Box–Behnken Design
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In statistics, Box–Behnken designs are
experimental design The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associ ...
s for
response surface methodology In statistics, response surface methodology (RSM) explores the relationships between several explanatory variables and one or more response variables. The method was introduced by George E. P. Box and K. B. Wilson in 1951. The main idea of RSM ...
, devised by
George E. P. Box George Edward Pelham Box (18 October 1919 – 28 March 2013) was a British statistician, who worked in the areas of quality control, time-series analysis, design of experiments, and Bayesian inference. He has been called "one of the gr ...
and
Donald Behnken Donald is a masculine given name derived from the Gaelic name ''Dòmhnall''.. This comes from the Proto-Celtic *''Dumno-ualos'' ("world-ruler" or "world-wielder"). The final -''d'' in ''Donald'' is partly derived from a misinterpretation of the ...
in 1960, to achieve the following goals: * Each factor, or independent variable, is placed at one of three equally spaced values, usually coded as −1, 0, +1. (At least three levels are needed for the following goal.) * The design should be sufficient to fit a
quadratic model In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown (mathematics), unknown value, and , , and represent known numbers, where . (If and then the equati ...
, that is, one containing squared terms, products of two factors, linear terms and an intercept. * 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 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. (See "rotatability" in
Comparisons of response surface designs
.) Box-Behnken design is still considered to be more proficient and most powerful than other designs such as the three-level full factorial design, central composite design (CCD) and Doehlert design, despite its poor coverage of the corner of nonlinear design space. 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 In statistics, a full factorial experiment is an experiment whose design consists of two or more factors, each with discrete possible values or "levels", and whose experimental units take on all possible combinations of these levels across all ...
with an
incomplete block design In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of b ...
. 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. : The design for 8 factors was not in the original paper. Taking the 9 factor design, deleting one column and any resulting duplicate rows produces an 81 run design for 8 factors, while giving up some "rotatability" (see above). 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. Using Plackett–Burmans to construct a 16 factor design (see below) requires only 221 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 α = min(2, (int(1.5 + ''K''/4))1/2), for ''K'' factors, roughly to approximate original design points' distances from the centre. Plackett–Burman designs can be used, replacing the fractional factorial and incomplete block designs, to construct smaller or larger Box–Behnkens, in which case, axial points of length ''α'' = ((''K'' + 1)/2)1/2 better approximate original design points' distances from the centre. Since each column of the basic design has 50% 0s and 25% each +1s and −1s, multiplying each column, ''j'', by ''σ''(''X''''j'')·21/2 and adding ''μ''(''X''''j'') prior to experimentation, under a
general linear model The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regr ...
hypothesis, produces a "sample" of output Y with correct first and second moments of ''Y''.


References


Bibliography


George Box
, Donald Behnken, "Some new three level designs for the study of quantitative variables", Technometrics">George Box">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
handbook on engineering statistics
at
NIST The National Institute of Standards and Technology (NIST) is an agency of the United States Department of Commerce whose mission is to promote American innovation and industrial competitiveness. NIST's activities are organized into physical sc ...
{{DEFAULTSORT:Box-Behnken design Design of experiments