Second-order designs

The following sub-sections will describe some of the experimental designs that are commonly used to fit the first-order and second-order model. These designs will be called first-order and second-order designs, respectively. 

Some second-order designs, such as the uniform shell designs (Doehlert [28]), have been proposed which are not based on the central composite design. A more thorough treatment of additional second-order designs can be found in the texts mentioned earlier see Myers [11], Box and Draper [12], Khuri and Cornell [13]. 

A basic requirement in constructing designs of experiments is reduction of the number of experimental trials. Table 2.132 shows the total number of trials (N) for various second-order designs at a different number of factors. 

Table 2.132 Number of trials in second-order designs...

An increase in the number of replicated trials causes a decrease in reproducibility variance or experimental error as well as in the associated variances of regression coefficients. Design points-trials can be replicated in all points of the experiment or in some of them. An upgrade of the design of experiment may be realized by a shift from fractional to full factorial experiment, a switch to bigger replica (from 1/6 to 1 /2 replica), a switch to second-order design (when the optimum region is dose by), etc. 

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 ... 

D-optimality, B -designs and Hartley s Second-order Designs... 

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. 

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 ... 

Equation (2.171) obtains different forms depending on a specific experiment or design of experiments and number of replications. To replicate all trials of a design evenly (even number of replications) and for N0>1, we use Eq. (2.171). In the case of a rotatable second-order design, when trials are replicated in all points the same number of times, Eq. (2.171) becomes ... 

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

Rotatable second-order design Only in design center (2.168) and (2.130)... 

An experiment with composite rotatable second-order design, where trials have been replicated only in design center, is given in Example 2.45. Five factors and system response (N=32 k=5 n0=6 Nxn=N-(n0-l)=27) have been studied in the experiment. Outcomes are given in Table 2.140. By processing experimental outcomes, we have obtained the regression model (2.97). To check lack of fit of the obtained regression model, we used data from Table 2.1. 

In the next phase, there is a point in thinking about upgrading the basic design to a second-order design. 

These three solutions are possible 1) end of research 2) switch to second-order design and 3) upgrading half-replica to a full factorial experiment. 

The first solution is unacceptable since the difference between achieved and possible yields is large (20%). The second solution of upgrading half-replica to a second-order design is acceptable but requires a large number of additional trials. It is most reasonable to accept the third solution and upgrade half-replica to FUFE. 

The method of steepest ascent has proved to be successful since we obtained a yield of 88.0% in trial No. 5. The optimum area is dose by and we may switch to a second-order design. CCRD 22+2 2+5 has been realized, Table 2.199. 

Besides these three problems, one should also know how to switch from simplex to a second-order design that may describe the optimum area. This is the subject of sect. 2.5.4. The first problem in simplex optimization consists of constructing the matrix of a design of experiments for initial simplex where coordinates of experimental points-vertices are given. In solving this problem, different orientations of initial simplex to the coordinate system are possible. A simplex center is mostly set in the coordinate beginning, while the distance between simplex vertices (simplex sides) has a coded value of one. Simplex is, as a rule, oriented in a factor space in such a way that vertex l>k+I lies on the xk axis, while other vertices are distributed symmetrically with respect to coordinate axes. Simplexes of such a construction are shown in Figs. 2.50 and 2.51. 

Method of steepest ascent —> Second-order design... 

In solving the problem, the simplex lattice design 4.2 has been utilized. The second-order design matrix for the quaternary system and experimental results (each trial was repeated twice) are summarized in Table 3.24. By processing these outcomes, the following values of regression coefficients were obtained ... 

---