Model adequacy

ESTIMATION OF RETENTION MODELS ADEQUACY IN MICELLAR LIQUID CHROMATOGRAPHY... 

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. 

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. 

A number of replications under at least one set of operating conditions must be carried out to test the model adequacy (or lack of fit of the model). An estimate of the pure error variance is then calculated from ... 

Parameter estimates are tested on whether they differ significantly from zero at a certain probability level. If not, the parameter should be skipped from the model and the model should be redefined even if the test for model adequacy was positive. When the errors have constant variance, the random variable... 

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. 

Let us now consider models that have only more than one measured variable (w>l). The previously described model adequacy tests have multivariate extensions that can be found in several advanced statistics textbooks. For example, the book Introduction to Applied Multivariate Statistics by Srivastava and Carter (1983) presents several tests on covariance matrices. 

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 ... 

Step 6. Perform the appropriate model adequacy test (x2-test, F-test or Bartlett s -/2-test) for all rival models (just in case one of the models... 

Table 12.7 Chemostat Kinetics Results from Model Adequacy Tests Assuming af. is Known (yftest) Performed at a=0.0I Level of Significance...

With this book the reader can expect to learn how to formulate and solve parameter estimation problems, compute the statistical properties of the parameters, perform model adequacy tests, and design experiments for parameter estimation or model discrimination. 

In addition to the maximum point, inflection points of the rate data can be used for testing model adequacy (M6). 

The several modeling methods discussed in the accompanying sections are quite useful in testing the ability of a model to fit a particular set of data. These methods do not, however, supplant the more conventional tests of model adequacy of classical statistical theory, i.e., the analysis of variance and tests of residuals. 

If this linear analysis is to be used, the experimental conversion-spacetime data should first be taken at several pressure levels. Using the C2 analysis alone, then, the plots of Ct versus total pressure should be made for a preliminary indication of model adequacy. If several models are found to provide near-linear Cx plots, the complete linear analysis using the C2 plots should assist in the discrimination among the remaining rival models. If a model is adequate, both the Cl and C2 points should be correctable by a straight line with a common intercept, as demanded by Eqs. (85) and (86). If only one model is found to be adequate following the initial Cl analysis, the complete Ct and C2 analysis should still be carried out on this model to verify its ability to fit the high conversion data. 

Figure 30 portrays the grid of values of the independent variables over which values of D were calculated to choose experimental points after the initial nine. The additional five points chosen are also shown in Fig. 30. Note that points at high hydrogen and low propylene partial pressures are required. Figure 31 shows the posterior probabilities associated with each model. The acceptability of model 2 declines rapidly as data are taken according to the model-discrimination design. If, in addition, model 2 cannot pass standard lack-of-fit tests, residual plots, and other tests of model adequacy, then it should be rejected. Similarly, model 1 should be shown to remain adequate after these tests. Many more data points than these 14 have shown less conclusive results, when this procedure is not used for this experimental system.

The double summation is used to take each of the models as a reference in order to avoid mislocations of optimal discrimination conditions Since the model adequacy criteria and the design criterion are independent of each other, any type of design criterion can be used An alternative one is [9] given by eq 59... 

A comparison between the kinetic parameters obtained for both impro-moted PC and Fe-assisted PC is shown in Figure 14 (they are displayed in the same order as shown in the tables). One can observe that for the simultaneous concentrations of 20, 30, and 40 ppm C in phenol, the estimates obtained for the Fe-assisted PC oxidation are also higher than those estimated for the unpromoted PC reaction. This comparison allows to corroborate the model adequacy by producing estimates consistent with the experimental observations. That is, kinetic parameters in the Fe-assisted PC oxidation must be larger than in the unpromoted PC reaction. 

In this case, the value of N for Oj and U2 is the same and it is equal to the number of experiments accepted for the statistical calculations. Coming back to the problem of the model adequacy, it is clear that the zero hypothesis has been transformed into the following expression Hom s , = s, ... 

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