D-optimal designs

Factorial design methods cannot always be applied to QSAR-type studies. For example, i may not be practically possible to make any compounds at all with certain combination of factor values (in contrast to the situation where the factojs are physical properties sucl as temperature or pH, which can be easily varied). Under these circumstances, one woul( like to know which compounds from those that are available should be chosen to give well-balanced set with a wide spread of values in the variable space. D-optimal design i one technique that can be used for such a selection. This technique chooses subsets o... 

In reference 20, a typical robustness test is not performed, but a study on the influence of peak measurement parameters is reported on the outcome. The study is special in the sense that no physicochemical parameter in the experimental runs is changed, but only data measurement and treatment-related parameters. These parameters can largely affect the reported results, as shown earlier, and in that sense they do influence the robusmess of the method. The different parameters (see above) were first screened in a two-level D-optimal design (9 factors in 10 experiments). The most important were then examined in a face-centered CCD, and conclusions were drawn from the response surfaces plots. 

J. Ferre and F.X. Rius, Constructing D-optimal designs from a list of candidate samples, TrAC Trends Anal Chem., 16(2), 70-73 (1997). 

A.S. El-Hagrasy, F.D Amico and J.K. Drennen III, A process analytical technology approach to near-infrared process control of pharmaceutical powder blending. Part I D-optimal design for characterization of powder mixing and preliminary spectral data evaluation, J. Pharm. Sci, 95(2), 392 06 (2006). 

The optimal formulations were obtained using a statistical approach (D-optimal design) and the particles obtained with these formulations had high relaxivities (20-25 s mM ) and small particle sizes (80-100 nm). These formulations appeared to be highly stable in blood, since no change in Ti relaxivity was observed when they where mixed with whole blood. 

Giraud, E., Luttmann, C., Lavelle, F., Riou, J. F., Mailliet, P., and Laoui, A. (2000) Multivariate data analysis using D-optimal designs, partial least squares and... 

Efficient algorithms have been developed that construct D-optimal designs for a given response model, candidate design points, and number of runs (see, for example, Mitchell [23]). 

Example 3.10.2 Approximate D - optimal design for estimating Michaelis-Menten parameters... 

Starting with the substrate values x = [S ] in Table 3.4, we construct a nearly D - optimal design to estimate the parameters of the response function... 

The designs are called D-optimal if the volume of elliptical dispersion of parameter estimates is minimal. D-optimal designs correspond to designs that minimize the variance of response estimate (y J in the associated space. In practice, it is difficult to find a design that simultaneously satisfies several optimality criteria. It is therefore recommended in each individual case to ... 

Design B4, which requires only 24 trials, is recommended for k=4. The design is symmetrical and has certain advantages to the D-optimal design. There is sense in using Hartley s design too. 

Kono s designs [46] are attempts to reduce the number of design points in continuous D-optimality designs (constructed on a hypercube), by replacing all points in centers of two-dimensional planes with one point in the hypercube center. The number of design points by Kono s designs is defined by this expression ... 

In practice, we often use designs that are very similar to D-optimality designs in their properties, but that contain a smaller number of design points. Such designs are known as Bi< and Hartley s designs. 

Table 3.38 tabulates the D-optimal design for the derivation of a ternary system third-degree polynomial. Following this design the coefficients are obtained for a third-degree polynomial having the same form as that from a conventional simplex lattice ... 

Table 3.38 D-optimal design for a ternary system third-degree polynomial 3,3 ...

The adequacy test and the assignment of confidence intervals using a D-optimal design (Table 3.38) are accomplished along the same lines, as in the simplex-lattice method. The variation of with composition, are given in the reference literature [12], In constructing the fourth-order polynomial for the ternary system, the design will be D-optimal at ... 

Moreover, in the fourth-order D-optimal design there are points with coordinates ... 

In Table 3.39 a fourth-order D-optimal design for a ternary system is presented. Table 3.39 D-optimal design for a ternary system fourth-degree polynomial 3,4 ... 

Figure 3.20 shows the arrangement of points in D-optimal designs for ternary systems. 

Figure 3.20 Arrangement of points in the D-optimal designs of a) second-order b) incomplete third-order c) third-order d) fourth-order...

The third-order D-optimal design is prepared relative to pseudocomponents Zj, Z2 and Z3 and the content of initial components at the design points is determined by Eq. (3.84). Table 3.40 presents the experimental conditions both in terms of pseudocomponents and on the natural scale (per cent). The sample variance here is S 0.53 and the number of degrees of freedom is f=13. From Eq. (3.105) for viscosity at 0 °C the coefficients have been calculated for the third-order regression equation ... 

A fourth-order D-optimal design is produced with reference to pseudocomponents Zi Z2 and Z3 - Table 3.42. The pseudocomponents satisfy the principal condition for Scheffe s designs. The conversion to initial components at any point within the local simplex studied is carried out from Eq. (3.84). According to this design, an experiment is run with mixtures, each observation being repeated twice. Using Eqs. (3.109)-(3.113) the coefficients of fourth-order regression equation are calculated in pseudocomponents... 

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