Multiparameter model

To conclude, models of USC, and particularly the multiparameter models, are no better than the data to support them. The higher order the fit, the more demands on the data with the continual temptation to use data from a dissimilar system simply because the data are available. 

Different reactor networks can give rise to the same residence time distribution function. For example, a CSTR characterized by a space time Tj followed by a PFR characterized by a space time t2 has an F(t) curve that is identical to that of these two reactors operated in the reverse order. Consequently, the F(t) curve alone is not sufficient, in general, to permit one to determine the conversion in a nonideal reactor. As a result, several mathematical models of reactor performance have been developed to provide estimates of the conversion levels in nonideal reactors. These models vary in their degree of complexity and range of applicability. In this textbook we will confine the discussion to models in which a single parameter is used to characterize the nonideal flow pattern. Multiparameter models have been developed for handling more complex situations (e.g., that which prevails in a fluidized bed reactor), but these are beyond the scope of this textbook. [See Levenspiel (2) and Himmelblau and Bischoff (4).]... 

The dispersion and stirred tank models of reactor behavior are in essence single parameter models. The literature contains an abundance of more complex multiparameter models. For an introduction to such models, consult the review article by Levenspiel and Bischoff (3) and the texts by these individuals (2, 4). The texts also contain discussions of the means by which residence time distribution curves may be used to diagnose the presence of flow maldistribution and stagnant region effects in operating equipment. 

This chapter has outlined specifically how quantitative data on somewhat idealized reaction systems can be used as a basis for demonstrating the validity of our empirical electronic models in the field of reactivity. The multiparameter statistical models derived for the systems studied (PA, acidity, etc.) have limited direct application in EROS themselves. The next section develops the theme of applying the models in a much more general way, leading up to general reactivity prediction in EROS itself. 

The relevance of size-related properties of hERG-blocking molecules was also detected in a 2D QSAR model developed by Coi et al. [22] after the analysis of 82 compounds through the CODESSA method. These authors developed two multiparameter models with strong predictive properties, from which, besides the involvement of hydrophobic features, the importance of linearity as opposed to globularity of the hERG blockers emerged. 

Huang JC (2005) Multiparameter solubility model of fullerene C60. Fluid Phase Equil. 237 186-192. 

Moody RP, MacPherson H (2003) Determination of dermal absorption QSAR/QSPRs by brute force regression multiparameter model development with Molsuite 2000. J Toxicol Env Healt A 66 1927-1942. 

Multiparameter equations, such as Equation 4, obtained through MLRA are the simplest form of parallel connection of several models. Each model has been parameterized from its own source of primary data. Combined application can reproduce new types of data and lead to new information and knowledge. 

As shown in Figures 4.6-4.S, this difficulty also exists for other multiparameter models containing more degrees of freedom in the set of items to be estimated than there are degrees of freedom in the data set. The other models are... 

Uncertainty analysis for multiparameter models may require assigning sampling distributions to many random parameters. In which case, a single value is drawn from each of the respective sampling distributions during each Monte Carlo iteration. After each random draw, the generated values of the random parameters... 

For example, the distribution from which the samples are drawn is assumed to be the true distribution of the parameter of interest. To the degree that the sample distribution differs from the actual distribution (which is generally assumed unknown by the classical statistician), the confidence in the Monte Carlo results is decreased. Just how close these distributions must be is a complicated statistical issue that is frequently unclear. In a practical sense, if misspecification of a sampling distribution occurs for a very sensitive parameter in a multiparameter model, then the confidence in the Monte Carlo results is greatly diminished because the model prediction is greatly influenced by that parameter. 

The use of estimates of treatment effect based on indirect comparisons when there is a common comparator has recently been shown on many occasions to agree with the results of head-to-head clinical trials (Song et al. 2003). Clearly a more challenging situation exists where there is not a common parameter, for example, in a recent study of the relative cost effectiveness of newer drugs for treatment of epilepsy (Wilby et al. 2003). In this study, Bayesian Markov chain Monte Carlo models for multiparameter synthesis were used (Ades 2003). Here, complex models were used to analyze a set of clinical studies involving a series of clinical alternatives, including the two alternatives of interest. 

Ades, A. 2003. A Chain of Evidence with Mixed Comparisons Models for Multiparameter Syntheses and Consistency of Evidence. Statistics in Medicine 22 2995-3016. 

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

In the previous chapter, it was seen that a single observation of response from a system does not provide sufficient information to fit a multiparameter model. A... 

Section 9.6 shows how to test the significance of a set of parameters in a model. This set could contain just one parameter. Is it possible to fit a large, multiparameter model, and then test each parameter in turn, eliminating those parameters that do not have a highly significant effect, until a concise, best model is obtained Is it possible to start with a small, single-parameter model, and then add... 

Another approach for the determination of the kinetic parameters is to use the SAS NLIN (NonLINear regression) procedure (SAS, 1985) which produces weighted least-squares estimates of the parameters of nonlinear models. The advantages of this technique are that (1) it does not require linearization of the Michaelis-Menten equation, (2) it can be used for complicated multiparameter models, and (3) the estimated parameter values are reliable because it produces weighted least-squares estimates. 

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