A three-level full factorial design

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. 

It is obvious that to be able to estimate the quadratic coefficients, p, in equation (14), it is necessary to have at least three distinct levels or settings for the variables. This suggests that a suitable design for estimating the coefficients of the second-order model would be a single replicate of a three-level full factorial design in p variables. This design... 

Figure 5.5 shows a three level full factorial design for the HPLC example which tests three factors, centred around the method conditions. This design allows the testing of changes from the method conditions to extremes on either side. 

The design, which is illustrated in Figure 5.8, gives the most comprehensive evaluation of the response surface using a given number of experiments. It provides greater efficiency than a three level full factorial design yet essentially obtains the same information. However, as k increases the number of required experiments quickly becomes impractical [22]. 

The exact structure of each of the functions/ x),..., fk(x) depends on the transformation or factor coding used. For example, the F matrix for a three-level full factorial design for two process variables and a second-order model is shown in Table 8.4. 

Three-Level Full Factorial Designs. A three-level full factorial design contains all possible combinations between the /factors and their levels L = 3, and the number of experiments thus is A = = 3L These three-level... 

A three-level full factorial design contains all possible combinations of / factors at their 3 levels (-1,0, + 1). In total, f experiments will be required to examine the /factors (9 experiments for 2 factors 27 for 3 factors, etc.). 

A larger three-level full factorial design... 

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

FIGURE 2.8 Coded levels for the experiments required by a two-level full factorial design with three factors. 

If there are only p=2, or p=3 variables then a full factorial design is often feasible. However, the number of runs required becomes prohibitively large as the number of variables increases. For example, with p=5 variables, the second-order model requires the estimation of 21 coefficients the mean, five main effects, five pure quadratic terms, and ten two-factor interactions. The three-level full factorial design would require... 

In a full factorial design all combinations between the different factors and the different levels are made. Suppose one has three factors (A,B,C) which will be tested at two levels (- and +). The possible combinations of these factor levels are shown in Table 3.5. Eight combinations can be made. In general, the total number of experiments in a two-level full factorial design is equal to 2 with /being the number of factors. The advantage of the full factorial design compared to the one-factor-at-a-time procedure is that not only the effect of the factors A, B and C (main effects) on the response can be calculated but also the interaction effects of the factors. The interaction effects that can be considered here are three two-factor interactions (AB,... 

Figure 3.18 displays the design matrix and the layout of a two-factor three-level full factorial design. The central point of the design [i.e. (0, 0)] is often replicated to estimate the experimental error for the subsequent calculations. 

Central composite designs (CCD) are the most frequently applied response surface designs. They consist of a two-level full factorial design (2 experiments), a star design (2/ experiments) and a central point. As a consequence the CCD requires 2 + 2/+ 1 experiments to examine / factors (9 experiments for two factors 15 experiments for three factors, etc.). The experiments of the full factorial are situated at levels -1 and +1, those of the star designs at levels -a and +a for each factor, and the central point has all levels at 0 (see Figure 3.19). The central point of the design (0, 0) is replicated commonly to estimate the experimental error. 

A two level full factorial experimental design with three variables, F/P molar ratio, OH/P wt %, and reaction temperature was implemented to analyses the effect of variables on the synthesis reaction of PF resol resin. Based on the composition of 16 components of 10 samples, the effect of three independent variables on the chemical structure was anal3 ed by using 3 way ANOVA of SPSS. The present study provides that experimental design is a very valuable and capable tool for evaluating multiple variables in resin production. 

Figure 14.2 A three-factor two-level full factorial design in factors A, B, and C. The open circle locates the center of the design.

Three-Level, Full-Factorial Experiment Design, Interaction Model Between Pad A and Pad B... 

Figure 3.9. Two-level, full factorial designs for (a) two and (b) three factors. The low value of each factor is designated by a contrast coefficient of— and the high value designated by +.

FIGURE 8.12 Two different presentations of a 12-point design for three mixture variables and two process variables, (a) The three-point 3,1 simplex-lattice design is constructed at the position of each of the 22 points of two-level full factorial design, (b) The 22 full factorial design is repeated at the position of each point of the 3,f simplex-lattice design. The way of representation is related to the order chosen for the variables. 

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