Statistics data extrapolation

The model was forced with agricultural application data of the insecticide DDT compiled by Semeena and Lammel (2003). Statistical data of DDT consumption reported by member of the UN states to Food and Agriculture Organisation (FAO) were combined with other published data (details in Semeena and Lammel (2003)). The emission inventory assumed 100 % of p,p -DDT. After scaling the DDT consumption with crop land distribution, the data were extrapolated to the model grid. The result was a data set with spatially and temporally varying applications (accumulated application and temporal evolution shown in Figure 3.1). No seasonal or diurnal variation of the applications is considered. 

Quality assessments are recorded as statistical data, which are used to maintain or control the specific process from which they were obtained. In the cast magnesium rotor example, if casting flaws frequently occurred in a specific location on the rotor, the casting process would be adjusted to eliminate the flaw or reduce the frequency of its occurrence. In other processes with high throughput (output over an extended period of time), it is not feasible to examine every single unit. In such cases, a random selection of individual outputs is tested and their conformance to the ideal is extrapolated to the entire output. This method relies on the output history as the basis for comparison, and variances in the output are tracked very closely to ensure that, overall, the individual components of the output stream remain within the parameters set in the design standard. 

From UNHCR figures, it is estimated that, at the endof 2011, no more than 3 per cent of all refiigees, Le.some 315,000 for a population of 10,400,000, were aged 60 years or more (UNHCR 2012).2 The same figures can be extrapolated for the internally displaced and for other persons in refugee-like situations. At the outset, it appears to be a very low proportion moreover, one should bear in mind that the proportion of older refiigees is a dynamic phenomenon. It should not be reduced to a matter of statistical data, whether accurate or inaccurate, but rather should be tackled as a qualitative reflection of a series of factors—such as when, for example, exile took place, at what age the person was exiled, and for how long. 

The primary purpose for expressing experimental data through model equations is to obtain a representation that can be used confidently for systematic interpolations and extrapolations, especially to multicomponent systems. The confidence placed in the calculations depends on the confidence placed in the data and in the model. Therefore, the method of parameter estimation should also provide measures of reliability for the calculated results. This reliability depends on the uncertainties in the parameters, which, with the statistical method of data reduction used here, are estimated from the parameter variance-covariance matrix. This matrix is obtained as a last step in the iterative calculation of the parameters. 

From a statistical viewpoint, there is often little to choose between power law and hyperbohc equations as representations of data over an experimental range. The fact, however, that a particular hyperbolic equation is based on some land of possible mechanism may lead to a belief that such an equation may be extrapolated more safely outside the experimental range, although there may be no guarantee that the controlling mechanism will remain the same in the extrapolated region. 

Fatal reactions to Hymenoptera stings are rare they range from 0.09 to 0.48 per million inhabitants and year [31, 32]. However, the true number may be underestimated in one study, specific IgE antibodies to Hymenoptera venoms were detected in 23% of postmortem serum samples from patients who died outdoors from unknown reasons [33]. Between 1961 and 2004, 140 fatal Hymenoptera sting reactions were registered by the federal administration for statistics in Switzerland with about 7.5 million inhabitants, resulting in an average annual fatality rate of 3.18. If these data are extrapolated to Europe with a population of around 500 million, the annual death rate in Europe would amount to about 200. 

A first evaluation of the data can be done by running nonparametric statistical estimation techniques like, for example, the Nadaraya-Watson kernel regression estimate [2]. These techniques have the advantage of being relatively cost-free in terms of assumptions, but they do not provide any possibility of interpreting the outcome and are not at all reliable when extrapolating. The fact that these techniques do not require a lot of assumptions makes them... 

Alternatively, methods based on nonlocal projection may be used for extracting meaningful latent variables and applying various statistical tests to identify kernels in the latent variable space. Figure 17 shows how projections of data on two hyperplanes can be used as features for interpretations based on kernel-based or local methods. Local methods do not permit arbitrary extrapolation owing to the localized nature of their activation functions. 

The following example is based on a risk assessment of di(2-ethylhexyl) phthalate (DEHP) performed by Arthur D. Little. The experimental dose-response data upon which the extrapolation is based are presented in Table II. DEHP was shown to produce a statistically significant increase in hepatocellular carcinoma when added to the diet of laboratory mice (14). Equivalent human doses were calculated using the methods described earlier, and the response was then extrapolated downward using each of the three models selected. The results of this extrapolation are shown in Table III for a range of human exposure levels from ten micrograms to one hundred milligrams per day. The risk is expressed as the number of excess lifetime cancers expected per million exposed population. 

H.A. Guess and K.S. Crump. "Low Dose Extrapolation of Data from Animal Carcinogenicity Experiments - Analysis of a New Statistical Technique." Math. Biosciences, 32, 1976, pp. 15-36. 

The mathematical methods used for interpolation and extrapolation of the data obtained from accelerated tests, as described in Chapters 8 and 9, include both the mechanistic and the empirical. Arrhenius formula, based on chemical rate kinetics and relating the rate of degradation to temperature, is used very widely. Where there are sufficient data, statistical methods can be applied and probabilities and confidence limits calculated. For many applications a high level of precision is unnecessary. The practitioners of accelerated weathering are only too keen to tell you of its quirks and inaccuracies, but this obscures... 

Our statistical analysis reveals a large improvement from cc-pCV(DT)Z to cc-pCV(TQ)Z see Fig. 1.4. In fact, the cc-pCV(TQ)Z calculations are clearly more accurate than their much more expensive cc-pcV6Z counterparts and nearly as accurate as the cc-pcV(56)Z extrapolations.The cc-pCV(TQ)Z extrapolations yield mean and maximum absolute errors of 1.7 and 4.0 kJ/mol, respectively, compared with those of 0.8 and 2.3 kJ/mol at the cc-pcV(56)Z level. Chemical accuracy is thus obtained at the cc-pCV(TQ)Z level, greatly expanding the range of molecules for which ab initio electronic-structure calculations will afford thermochemical data of chemical accuracy. 

The latter danger is, of course, potentially present any time any data interpretation is attempted, particularly if nature is assumed always to follow Eq. (1). The only course of action is to attempt to include as much theory in the model as possible, and to confirm any substantial extrapolation by experiment. It is erroneous, however, to presume that kinetic data will always be so imprecise as to be misleading. The use of computers and statistical analyses for any linear or nonlinear reaction rate model allows rather definite statements about the amount of information obtained from a set of data. Hence, although imprecision in analyses may exist, it need not go unrecognized and perhaps become misleading. 

Because it is derived from the upper confidence bound on risk, the BMD is actually the lower confidence bound on the dose corresponding to a 10% risk. Statistical confidence bounds are used to account for expected variability in observed data. Their use adds an element of additional caution to the extrapolation process. See later. 

However, there are prices to pay for the advantages above. Most empirical modeling techniques need to be fed large amounts of good data. Furthermore, empirical models can only be safely applied to conditions that were represented in the data used to build the model (i.e., extrapolation of such models is very dangerous). In addition, the availability of multiple response variables for building a model results in the temptation to overfit models, in order to obtain artificially optimistic results. Finally, multivariate models are usually much more difficult to explain to others, especially those not well versed in math and statistics. 

Contrary to the practical results reviewed above, statistics from correlation work revealed a serious deficiency in the accuracy with which Phase I Equations 3 and 4 predicted -for the Phase II dataset r for Equation 3 predictions for the 103 compound Phase II data was only 0.45 r for Equation 4 predictions for the Phase II dataset was only 0.44. An analysis of the residuals for the Phase II dataset [Potency(observed)-Potency(predicted by Phase I models)] immediately Identified the source of the problem of the 26 Phase II compounds having DICARB >4, 17 had potency for adult observed more than one log unit better than predicted 15 had egg potency observed more than one log unit better than predicted. As schematically shown in Figure 2B, the parabolic functions for DICARB for the Phase I models underpredict at values of DICARB extrapolated beyond those represented in the Phase I dataset. 

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