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Math Details

This page gives how a Box Behnken design is constructed: the groups of factors it varies together, how many rows each group contributes, the center points, and the run count that follows. It also records which factor counts the product supports and which it does not.

Notation

Term Description
\(k\) the number of factors
\(G\) the number of factor groups the design is built from
\(m\) the number of factors varied together within one group
\(C\) the number of center points
\(N\) the number of rows in the design

How the rows are formed

The design is built from a stored list of factor groups. Each group names \(m\) of the \(k\) factors. For one group, the design writes out every combination of \(+1\) and \(-1\) on those \(m\) factors, which is \(2^m\) rows, and holds every factor outside the group at \(0\) in all of them.

So for three factors the first group names factors 1 and 2, and contributes the four rows

\[ (-1, -1, 0), \quad (-1, +1, 0), \quad (+1, -1, 0), \quad (+1, +1, 0) \]

with the third factor held at its middle level throughout. The remaining groups do the same for the other pairs.

Every row produced this way is tagged as an edge centroid, which is what it is: a point at the middle of an edge or a face of the cube rather than at a corner. No run of a Box Behnken design has every factor at an extreme, because every row holds at least \(k - m\) of the factors at \(0\). That is the structural difference between this design and a central composite design, whose factorial portion visits the corners and whose axial portion goes beyond them.

Each factor takes exactly three coded values, \(-1\), \(0\) and \(+1\), so no run asks for a setting outside the low and high typed into the dialog.

The order the \(2^m\) combinations appear in follows the same run order setting the two level designs use, and changes the sequence of the rows and nothing else.

The groups, and the run count that follows

The group list is stored per factor count, not derived. Each group contributes \(2^m\) rows, all groups of one design share the same \(m\), and the center points are added after them, so an unblocked design holds

\[ N = G \cdot 2^{m} + C \]

That formula and the run counts below are for the unblocked design. A design blocked in the design distributes its center points across the blocks, and the count it arrives at is not given here.

The table below is the stored group list for every supported factor count, with the run count each one produces:

Factors \(k\) Groups \(G\) Factors per group \(m\) Rows per group \(2^m\) Edge centroid rows Default \(C\) Runs \(N\)
3 3 2 4 12 3 15
4 6 2 4 24 3 27
5 10 2 4 40 6 46
6 6 3 8 48 6 54
7 7 3 8 56 6 62
9 15 3 8 120 10 130
10 10 4 16 160 10 170

Two things in that table are worth reading off directly.

The number of factors varied together is not always two. The groups are pairs at 3, 4 and 5 factors, triples at 6, 7 and 9, and quadruples at 10. The rows per group follow, so a 10 factor design contributes 16 rows per group where a 5 factor design contributes 4.

Eight factors is not supported. The supported counts are 3, 4, 5, 6, 7, 9 and 10. There is no stored group list for 8, and asking for it is refused rather than approximated.

The default number of center points

The default is read from the factor count alone:

\[ C = \begin{cases} 3, & k < 5 \\ 6, & 5 \le k < 8 \\ 10, & k \ge 8 \end{cases} \]

which gives the Default \(C\) column above. The dialog can be given a different number, and the run count changes with it by the formula above.

Every factor of a Box Behnken design has a middle level by construction, so a center point is always a single row here. It does not multiply the way it does in a two level design holding categorical factors.

Blocking

The block counts available are stored per factor count and are not derived:

Factors Block counts offered
3 none
4 3
5, 6, 7, 10 2
9 5 or 10

A three factor design cannot be blocked. Any factor count outside the supported list is refused before the question of blocking arises.

Blocking in the design divides the edge centroid rows among the blocks and distributes the center points across them, so the total row count of a blocked design is not the \(N\) given above. The run counts in the table are for the unblocked design.

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
Design coding turning \(-1\), \(0\) and \(+1\) into the values written on the design sheet Design Coding

See Also

References

  1. Box, G. E. P. and Behnken, D. W. (1960). Some New Three Level Designs for the Study of Quantitative Variables. Technometrics, 2(4), 455 to 475.
  2. Montgomery, D. C. (2013). Design and Analysis of Experiments, 8th edition. Wiley.
  3. Myers, R. H., Montgomery, D. C., and Anderson-Cook, C. M. (2016). Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 4th edition. Wiley.