:
A center point here is always a single row. It does not multiply the way it does in a two level factorial holding categorical factors.
Coding and the levels each factor takes¶
Every entry of the design is \(+1\) or \(-1\), and a center point row is \(0\), so a factor takes two levels in the design proper and three when center points are present. Each is converted to the value written on the design sheet by the design's own coding.
Every quantitative factor is put on a common scale before the regression runs. One expression does it, whichever coding is in force, using a centre \(m_c\) and a scale \(s_c\) that the coding chooses:
A categorical factor is never coded this way. Its coded value is \(1\), and the level it sits at selects which coefficient is used instead.
An interaction's coded value is the product of the coded values of the factors in it.
What the aliasing report finds¶
The aliasing report for any design is built by comparing columns, as Aliasing and Resolution describes: two terms are placed on the same alias line when their columns agree entry by entry, or agree after one is negated.
That comparison tests exact equality, and the confounding a Plackett Burman design carries is not exact. A main effect column of one of these designs is not equal to any single interaction column, so the comparison does not pair it with one. What the design carries instead is partial confounding spread across many interactions at once, which is a different relationship from the exact column identity a fractional factorial's generator creates, and it is not what the alias table is built to report.
The practical consequence is that the aliasing block of a Plackett Burman design sheet does not read like that of a fractional factorial of similar size, and the absence of alias lines is a statement about exact column matches rather than a statement that the design separates every effect.
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 | Aliasing and Resolution |
| Design coding | turning \(-1\), \(0\) and \(+1\) into the values written on the design sheet | Design Coding |
See Also¶
- Plackett Burman Design
- Two Level Factorial Designs Math Details, the two level designs at power of two run counts
- Taguchi Designs Math Details, which includes arrays at 12, 18 and 36 runs
References¶
- Plackett, R. L. and Burman, J. P. (1946). The Design of Optimum Multifactorial Experiments. Biometrika, 33(4), 305 to 325.
- Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.