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Box-Behnken Designs

QXL DOE New > Create Design > Modeling > Create Box-Behnken Design

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. 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.

The page sequence

Select the Box-Behnken design > Enter Factor Names and Coding > Define outputs (responses) > Enter number of replicates and number of blocks > Center points.

Center points asks for one number. Its label depends on the design type and the replicate format, not on how many replicates or blocks the design has:

Design The label
2-level factorial, Stacked format Number of center points per block:
2-level factorial, Table format Number of center points per replicate:
every other design type Extra number of center points:

Below the box are two total lines, and both update as you type. With a single replicate and a single block they read Total number of center points: and Total number of runs:. Once the design is replicated or blocked they show the arithmetic instead, as Total center points: n replicates * m center points = t (or n blocks * ... in the stacked multi-block case) and Total runs: n replicates * m = t.

On this design the box is for EXTRA center points, and it starts at 0. Its label reads Extra number of center points:, and what you type is added to the default count in the table above. Typing 6 on a 3-factor design gives 9 center points and 21 runs, not 6 and 18. This differs from the CCD, whose Axial and center points page holds the total rather than an addition.

The design sheet records the block count followed by the word blocks.

See Also