Experimental designs coding

A five-level-five-factor CCRD was employed in this study, requiring 32 experiments (Cochran and Cox, 1992). The fractional factorial design consisted of 16 factorial points, 10 axial points (two axial points on the axis of each design variable at a distance of 2 from the design center), and 6 center points. The variables and their levels selected for the study of biodiesel synthesis were reaction time (4-20 h) temperature (25-65 °C) enzyme amount (10%-50% weight of canola oil, 0.1-0.5g) substrate molar ratio (2 1—5 1 methanol canola oil) and amount of added water (0-20%, by weight of canola oil). Table 9.5 shows the independent factors (X,), levels and experimental design coded and uncoded. Thirty-two runs were performed in a totally random order. 

It is to be stressed, however, that the geometric interpretation of the parameter estimates obtained using coded factor levels is usually different from the interpretation of those parameter estimates obtained using uncoded factor levels. As an illustration, pj (the intercept in the coded system) represents the response at the center of the experimental design, whereas Pq (the intercept in the uncoded system) represents the response at the origin of the original coordinate system the two estimates (Pq and Pq) are usually quite different numerically. This difficulty will not be important in the remainder of this book, and we will feel equally free to use either coded or uncoded factor levels as the examples require. Later, in Section 11.5, we will show how to translate coded parameter estimates back into uncoded parameter estimates. 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

Note that with this coded experimental design, the estimates of b, b 2, and will be independent of the other estimated parameters the estimates of b, bj, and b j will be interdependent. 

Full second-order polynomial models used with central composite experimental designs are very powerful tools for approximating the true behavior of many systems. However, the interpretation of the large number of estimated parameters in multifactor systems is not always straightforward. As an example, the parameter estimates of the coded and uncoded models in the previous section are quite different, even though the two models describe essentially the same response surface (see Equations 12.63 and 12.64). It is difficult to see this similarity by simple inspection of the two equations. Fortunately, canonical analysis is a mathematical technique that can be applied to full second-order polynomial models to reveal the essential features of the response surface and allow a simpler understanding of the factor effects and their interactions. 

Although it is true that the first three columns of plus and minus signs in Table 14.3 are equivalent to the abbreviated coded experimental design matrix D, the signs in Table 14.3 are used for a slightly different purpose than they were Table 14.2. In fact, as we will see, the eight columns of signs in Table 14.3 are equivalent to the matrix of parameter coefficients, X. 

When few factors (/ from two to four) are studied, the full factorial design is the most common approach. The full factorial scheme is the basis for all classical experimental designs, which may be used in more complex situations. For a general two-level full factorial design, each factor has to be considered at a low level (coded as —1) and a high level... 

Experimental Design. The first objective of the experiment was to determine which of the five factors listed above were of decisive importance, and to quantify their effects and their eventual interactions. The number of variables (k = 5) being limited, a screening was not required, and a 2k factorial design was chosen [130,136]. In this type of design, all experiments are performed with variables set at two different levels, which correspond to the limits of the experimental region and which are coded (-1) and ( +1) (coded variables are denoted Xf) (Table 2). 

In this example, orthogonality of all factor effects has been achieved by including additional center points in the coded rotatable design of Equation 11.81. Orthogonality of some experimental designs may be achieved simply by appropriate coding (compare Equation 11.26 with Equation 11.20, for example). Because orthogonality is almost always achieved only in coded factor spaces, transformation of... 

Figure 1 shows scatter plots of the raw data from a Latin hypercube experimental design (see McKay et al., 1979) with 500 runs of the Wonderland code. The output variable HDI is plotted against two of the input variables shown in Section 6 to be important economic innovation in the north (e.inov.n) and sustainable pollution in the south (v.spoll.s). 

Figure 28. Experimental design (A) spinning distance 7 cm and (B) spinning distance 5 cm. The values at the coordinate points show the mean fiber diameter (nm) of 100 measurements and coded values are shown in the brackets (electric field and concentration). NF no fiber formation [78].

Regression leads to a model estimating the relation between the Ai x 1 response vector y, and the Ai x r model matrix X (7,17,116) (Eq. 2.27). N is the number of design experiments, and t the number of terms included in the model. For example, in Equation 2.26, the number of terms equals six, since one intercept, two main effect terms, one interaction term, and two quadratic effect terms were included. The model matrix X is obtained by adding a row of ones before the Aix (r-1) design matrix, which consists of the coded factor levels and columns of contrast coefficients, as defined by the chosen experimental design. 

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