Experimental three-level designs

Apart from good statistical properties of the central composite design, there is one experimental disadvantage. Because the star points are outside the hypercube, the number of levels that have to be adjusted for every factor is actually five instead of the three in a conventional three-level design. If the adjustment of levels is difficult to achieve, an alternative response surface design would be the design introduced by Box and Behnken. 

The absorption spectra of Aspt, Ace-K, Caf and Na-Benz were recorded from 190 to 300 nm. The calibration set was generated by a three-level full factorial design (4).The absorbance valnes were recorded eveiy 5 nm. The calibration samples were measured in random order, so that experimental errors due to drift were not introduced. 

For theexample discussed here, the calibration sets for classes A and B are selected gs hically, and for class C are selected as the extremes and centers of each ofdie three levels in the experimental design. The selection results in 15 samples in each of the calibration sets and 12 in each of the validation sets. A score pS>t of all samples in class A is shown in Figure 4.69 with the calibration set samples indicated by X and the validation samples indicated by O. Similarly, SCO plots of clas.es B and C with calibration and validation samples identifiedsre shown in Figures 4.70 and 4.71, respectively. 

FIGURE 5.90. Concentration of component B versus concentration of component A in a three-level, full-factorial experimental design. 

Let us now consider an experimental arrangement where the subplot levels are assigned randomly in strips across each block of whole-plot levels. Such arrangements are frequently called strip-block designs. As an illustration of this arrangement, suppose that we have a whole-plot variable with three levels, a, a, and a, a subplot variable with two levels, b and... 

THE EXPERIMENTAL CONDITIONS FOR A FULL FACTORIAL TWO LEVEL DESIGN TO TEST THREE HPLC FACTORS... 

These limitations can be seen by comparing the experimental procedure for a three factor, three level star design, shown in Table 5.10, with the experimental procedure for a reflected saturated fractional design, which also tests three factors at three levels, shown in Table 5.11. 

The experimental scheme for a three level reflected saturated fractional design for seven factors is shown in Table 5.15 ( note that one factor was retained as a dummy factor to be used as an additional error check). The experimental order of the scheme was sorted on acid type as this required long equilibration times, this ordering loses some of the features of the initial design but is a compromise that can be justified on the fact that... 

Or, looking at the experimental design, we see three levels of xv This may lead to the question Is there curvature in the relationship of y with x Then an equation of one of the following forms might be helpful ... 

In setting the number of levels to be studied of any one variable factor, the type of effects which it is likely to have on the functional properties to be studied is very important. If its effects are known to be linear and that factor is of secondary Importance to the researcher, then two levels (one at each end of some practical range of levels) may be sufficient. If, on the other hand, it is known that the effects of this factor are curvilinear and/or discontinuous at some point, then at least three levels should be included in the experimental design. If the Interaction of a factor with other factors is known to be significant, then this too could be sufficient reason to Include more than two levels of that factor in the design. 

The design factors require 12 df so an Lig orthogonal array (with 15df) was selected. Hence we can study a factor at two levels and seven factors at three levels each. The matrix is adapted to our needs by discarding column 1 (designed for a variable with two levels) and column 7 (not needed in this example). This yields 3 df to calculate the residuals. Hence the experimental matrix is as presented in Table 2.14. 

For the second retention surface, data were collected according to a three-level, two-factor (density and temperature) experimental design. Each factor was assigned three different values (0.2,0.3 and 0.4 g/mL 75,100 and 125°C), and experiments were conducted at the nine combinations. Data were fit to the model by multiple regression, and these retention surfaces were used to calculate the response surface. 

Certain circumstances may force us to follow the opposite direction and to burden ourselves with additional experimental expense. Models of higher order may be unavoidable if the response variable follows a nonlinear function of the primary variables, or factor variables. Then one can utilize designs which take into account more than two levels of each factor. As an example, in 3" designs one has n factors with three levels each. 

Sukigara et al. [78] designed a factorial experiment by using two factors (electric field and concentration). For a quadratic model, experimentsmust be performed for at least three levels of each factor. These levels are best chosen equally spaced. The two factors (silk concentration and electric field) and three levels resulted in nine possible combinations of factor settings. A schematic of the experimental design is shown in Figure 28(A) and (B). 

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