Statistical Fits

It is apparent that there are a considerable number of parameters to be determined. According to equation (8) and equations (2-7) there are 6N+2 parameters where N is the number of relaxations present (it is not 8N because the relaxed modulus of one process is equal to the unrelaxed modulus of the next process in a sequence). In practice, it is found that with the large number of experimental points available in a scan (typically 50-100) the determinaton usually proceeds satisfactorily. However, in coitimon with many statistical fitting situations, it can happen that parameter determination is not unique. Our experience has shown that problems can arise when the relaxation strength is small or when only part of a peak is recorded. The problem with small relaxaton strength is associated with equation (1) where it is seen that the activation energy is related to the ratio of peak area and relaxation strength E(j- Ep. When the process is quite... 

Boric Acid. The statistical fits for the different rate expressions (Table III) continued to give high r values for the cellulose/H BC samples. Unlike the control samples, the 2nd-order equation generally gave the statistically best fit. 

The relationship between spot sales price and spot sales quantity can be modeled as a linear function within the feasible minimum and maximum quantities defined by the management of the company. Of course, the price-quantity relationship could also be modeled using a non-linear function depending on the actual price-quantity-bids the company receives. In this work the linear function showed a sufficient statistical fit based on the real data provided by the industry case. 

CHEMRev The Comparison of Detailed Chemical Kinetic Mechanisms Forward Versus Reverse Rates with CHEMRev, Rolland, S. and Simmie, J. M. Int. J. Chem. Kinet. 37(3), 119-125 (2005). This program makes use of CHEMKIN input files and computes the reverse rate constant, kit), from the forward rate constant and the equilibrium constant at a specific temperature and the corresponding Arrhenius equation is statistically fitted, either over a user-supplied temperature range or, else over temperatures defined by the range of temperatures in the thermodynamic database for the relevant species. Refer to the website http //www.nuigalway.ie/chem/c3/software.htm for more information. 

A more recent study for estimating Henry s law constants using the bond contribution method was provided by Meylan and Howard (1991). In this study, the authors updated and revised the method developed by Hine and Mookeijee (1975) based on new experimental data that have become available since 1975. Bond contribution values were determined for 59 chemical bonds based on known Henry s law constants for 345 organic compounds. A good statistical fit [correlation coefficient (r ) = 0.94] was obtained when the bond contribution values were regressed against known Henry s law constants for all compounds. For selected chemicals classes, r increased slightly to 0.97. 

In addition, mean sequence lengths may be useful in evaluating statistical fits. For example. 

Table VI. Statistical fitness of data of corrosion of Copper, Steel, Nickel and Tin inside a metallic box (heat trap conditions) respecting time and TOW according to ISO...

Multiple regression analysis was applied to the four species to seek a best statistical fit of the form ... 

Quantitative models for predicting quality can be classified into two categories (1) fundamental process models, which are based on physical and chemical events that occur in the autoclave, and (2) regression-type models, which are based on a statistical fit of the observed product quality to the input raw material properties and the process conditions used. 

Parameters determined from statistical fit of experimental data. 

For noncarcinogenic hazardous chemicals, NCRP believes that the threshold for deterministic effects in humans should be estimated using EPA s benchmark dose method, which is increasingly being used to establish allowable doses of noncarcinogens. A benchmark dose is a dose that corresponds to a specified level of effects in a study population (e.g., an increase in the number of effects of 10 percent) it is estimated by statistical fitting of a dose-response model to the dose-response data. A lower confidence limit of the benchmark dose (e.g., the lower 95 percent confidence limit of the dose that corresponds to a 10 percent increase in number of effects) then is used as a point of departure in establishing allowable doses. 

Fig. 3.6. Illustration of use of benchmark dose method to estimate nominal thresholds for deterministic effects in humans. The benchmark dose (EDio) and LEDi0 are central estimate and lower confidence limit of dose corresponding to 10 percent increase in response, respectively, obtained from statistical fit of dose-response model to dose-response data. The nominal threshold in humans could be set at a factor of 10 or 100 below LED10, depending on whether the data are obtained in humans or animals (see text for description of projected linear dose below point of departure).

Sorption Prediction Equations. Equations predicting radioelement distribution coefficients, K s, as arithmetic functions of component concentrations were obtained for sorption of strontium, neptunium, plutonium, and americium on two Hanford sediments. These equations, presented in Table VH and derived from statistical fits of Box-Behnken experimental designs, were generated for strontium in terms of sodium ion, HEDTA, and EDTA concentrations. Prediction equations for neptunium and plutonium sorption were derived from NaOH, NaA102, HEDTA, and EDTA concentrations. Americium sorption prediction equations were based on NaOH, HEDTA, and EDTA concentrations. 

Values for AE obtained from this analysis are plotted versus applied magnetic field at 2 and 4.2 K for these Mn2+ CdS nanocrystals in Fig. 32. AE was then analyzed using a model in which spatial distributions of Mn2+ ions within nanocrystals as well as the reduced magnetization from dimer superexchange up to the third-nearest-neighbor shell were taken into account explicitly. Dopants were assumed to be distributed throughout the nanocrystals statistically. Fitting the data within this model (dashed lines in Fig. 32) yielded a Mn2+ concentration of 0.16%. From this concentration, the QDs were estimated to contain an... 

Figure 5.1 (New PSD and data generated from traditional PSD functions). Table 5.1 (Parameters and statistical fits).

It is important to note that validation should not be confused with the statistical fit of a QSAR. The latter can be assessed by many easily available statistical terms (e.g., R2(adj), s, F, etc.), which reflect the ability of the QSAR to mimic the data. While a poor statistical fit to a QSAR results in a model with little or no predictive value, a significantly good statistical fit does not necessarily imply that the QSAR will predict toxic potency accurately for an untested compound. 

Also important to the validation process of QSARs is vertical validation. In this instance, quantitatively similar QSARs are developed with similar descriptors but using data for a different toxic endpoint. For example, the investigation of Karabunarliev et al. (1996b) modeled acute aquatic toxicity data for the fathead minnow Pimephales promelas. The compounds considered in the analysis were confined to substituted benzenes, and descriptors limited to log Kow and Amjx. The fish toxicity QSAR (log [LQ,]-1 = 0.62 log K, + 9.17 A - 3.21 n = 122 R2 = 0.83 i = 0.16 F = 292) of Karabunarliev et al. (1996b) was very similar in terms of slope, intercept, and statistical fit to the QSAR presented in Equation 12.2. The fact that different endpoints provide very similar QSARs indicates that the QSAR is valid across protocols. This shows the universality of the model. 

To validate the model presented by Equation 12.4, the 214 compounds included in equation (12.3) were used. For this second validation the model between observed and predicted toxicity showed r2 = 0.886, s = 0.379, and a slope of 1.091 ( 0.027). Because both the intercepts were very close to one, it may be assumed that both the models are capable of predicting correctly the toxicity of new compounds with descriptors within the defined ranges. It can be noted that Equation 12.3 performs a little better than Equation 12.4 because of the higher R2, lower s, and slope that is closer to 1 in the statistical fit between the observed and predicted toxicity. To take advantage of the availability of data for larger number of compounds, the model presented by Equation 12.5 was developed. The slopes of the regression line between the observed toxicity values and those calculated by Equation 12.5 for both the subsets (n = 174 and n = 214) were 0.959 ( 0.017) and 1.036 ( 0.024), respectively. Because this is not a real validation with external data sets, the validity of the latter model needs to be additionally demonstrated. 

As described in several chapters of the present book [1,7], the application of pharmacokinetic (PK) and pharmacodynamic (PD) methods is widely accepted in the pharmaceutical industry. A PK model typically predicts the availability of a drug in the blood and interstitial spaces at different times after the drug has been administered. The model is used to determine characteristic parameters of the absorption, distribution, metabolism, and excretion processes from experimentally observed time courses, or the model follows the rates of formation and removal of various metabolites. PD models describe the effects of the drug (and its metabolites) as a function of time, again based on statistical fits to experimental results. 

Using a one-dimensional Monte Carlo analysis to estimate population exposure and dose uncertainty distributions for particulate matter, where model inputs and parameters (e.g. ambient concentrations, indoor particulate matter emission rates from environmental tobacco smoke, indoor air exchange rates, building penetration values, particle deposition rates) are represented probabilistically with distributions statistically fitted to all available relevant data. 

An excellent statistical fit to data is therefore insufficient to render a packing pressure drop correlation suitable for design. In addition to a good fit to data, the correlation limitations must be fully explored. Most published packing pressure drop correlations fail miserably here their limitations are often unknown, and if known, are seldom reported. 

Although the above method can give a simple evaluation of Peclet number for the system, the tailing in the RTD curve can cause significant inaccuracy in the evaluation of the Peclet number. Michell and Furzer67 suggested that a better estimation of the axial dispersion coefficient is obtained if the observed RTD is statistically fitted to the exact solution of the axial dispersion model equation with appropriate boundary conditions. For example, a time-domain solution to the partial differential equation describing the dispersion model, i.e.,... 

Figure 8. The scatter in the loss tangent data for 87-1 poly(urea-urethane) makes identification of the onset of gelation or hard segment vitrification impossible. Although we have drawn peaks in some of the isotherms. Statistical fits do not justify them.

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