Adequacy of models

In Section 5.5 a question was raised concerning the adequacy of models when fit to experimental data (see also Section 2.4). It was suggested that any test of the adequacy of a given model must involve an estimate of the purely experimental uncertainty. In Section 5.6 it was indicated that replication provides the information necessary for calculating the estimate of (. We now consider in more detail how this information can be used to test the adequacy of linear models [Davies (1956)]. 

It should be stated at the beginning that despite quite numerous partial successes the present state of the quantum-chemical theory of reactivity is far from satisfactory. First of all, it is not clear whether the present shortcomings are due to the non-adequacy of models of activated complexes or to the drastic approximations made in the calculation of the energy of the activated complex and the reactants. On the other hand, some other deficiencies of most of the reported attempts to interpret reactivity in terms of the theory are obvious the very nature of the HMO method is thought148 to make it necessary to treat as large sets of theoretical and experimental data as possible and, in addition, to respect the distinction in properties of the three classes of positions mentioned (in this connection we do not refer to the difference in the stereochemistry of these positions). 

These formulas are useful when seeking to diagnose the adequacy of model fit. 

Sixth, and finally, the adequacy of model structure as well as parameter values should be evaluated based on comparison of mode predictions with experimental data that had not been used for calibration purpose. This process essentially evaluates whether the PBPK model is capable of providing reliable predictions of the various dose metrics of potential use in a cancer risk assessement. The model should not only reprodnce consistently the shape of the pharmacokinetic time-course curve (i.e., including bnmps and valleys) and not jnst provide satisfactory fit only to a portion of the cnrve. Evaluation or validation of PBPK models should be regarded... 

Before closing this topic, we must stress the importance of the thermodynamic data in getting reliable RCMs. The adequacy of models and the accuracy of interaction parameters must be checked always. Wilson is very accurate for homogeneous... 

The kinetic parameters of the equations were determined by the least square method, where the sum of Z (X exp-X modei) has teen minimized. In order to test the adequacy of models two statistica tests have been realized as it is proposed by [Froment and Bishoff, (1990)]. Lack of fit (F) test was applied as well as 95% confidence interval of the parameters was estimated. Models that show lack of fit or whose parameter(s) is non-significantly different from zero or negative were rejected A block diagram of model discrimination is shown in the Figure 3. 

The shape of Cj(t) also has implications for the adequacy of models (14) used to describe reorientation. Models which are based on assumptions about the physical processes governing reorientation make predictions about Cj(t) in addition to giving the reorientational functions themselves. Approximations, such as a truncated memory function expansion or Steele s torque approximation, do not give predictions about Cj(t). Both Fokker-Planck and J diffusion models (14) assume uncorrelated collisions, which we have seen is an inappropriate assumption for most fluids. Some variant of the itinerant oscillator model may well prove to be more realistic, but it is easy to overestimate the librational character of the motion. 

The predictability of a computer code and the adequacy of modeling of a plant used in the assessment of the anticipated transients and the accidents to assess the adequacy of the safety systems design must be validated through comparisons between predictions and the startup test data or system effect experiments. The adequacy of each model incorporated in a code to describe each of events included in the transients and the accidents must be validated based on information provided by separate effect experiments. Conservative assumptions may be made whenever those information to validate a code and modeling of a plant are not available. 

In summary, a combination of the plot based on equation (10.6), using any single substance, and determination of the asymptote (10.14), using any pair of substances, provides a sound means of evaluating the parameters K, tC and. Having found these, further experimental points on (10.6) and (10.15), and possibly also (10.7), provide a check on the adequacy of the dusty gas model. Provided attention is limited to binary mixtures, this check can be quite comprehensive. In their published paper Gunn and King... 

Validation and Application. VaUdated CFD examples are emerging (30) as are examples of limitations and misappHcations (31). ReaUsm depends on the adequacy of the physical and chemical representations, the scale of resolution for the appHcation, numerical accuracy of the solution algorithms, and skills appHed in execution. Data are available on performance characteristics of industrial furnaces and gas turbines systems operating with turbulent diffusion flames have been studied for simple two-dimensional geometries and selected conditions (32). Turbulent diffusion flames are produced when fuel and air are injected separately into the reactor. Second-order and infinitely fast reactions coupled with mixing have been analyzed with the k—Z model to describe the macromixing process. 

Quantitative controllable variables are ftequentiy related to the response (or performance) variable by some assumed statistical relationship or model. The minimum number of conditions or levels per variable is determined by the form of the assumed model. For example, if a straight-line relationship can be assumed, two levels (or conditions) may be sufficient for a quadratic relationship a minimum of three levels is required. However, it is often desirable to include some added points, above the minimum needed, so as to allow assessment of the adequacy of the assumed model. 

The determination of piezoelectric constants from current pulses is based on interpretation of wave shapes in the weak-coupling approximation. It is of interest to use the wave shapes to evaluate the degree of approximation involved in the various models of piezoelectric response. Such an evaluation is shown in Fig. 4.5, in which normalized current-time wave forms calculated from various models are shown for x-cut quartz and z-cut lithium niobate. In both cases the differences between the fully coupled and weakly coupled solutions are observed to be about 1%, which is within the accuracy limits of the calculations. Hence, for both quartz and lithium niobate, weakly coupled solutions appear adequate for interpretation of observed current-time waveforms. On the other hand, the adequacy of the uncoupled solution is significantly different for the two materials. For x-cut quartz the maximum error of about 1%-1.5% for the nonlinear-uncoupled solution is suitable for all but the most precise interpretation. For z-cut lithium niobate the maximum error of about 8% for the nonlinear-uncoupled solution is greater than that considered acceptable for most cases. The linear-uncoupled solution is seriously in error in each case as it neglects both strain and coupling. 

With the adequacy of lipid bilayer membranes as models for the basic structural motif and hence for the ion transport barrier of biological membranes, studies of channel and carrier ion transport mechanisms across such membranes become of central relevance to transport across cell membranes. The fundamental principles derived from these studies, however, have generality beyond the specific model systems. As noted above and as will be treated below, it is found that selective transport... 

At present, it is known that the structures of the ECC type (Figs 3 and 21) can be obtained in principle for all linear crystallizable polymers. However, in practice, ECC does not occur although, as follows from the preceding considerations, the formation of linear single crystals of macroscopic size (100% ECC) is not forbidden for any fundamental thermodynamic or thermokinetic reasons60,65). It should be noted that the attained tenacities of rigid- and flexible-chain polymer fibers are almost identical. The reasons for a relatively low tenacity of fibers from rigid-chain polymers and for the adequacy of the model in Fig. 21 a have been analyzed in detail in Ref. 65. 

The structure and mathematical expressions used in PBPK models significantly simplify the true complexities of biological systems. If the uptake and disposition of the chemical substance(s) is adequately described, however, this simplification is desirable because data are often unavailable for many biological processes. A simplified scheme reduces the magnitude of cumulative uncertainty. The adequacy of the model is, therefore, of great importance, and model validation is essential to the use of PBPK models in risk assessment. 

Testing the adequacy of a model with respect to its complexity by visually checking for trends in the residuals, e.g., is a linear regression sufficient, or is a quadratic polynomial necessary ... 

Test of Model Adequacy. The final step is to test the adequacy of the model. Figure 4 is a plot of the residual errors from the model vs. the observed values. The residuals are the differences between the observed and predicted values. Random scatter about a zero mean is desireable. 

The way in which the family of models is selected depends on the main purpose of the exercise. If the purpose is just to provide a reasonable description of the data in some appropriate way without any attempt at understanding the underlying phenomenon, that is, the data-generating mechanism, then the family of models is selected based on its adequacy to represent the data structure. The net result in this case is only a descriptive model of the phenomenon. These models are very useful for discriminating between alternative hypotheses but are totally useless for capturing the fundamental characteristics of a mechanism. On the contrary, if the purpose of the mode-... 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

However, regression coefficients cannot be used to determine the adequacy of a model choice, as discussed previously. 

There is a plethora of model adequacy tests that the user can employ to decide whether the assumed mathematical model is indeed adequate. Generally speaking these tests are based on the comparison of the experimental error variance estimated by the model to that obtained experimentally or through other means. 

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