Special cubic model

Figure 4.16 Simplex lattice design for a special cubic model with ten...

With the design used it is not possible to construct a complete third order model. The most complex third order model which can be used contains 38 terms. This includes third order blending effects (the so called special cubic model terms [10], e.g. d m c) and temperature and relative humidity dependent blending effects (e.g. d m t and e t h). 

CALCULATED REGRESSION COEFFICIENTS AND MODEL VALIDATION CRITERIA OF THE SPECIAL CUBIC MODEL (EQUATION (3)) FOR THE LIQUID-LIQUID EXTRACTION OF 9 SULPHONAMIDES WITH... 

FIGURE 8.8 Examples of simplex lattices for (a) linear, (b) quadratic, (c) full cubic, and (d) special cubic models. 

Our objective is to generate response surfaces which will allow the prediction of resolution at any point within the regions ABC and BCD. The response surface is described mathematically by the special cubic model shown in equation 2. 

Somewhat more complex designs have been used in the literature and software has been made commercially available. They require seven or ten chromatograms (Fig. 6.27) obtained with the experimental conditions of Table 6.14. The respective models are called the reduced or special cubic model and the complete cubic model. 

The special cubic model describes a eertain third-order eurvature in the response surface ... 

The disintegration times of the tablets were also measured. Mean values (5 tablets) for each experiment were given in table 9.2, along with the previously discussed hardness data. The disintegration times were also analysed according to the special cubic model. It was noted that there were (a) moderate differences between calculated and measured values in the case of the test points, and (b) negative disintegration times were predicted within part of the domain. 

For example, in the case of the 3 component mixture we may assume that the response is well or at least adequately modelled by a third-order polynomial. However if we are not sure of the best model, we may prefer a step-wise approach, first building up a second-order design, and then augmenting it as necessary. The possibility of such a sequential approach to experimental design is as valuable for mixtures, as it is for independent variables. Thus the ternary test point 7 is added to the 3 component second-order design (figure 9.3) and the special cubic model determined. Extra test points may be added but these points do not allow fitting of the full cubic model. 

It would not be possible to choose between the second-order polynomial model and the additive blending model of Becker, using only the second-order simplex lattice design of 6 experiments. Measuring the response at a test point, the Vs, Vs, Vs point, would show the inadequacy of the second-order polynomial and would suggest tlie use of the special cubic model (addition of a term to equation 9.2). A... 

Since this equation has 10 terms, we will have to perform at least 10 distinct runs to determine the values of all the coefficients. Often, however, adding a single cubic term is sufficient to transform the model into a satisfactory description of the experimental region. Eliminating the terms in Eq. (7.18), we arrive at the expression for the special cubic model, which contains only one more term than the quadratic model, and therefore requires only one more run ... 

The experimental design normally used to determine the values of the coefficients of the special cubic model is called the simplex centroid, which we obtain by simply adding a center point, corresponding to a 1 1 1 ternary mixture, (xi,X2,X3) = (5,5,5) to the simplex lattice design. The coefficient of the cubic term is given by... 

Eliminating the terms with non-significant coefficients, we reduce the special cubic model to the equation... 

Fig. 7.6. Simplex centroid design, with average responses for the mixtures represented by the points and contour curves for the special cubic model, Eq. (7.20). The three boxed response values close to each vertex were used to test the quality of the fit.

Exercise 7.11. The cubic model can also be determined from a matrix equation. Write the matrix equation that we should solve to obtain the values of the seven coefficients of the special cubic model for the membrane study. 

Analysis of variance for the fits of the quadratic and special cubic models to the data in Table 7.1, augmented by the results observed for the mixture with Xi = X2 = xs = 1/3 (duplicate response values with an average of 3.50 cm), which increases the number of observations to 17... 

Note. Values within parentheses refer to the special cubic model. 

In our example, I is the quadratic model, II the special cubic model and d = 1. We simply have... 

Study of a membrane for a selective electrode. Comparison between the average observed analytical signals (yobg) and the values predicted by the quadratic (yquad) and special cubic models (Pcub)... 

Analyses of variance for the quadratic and special cubic models. The number of distinct mixtures is 10, which permits us to test for lack of fit of the two models... 

Substituting Eq. (7.24) into the equation for the special cubic model in terms of pseudocomponents, Eq. (7.20), we can express the response as an explicit function of the proportions of p3u-role, KCl and PCtFe(CN)e solutions in the mixture ... 

Fig. 7.7. Response contour curves for the special cubic model as a function of the proportions of pyrrole, KCl and K4Fe(CN)6 solutions.

To define the design, we need to consider what models could be appropriate to describe the behavior of the two responses — film elongation and swelling. Unless we already know a good deal about the system imder study, this usually cannot be done before the experiments are made. Also, different responses may require different models. Since it is possible that a description of the results will require a special cubic model, we had better choose a design with at least seven different mixtures. 

Equation (5.12) was used to fit linear, quadratic and special cubic models to the data from each response. The analysis of the results led to the following conclusions ... 

The special cubic model for four-component mixtures has 14 terms, and its coefficients can be estimated from the design shown in Fig. 7.10b. The points on each face now reproduce the arrangement corresponding to the simplex centroid design, which we used to fit the special cubic model for three-component mixtures. 

The analysis of variance shows that the linear model is unsatisfactory, and that the other two models do not show lack of fit (Table 7A.4). This time, however, the special cubic model seems superior. The explained variation is larger, the MS of/MS ratio is smaller, and the cubic term is statistically significant. The contour curves for the cubic model are presented in Fig. 7A.3. The largest tensile strength values are obtained close to the base of the triangle, toward the left vertex, which corresponds to a blend that is predominantly PVDF, with httle or no polystyrene. 

Surely you noticed that the response for one of the three replicates of run no. 6 is much lower than the other two. If this value is removed from the data as a suspected outlier, the pure error mean square drops from 17.4 to only 3.6. The downside of this decision is that in comparison with the smaller error value even the special cubic model shows lack of fit. This suggests that a more extensive model is needed to adequately represent these data. 

The linear model shows lack of fit. The quadratic and special cubic models are represented by the expressions... 

The coefficients of the statistically significant terms of the fitted models for the five responses are given in Table 7A.8. The special cubic model is the one that best fits all properties except cohesivity, for which the quadratic model is sufficient. Of all fitted models, only that for firmness shows some lack of fit, but the number of level combinations in the design is not sufficient for fitting a full cubic model. The coefficient values clearly indicate that starch (xi) is the most important component, but its effect depends on the proportions of the other two ingredients, sugar and powdered milk. 

Notes The values for the special cubic model are within parentheses. 

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