Each factor, or independent variable, is placed at one of three equally spaced values, usually coded as −1, 0, +1.
The design should be sufficient to fit a quadratic model, 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.
The estimation variance should more or less depend only on the distance from the centre, and should not vary too much inside the smallest 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 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 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". Designs for other numbers of factors have also been invented. A design for 16 factors exists having only 256 factorial points. Using Plackett–Burmans to construct a 16 factor design requires only 221 points. Most of these designs can be split into groups, 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, for K factors, roughly to approximate original design points' distances from the centre. Plackett–Burman designs can be used to construct smaller or larger Box–Behnkens, in which case, axial points of length α = 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·21/2 and adding μ prior to experimentation, under a general linear model hypothesis, produces a "sample" of output Y with correct first and second moments of Y.
