Determination of formulas on the orthogonal central composite design (Box)

Abstract

A theoretical study was devised to determine the formula of ʆ which makes orthogonal the central composite design (Box), with P + 1 central points and maximum of four factors. The following expressions of α was obtained:

α = [-2^k-1+ [2^2k-2 + 2^k-2 (2k + 1 + P)]^1/2]^1/2

it was equally verified that in the case of the orthogonal central composite design, with P + 1 central points, the formulas which estimated the variances of the estimates of the polynomial equation coefficients, could be calculated through the following expressions:

V (β_i) = 1/r (2^k+ 2α²) ^*α-2 V (β_ij) = 1/r2^{k *α-2}

V (β_ii) = 1/r * [2^k+ 2α⁴ - (2^k+ 2α2)²/2^k+ 2k + 1 + P]^-1*α-2

Finally, it was verified that when the orthogonal central composite design has eight central points, the coefficients β_ii of the squared terms are estimated as precise as the coefficients β_i and more precise than the coefficients β_ij.