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Box-Behnken Designs

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A Box-Behnken design estimates curvature using three levels per factor and no runs at the extreme corners. Each run moves a small group of factors together and holds the rest at their middle setting. How many move together depends on the factor count, and it is not always two:

Factors Factors moved together per run
3, 4, 5 two
6, 7, 9 three
10 four

What this design gives you

Like a CCD it can fit quadratic terms, and unlike a CCD it never asks for a setting outside the low to high range you gave, because there are no axial points. Nothing in it is run at all factors low or all factors high.

Factor counts and run counts

Box-Behnken designs exist for seven factor counts only. There is no 8-factor design and none above 10, and the page offers only what exists. The first page of the builder is a single list of those seven, each entry written as its factor count and its run count, 3 Factors, 15 runs through 10 Factors, 170 runs. Asking for anything else gives Invalid number of factors for the Box-Behnken Design.

Factors Runs, including the default center points Default center points In-design block counts offered
3 15 3 none
4 27 3 3
5 46 6 2
6 54 6 2
7 62 6 2
9 130 10 5 or 10
10 170 10 2

The run counts include the default center points, so the 3-factor design is 12 design points plus 3 center points. Adding center points moves the total up with it.

Every factor in a Box-Behnken design is quantitative. Its factor page offers no Categorical column, so there is no way to enter a named-level factor.

Blocking

The last column of the table is the in-design block counts, the ones formed from the design itself. The blocking list can offer more than that. Once the replicate count is above 1, block-with-replicate entries are added for every divisor of it, so a 3-factor design with 2 replicates does offer 2 blocks even though its in-design list is empty.

Invalid number of blocks for the Box-Behnken Design. is raised in one case only: a 9-factor design whose block count is neither 5 nor 10. For the other factor counts the dialog simply does not offer an unsupported in-design count.

How the blocks are formed differs by factor count. For 4 and 5 factors the runs are divided into block groups. For 6, 7 and 10 factors the split is made on aliasing. The 9-factor design does both: 5 blocks divides it into block groups, and 10 blocks divides it into 5 groups first and then splits each group in two on aliasing.

Center points are distributed between the blocks rather than all landing in one. The split is even, by integer division, and any remainder goes to the last blocks. So a count below the number of blocks leaves the first blocks with no center point at all: three blocks and two center points puts one each in blocks 2 and 3 and none in block 1.

Every control

Every control on every page this builder shows for this design is on Options.

See Also