Box–Behnken design

In statistics, Box–Behnken designs are experimental designs for response surface methodology, devised by George E. P. Box and Donald Behnken in 1960, to achieve the following goals:

• Each factor, or independent variable, is placed at one of three equally spaced values. (At least three levels are needed for the following goal.)
• The design should be sufficient to fit a quadratic model, that is, one containing squared terms and products of two factors.
• The ratio of the number of experimental points to the number of coefficients in the quadratic model should be reasonable (in fact, their designs kept it in the range of 1.5 to 2.6).
• The estimation variance should more or less depend only on the distance from the centre (this is achieved exactly for the designs with 4 and 7 factors), and should not vary too much inside the smallest (hyper)cube containing the experimental points.

The design with 7 factors was found first while looking for a design having the desired property concerning estimation variance, and then similar designs were found for other numbers of factors.

Each design can be thought of as a combination of a two-level (full or fractional) factorial design with an incomplete block design. In each block, a certain number of factors are put through all combinations for the factorial design, while the other factors are kept at the central values. For instance, the Box–Behnken design for 3 factors involves three blocks, in each of which 2 factors are varied through the 4 possible combinations of high and low. It is necessary to include centre points as well (in which all factors are at their central values).

In this table, m represents the number of factors which are varied in each of the blocks.

factors	m	no. of blocks	factorial pts. per block	total with 1 centre point	typical total with extra centre points	no. of coefficients in quadratic model
3	2	3	4	13	15, 17	10
4	2	6	4	25	27, 29	15
5	2	10	4	41	46	21
6	3	6	8	49	54	28
7	3	7	8	57	62	36
8	4	14	8	113	120	45
9	3	15	8	121	130	55
10	4	10	16	161	170	66
11	5	11	16	177	188	78
12	4	12	16	193	204	91
16	4	24	16	385	396	153

The design for 8 factors was not in the original paper. Designs for other numbers of factors have also been invented (at least up to 21). A design for 16 factors exists having only 256 factorial points.

Most of these designs can be split into groups (blocks), for each of which the model will have a different constant term, in such a way that the block constants will be uncorrelated with the other coefficients.

Extended uses

These designs can be augmented with positive and negative "axial points", as in Central composite designs, but, in this case, to estimate univariate cubic and quartic effects, with length "alpha" = min(2, (int(1.5 + K/4))^1/2), for K factors, to approximate original design points' distance from the centre.

See also

• Extending Plackett–Burman designs to construct smaller or larger Box–Behnkens in which case, axial points of length alpha = (K+1)/2)^1/2 better approximate original design points' distance from the centre.

References

• George Box, Donald Behnken, "Some new three level designs for the study of quantitative variables", Technometrics, Volume 2, pages 455–475, 1960.
• Box–Behnken designs from a handbook on engineering statistics at NIST