Orthogonal second-order design

Orthogonal Second-order Design (Box-Benken Design)... 

This example refers to response dependence on two factors (k=2). Orthogonal second-order design in this case, according to Table 2.164, has nine design points (N=9). The design matrix with outcomes of design points is shown in Table 2.165. The same case has been elaborated in the previous section, in Example 2.43, by application of rotatable second-order design. However, the connection between coded and real values of factors for the same null point is now different ... 

To process data obtained by application of orthogonal second-order designs, regression coefficient significances are checked by expression (2.144), along with previous calculations of variance or error in determining regression coefficients ... 

When nc=0 or n0=l, as is the case in orthogonal second-order designs, B4 and Hartley s designs, Eq. (2.157) becomes simpler ... 

Orthogonal second-order design Equal replication (2.170) and (2.128)... 

The experimental error is by analogy determined also for second-order designs with no replications in null point or experimental center (n(l=0 or n(l=1), for orthogonal B4, Hartley s designs, etc. 

Central composite orthogonal designs Quantitative Regression models of second order... 

Rotatable designs are most efficient for k=3. Rotatable designs of second order are not orthogonal and they do not minimize the variance of estimates of regression coefficients. They are efficient in solving research problems when trying to find an optimum. 

The second-order orthogonal design for k=3, is shown in Table 2.166. Based on design-point outcomes, we can calculate regression coefficients ... 

Kono s design matrix corresponds to second-order orthogonal design matrices so that in this case experimental outcomes from Example 2.51, Table 2.165, were used. The value of regression coefficient b0 is determined by adding coefficient yields of the associated column to the response column ... 

The three-factor and five-level second-order regression for rotation orthogonal composite design is consisting of 20 experimental runs, which was employed at the center point, including 6 replicates. The experimental design with observed endoinulinase activity is shown in Tables 3,6. 

In general, if an /-level factorial design is of interest, then the orthogonal basis for this factorial design will be formed by the set of polynomials from 0 to / - 1, evaluated at each of the treatment levels. The zeroth-order polynomial, Lq, will be used for the zero-order interaction (/ o)- The first-order polynomial, Lj, will be used for the factors whose powers are 1. Similarly, the second-order polynomials, L2, will be used for the factors whose powers are 2. The results can be obtained by solving Eqs. (4.35) and (4.36). In the following sections, the results for 1 = 2, 3, and 4 will be provided, as well as generalised orthonormal polynomials for 1 = 2 and 3. Examples will be provided as appropriate. 

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