Adequacy of fit

On the a priori assumption that only random errors are present, Eq. (4.5), or (4.2) for linear models, provides an unbiased (or asymptotically unbiased) estimator of the error variance If the calculated estimate proves to be unbiased, we say that the model adequately fits the experimental data or simply the model is adequate. Otherwise, the bias in the variance estimate indicates the presence of systematic errors. This means that the model is inadequate and neither the parameter estimates nor the confidence region are valid. 

In order to test the unbiasedness of the estimate, the exact value of a must be known. In practice, it is not. The usual approach is to estimate the error variance by replicated experiments, designed so that systematic errors of instrumentation (called instrumental errors, such as unbalanced scales, wrong 

There are situations when several rival models are suggested to explain the same experimental facts. Certain discriminatory criteria can be derived for choosing the most probable model. The two most commonly used approaches, the likelihood and the Bayesian, are reviewed by Singh and Rao (1981). 

It should be emphasized that the results of the inference tests must be interpreted in their statistical sense. Thus, when a model passes the F test or is chosen by a discriminatory criterion, we should say that with a given probability this model cannot be rejected and by no means does it prove the uniqueness of the model. When the results of a test are marginal, the experimenter should abstain from making any conclusions. Further experimental research directed towards an increase in the power of the criterion is required. 

The goal of the researcher in conducting experiments is to acquire desired information. It is possible to increase the amount of information obtained per data point if the experiments are carried out in accordance with a prearranged plan. Let us consider the following example borrowed from Box et al. (1954), pp. 445-446. 

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Table 3.30 summarizes the results of the regression equation testing for adequacy of fit ... 

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 ... 

Coulson concluded that the most important contribution to H-bonding is ionic resonance (5.29a). However, generations of empirical modelers have found it convenient to employ simple pairwise-additive Coulombic formulas with empirically fitted point charges to model H-bonds, and such empirical models have tended to encourage uncritical belief in the adequacy of a classical electrostatic picture of H-bonding. 

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. 

An analysis of variance can also be used to test the adequacy of more theoretical models. For example, two models considered in Section III for pentane isomerization are the single-site and dual-site models of Eqs. (30) and (31). These were linearized to provide Eqs. (32) and (33). The overall fit of these equations to the data may now be judged by an analysis of variance, reported in Tables V and VI (K3). It is seen that Eq. (33) fits the data quite... 

This is, then, the regression sum of squares due to the first-order terms of Eq. (69). Then, we calculate the regression sum of squares using the complete second-order model of Eq. (69). The difference between these two sums of squares is the extra regression sum of squares due to the second-order terms. The residual sum of squares is calculated as before using the second-order model of Eq. (69) the lack-of-fit and pure-error sums of squares are thus the same as in Table IV. The ratio contained in Eq. (68) still tests the adequacy of Eq. (69). Since the ratio of lack-of-fit to pure-error mean squares in Table VII is smaller than the F statistic, there is no evidence of lack of fit hence, the residual mean square can be considered to be an estimate of the experimental error variance. The ratio... 

An intrinsic parameter is one that is inherently present in or arises naturally from a reaction-rate model. These parameters, which are of a simpler functional form than the entire rate model, facilitate the experimenter s ability to test the adequacy of a proposed model. Using these intrinsic parameters, this section presents a method of preparing linear plots for high conversion data, which is entirely analogous to the method of the initial-rate plots discussed in Section II. Hence, these plots provide a visual indication of the ability of a model to fit the high conversion data and thus allow a more... 

Before the analysis is complete, we must examine the adequacy of the model to fit the data. Since this procedure is described in Section IV as well as in Reference (B14), this aspect of the fitting will not be pursued here. We have simply demonstrated here how the experimenter can be led from an inadequate model to an adequate model by the data analysis. 

A comparison of the predicted rates of Eqs. (7) and (135) is shown in Fig. 24. Since the bulk of the data were taken above 5 % propylene, it is apparent that both models fit reasonably well. However, at lower concentrations of propylene, Eq. (135) will deviate widely from the data. Undoubtedly, deviations will also occur when extrapolating other directions from the data base as well. Checks on the adequacy of the transformation and calculations of the confidence regions for all parameters may also be carried out (B11, B17, H2). 

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 approach described above is by no means complete or exclusive. For example, Lamb et al. (1975) have proposed an alternative route to assess the adequacy of the atmospheric diffusion equation. Their approach is based on the Lagrangian description of the statistical properties of nonreacting particles released in a turbulent atmosphere. By employing the boundary layer model of Deardorff (1970), the transition probability density p x, y, z, t x, y, z, t ) is determined from the statistics of particles released into the computed flow field. Once p has been obtained, Eq. (3.1) can then be used to derive an estimate of the mean concentration field. Finally, the validity of the atmospheric diffusion equation is assessed by determining the profile of vertical dififiisivity that produced the best fit of the predicted mean concentration field. 

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)]. 

Plots of Residuals. Residuals can be plotted in many ways overall against a linear scale versus time that the observations were made versus fitted values versus any independent variable (3 ). In every case, an adequate fit provides a uniform, random scatter of points. The appearance of any stematic trend warns of error in the fitting method. Figures 4 and 5 shows a plot of area versus concentration and the associated plot of residuals. Also, the lower part of Figure 2 shows a plot of residuals (as a continuous line because of the large number of points) for the fit of the Gaussian shape to the front half of the experimental peak. In addition to these examples, plots of residuals have been used in SBC to examine shape changes in consecutive uv spectra from a diode array uv/vis spectrophotometer attached to an SBC euid the adequacy of linear calibration curve fits (1). 

While this review of the literature shows increases in trends in the use of medications across all classes, most questions about the adequacy of treatment practices, and the fit between diagnosis and treatment have not been addressed by most studies to date (with the possible exceptions of Jensen et ah, 1999b, and Angold et ah, 2000). Consequently, strong conclusions about the meaning of the available findings are not possible, and the extant data leave several major questions unanswered. 

A global multiresponse non-linear regression was performed to fit Eq. (57) to all the runs with both 2% and 6% v/v 02 feed content to obtain the estimates of the kinetic parameters (Nova et al., 2006a). Figure 37 (solid lines) illustrates the adequacy of the global fit of the TRM runs with 2 and 6% 02 the MR rate law can evidently capture the complex maxima-minima NO and N2 traces (symbols) at low T at both NH3 startup, that a simple Eley-Rideal (ER), approach based on the equation... 

Here, we want to emphasize that one is able to calculate the fraction of the experimental error only if replicate measurements (at least at one point x ) have been taken. It is then possible to compare model and experimental errors and to test the sources of residual errors. Then, in addition to the GOF test one can perform the test of lack of fit, LOF, and the test of adequacy, ADE, (commonly used in experimental design). In the lack of fit test the model error is tested against the experimental error and in the adequacy test the residual error is compared with the experimental error. 

A probability distribution is a mathematical description of a function that relates probabilities with specified intervals of a continuous quantity, or values of a discrete quantity, for a random variable. Probability distribution models can be non-parametric or parametric. A non-parametric probability distribution can be described by rank ordering continuous values and estimating the empirical cumulative probability associated with each. Parametric probability distribution models can be fit to data sets by estimating their parameter values based upon the data. The adequacy of the parametric probability distribution models as descriptors of the data can be evaluated using goodness-of-fit techniques. Distributions such as normal, lognormal and others are examples of parametric probability distribution models. 

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