D-optimality criterion

Table 2 shows the design layout in terms of actual factor values and viscosity results from each experiment at room temperature. A set of candidate points in the design space is selected using the D-optimal criterion. In this work, 25 candidate points have been selected. 

The statistical mixture design for 5-components was carried out by using Design Expert, D-Optimal criterion (Version 6.10, Stat-Easy Inc., Minneapolis USA). In this study, there are restriction on the component proportions Xj that take the form of lower Lj and upper Uj constraint as Lj experimental results of the previous study [2,5]... 

The MUF resin formulation is built up from combination of certain amount of formalin, melamine and urea (in initial and post refluxing stages) and also sorbitol. Variation on the formulation gives different resin properties. The optimum resin properties give the optimum MUF resin formulation. From the properties analysis data, the optimum formulation is determined by using Mixture Experimental Design D-optimal criterion. The selective criteria... 

A criterion that is closely related to D-optimality is E-optimality. The D-optimality criterion minimizes the volume of the confidence ellipsoid of the regression coefficients. Hence, it minimizes the overall uncertainty in the estimation of the regression coefficients. The E-optimality criterion minimizes the length of the longest axis of the same confidence ellipsoid. It minimizes the uncertainty of the regression coefficient that has the worst estimate (highest variance). 

D-optimal criterion being the most usual, based on optimization of the overall precision of estimation of the coefficients of the modelJ This method and type of design is extremely flexible because ... 

Classical designs in which experiments have failed to give a result may be repaired by redefining the domain and finding the best experiments (aeeording to the D-optimal criterion) to replaee the experiment(s) which failed ... 

The model matrix X is defined from the postulated model and the design matrix. Provided that a reasonable model can be suggested, it will thus be possible to select a series of experiments in such a way that X X has a maximum value. These experiments will then allow the parameters of the suggested model to be estimated with a maximum precision. Such a design is said to be D-optimal (D stands for "determinant"). The D-optimality criterion can be used to select experiments for screening. 

It is evident, that a random selection of twelve experiments runs the risk of being a very poor selection. To obtain a sub-set of twelve runs which can give good estimates of the model parameters the D-optimality criterion was used. 

To use the D-optimality criterion to select a design it is required that the experimenter can specify a plausible model. To do so, a good portion of chemical intuition is required. As intuition is a function of experience, this can be difficult with... 

D-optimality. The D-optimality criterion will therefore specify that the product of the dispersion matrix eigenvalues should be as small as possible. 

It is thus easy to determine the extreme vertices and related designs and they have been used quite widely (10-13). These designs enable the models to be determined with adequate precision, but they often require a far larger number of experiments than the number of coefficients. We therefore need a method to select those experiments which carry the most information. In chapter 8 we described how the exchange algorithm may be used to reliably select experiments, often in irregularly shaped domains, of independent variables according to the D-optimal criterion. The method is equally applicable to the mixture problem and will be demonstrated for the same examples as the extreme vertices. Other examples are to be found in the pharmaceutical literature (14-17). 

Typically optimal designs are planned for quadratic models. Probably the most common optimality criterion is the D-optimality criterion. A D-optimal design is a design that... 

Recent work by Hassan et al. [43] has indicated that this method tends to increase the rank of the selected subset at the expense of spread. They used a Monte-Carlo method to maximize a diversity objective function based on the D-optimal criterion. The resulting sets of compounds were biased toward the periphery of the property space. This bias was especially evident when the number of compounds selected far exceeded the dimensionality of the space. It must be noted that the design matrix used in this study did not include higher-order combinations of properties, which may partially account for the redundancy of the results. 

When looking for the best design a minimum and maximum number of experiments is predefined (in our case it could be 12 and 30). For each number of experiments the algorithm looks for the best subset according to the D-optimality criterion. The value of the determinant is then converted into the normalized determinant, and a plot like the one shown in Figure 16 is obtained. 

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