Mathematical models sequential designs

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

Toxicokinetics studies are designed to measure the amount and rate of the absorption, distribution, metabolism, and excretion of a xenobiotic. These data are used to construct predictive mathematical models so that the distribution and excretion of other doses can be simulated. Such studies are carried out using radiolabeled compounds to facilitate measurement and total recovery of the administered dose. This can be done entirely in vivo by measuring levels in blood, expired air, feces, and urine these procedures can be done relatively noninvasively and continuously in the same animal. Tissue levels can be measured by sequential killing and analysis of organ levels. It is important to measure not only the compound administered but also its metabolites, because simple radioactivity counting does not differentiate among them. 

Mathematical Models. Secondary variable interactions quantify the synergies which are common in food chemistry. These interactions cannot be computed from pooled primary variable/sequential design studies and interpolations from such pooled data would lack the information given by the secondary interaction terms. Prob > t is an estimate of the relative importance of each model term. Terms with the lowest Prob > t could well be the driving force of the reaction processes accounting for the quantity of the volatiles found. From Table IV, about 25% of the model terms present at >0.05 Prob > t are seen to be interaction terms. 

The simplex method has been used widely over the past 30 years, its success as much owing to its simplicity as to its efficiency. Unlike the other methods described in this chapter and most of the others in this book, it assumes no mathematical model for the phenomenon or phenomena being studied. The often long and costly phase of determination of a model equation may therefore be avoided, and the method is thus economical in principle. It is sequential because the experiments are analysed one by one, as each is carried out. Because the method is not model-based we will not describe it in detail, but well indicate how it can "fit in" and complement statistical experimental design. 

Like the centroid designs, the Scheff6 simplex lattice designs, described below, are easy to constmct. Their R-efficiencies are equal to 100%, as the number of experimental points is equal to the number of coefficients in the corresponding mathematical model. They may be resolved directly, that is the coefficients may be calculated directly without using a computer for least squares regression. They can sometimes be built up sequentially. The precision of the calculated response is optimal, that is, the variance over the design space is minimal for the number of experiments. 

Efficient experimentation is based on the methods of experimental design and its quantitative evaluation. The latter can be performed by means of mathematical models or graphical representations. Alternatively, sequential methods are apphed, such as the simplex method, instead of these simultaneous methods of experimental optimization. There, the optimum conditions are found by systematic search for the objective criterion, for example, the maximum yield of a chemical reaction, in the space of all experimental variables. 

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