True model

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

However, an important question that needs to be answered is "what constitutes a satisfactory polynomial fit " An answer can come from the following simple reasoning. The purpose of the polynomial fit is to smooth the data, namely, to remove only the measurement error (noise) from the data. If the mathematical (ODE) model under consideration is indeed the true model (or simply an adequate one) then the calculated values of the output vector based on the ODE model should correspond to the error-free measurements. Obviously, these model-calculated values should ideally be the same as the smoothed data assuming that the correct amount of data-filtering has taken place. 

Although all the underlying assumptions (local linearity, statistical independence, etc.) are rarely satisfied, Bartlett s jf-test procedure has been found adequate in both simulated and experimental applications (Dumez et al., 1977 Froment, 1975). However, it should be emphasized that only the x2-test and the F-test are true model adequacy tests. Consequently, they may eliminate all rival models if none of them is truly adequate. On the other hand, Bartlett s x2-test does not guarantee that the retained model is truly adequate. It simply suggests that it is the best one among a set of inadequate models ... 

We stop before the interminable threshold of the artist s look poised for the return of that which seems to have vanished but really is simply waiting for its next term of presence. The heart is a true model of the potential planetary unity of the future humanity, an organic clot of cosmos, a mechanism of created and indestructible life, a concentration of human love and its boundless energy, the iconology of future culture. 

As outlined in the previous section, there is a hierarchy of possible representations of metabolism and no unique definition what constitutes a true model of metabolism exists. Nonetheless, mathematical modeling of metabolism is usually closely associated with changes in compound concentrations that are described in terms of rates of biochemical reactions. In this section, we outline the nomenclature and the essential steps in constructing explicit kinetic models of metabolic networks. 

Although it is beyond the scope of this presentation, it can be shown that if the model yj. = 0 + r, is a true representation of the behavior of the system, then the three sui.. s of squares SS and divided by the associated degrees of freedom (2, 1, and 1 respectively for this example) will all provide unbiased estimates of and there will not be significant differences among these estimates. If y, = 0 + r, is not the true model, the parameter estimate will still be a good estimate of the purely experimental uncertainty, (the estimate of purely experimental uncertainty is independent of any model - see Sections 5.5 and 5.6). The parameter estimate however, will be inflated because it now includes a non-random contribution from a nonzero difference between the mean of the observed replicate responses, y, and the responses predicted by the model, y, (see Equation 6.13). The less likely it is that y, - 0 + r, is the true model, the more biased and therefore larger should be the term Si f compared to 5. ... 

If yij = 0 + r,j is the true model, then and should both be good estimates of and there should not be a significant difference between them we would expect be equal to that is, = 0. 

If the true model contains quadratic terms then the estimate of the intercept, Pq, of the first-order model will be biased. The lack-of-fit of the first-order model due to quadratic effects can be tested by adding center points to the design. 

Shoemaker et al. [37] give several examples of the reduction in the number of experimental runs that can occur when it is assumed that some of the terms in the full second-order model are negligible. The reader is warned, however, that assuming a term is negligible is not an assurance that it can be ignored. The presence of terms in the true model that were assumed negligible will bias the estimates of the other coefficients. 

It is not known if the effect of flexibility is an equilibrium or kinetic effect. The flexibility might allow the compounds to expand or contract to fill available space in the VP1 hydrophobic pocket. Alternatively, the flexibility may allow the compounds to achieve a conformation required to enter or leave the pocket, but this conformation would not be seen in the crystallographic experiment. If this is true, modeling of the equilibrium structure of compounds in the pocket will not be accurate predictors of compound potency. 

The far-infrared spectrum of Fe(EtPhDtc)3 as a function of pressure shows that the intensity of the band assigned to the 2 V2 state increases relative to that assigned to the 6A t state upon increasing pressure (93b) (38). For the Fe(n-Pr2 -Dtc)3 complex, the Fe—S vibration at 367 cm"1 was assigned to the low-spin-state (6i4j) isomer. On the basis of these results, the spin-state equilibrium was adopted as the true model with a spin-state interconversion rate lower than the vibrational time scale ( 10" 3 sec). 

Equation 6.13). The less likely it is that yu = 0 4- ru is the true model, the more biased and therefore larger should be the term lof compared to, v2c. 

Fig. 1.5 Illustration of the simulation and analysis of a virtual trial outcome. The solid line represent the true dose-response relationship based on a sampled set of parameters from the joint posterior distribution of the model parameters. The circles represent the simulated drug effects in the patients included in the trial on the basis of the true" model parameters and the errors bars...

Compare those required by the true model with the ranges estimated from the data... 

If all the requirements of the true model are met, accept the tentative model and propose alternatives... 

Suppose, for example, the true model is kjCa but the tentative... 

Admittedly, in the strictly theoretical sense, such a verification is a necessary but not sufficient requirement for the model to be the true model. It should be obvious to readers familiar with current research work on the nitration of aromatic compounds that the assumptions and mechanisms on which this model is based are under debate, albeit generally accepted. Exhaustive testing and verification of a model is usually not justifiable in a business... 

Miller (2002, pages 148-150) recommended fivefold to tenfold validation, so that effectively 80% to 90% of the data should be in the training set. Another recommendation is that n /4 of the data should make up a training set (randomly selected) and the rest predicted as test hold-out data see Shao (1993) for details. However, it is easy to show that use of the 3/4 rule does not perform well in settings such as drug discovery where prediction accuracy, rather than selection of the true model, is the objective. We are sometimes better off with a model that is not the true model but a simpler model for which we can make good estimates of the parameters (leading to more accurate predicted values). 

In this chapter, we discuss the choice of screening designs for model selection via the three elements of the design matrix D, the model /, and a criterion based on X. One important feature about the design screening problem is that the true model is usually unknown. If we denote the set of all possible models that might be fitted by T = [f1,..., / , where u is the number of all possible models, then the optimality criterion for design selection should be based on all possible models, rather than on a specific model in T. 

---