Goodness-of-Fit Parameters

Y data. The data set used for this example is from Miller and Miller ([1], p. 106) as shown in Table 58-1. This dataset is used so that the reader may compare the statistics calculated and displayed using the formulas and figures described in this reference with respect to those shown in this series of chapters. The correlation coefficient and other goodness of fit parameters can be properly evaluated using standard statistical tests. The Worksheets provided in this chapter series can be customized for specific applications providing the optimum information for particular method comparisons and validation studies. 

The first part of the output contains the principal component analysis of the correlation matrix discussed later in Section 3.5. In addition to the residuals, goodness-of-fit, parameter estimates and bounds, the Durbin-Wattson D statistics is also printed by the module. 

The goodness of fit parameter will approach a minimum which should correspond to the real error in the absorbance data and increase when a noise eigenvalue is added. Another test is the indicator function which also passes through a minimum when the correct number of eigenvectors are used. The indicator function is given by... 

With a four-phase model, the results are shown in Table 7.1 where the mean square error (MSE) represents the goodness-of-fit parameter. The uncertainties of the film thickness listed in Table 7.1 were obtained with a 90% confidence limit. To verify that the dispersion in the experimental spectral range (1.55-6.53 eV) is correct, we used the data inversion to obtain its pseudo-optical constants (< n ) and ). Theseconstants are calculated from the measured 4 and A using the film thickness obtained from the fitting. 

Importantly, the goodness-of-fit parameters are satisfactory and the coefficient of the term involving (M +tolal) is near unity in all cases. Also, it must be noted that the stability constants for the alkali cation-benzocrown ether interaction available in the literature [30,48] agree closely with the values extracted from Fig. 5 for the alkali cation (14) association. All of this very gratifyingly support the expectations of the fluorescent PET sensor design logic in terms of a supermolecule with modular behaviour [43],... 

The kinetic traces are obtained from stopped-flow experiments and fit to a single-exponential or to a sum-of-exponentials model. Strictly speaking, one does not know beforehand how many exponentials will be required to describe adequately a particular kinetic curve, so that one has to perform several separate fits, each using a different number of such terms. We therefore need criteria to use in choosing which model best describes the data. The standard ones are the form of (i) the residuals (ii) the autocorrelation function values and (iii) the value (statistical goodness-of-fit parameter). 

Figure 16. Decay-associated spectrum resulting from a global analysis of the decay of the porphyrin fluorescence of pentad 22 following excitation of a — lxlO M chloroform solution with a 590-nm laser pulse. Data were obtained using the single photon counting technique, and the instrument response time was 0.035 ns. Decays at 14 different wavelengths were analyzed simultaneously, and the goodness of fit parameter was 1.12.

The practical data analysis techniques for us include iterative methods in which estimates for the parameters are refined imtil we get an acceptable fit. The quality of the fit is judged by how closely the calculated fluorescence decay (generated from the instrument response function and an assumed biexponential decay law) and the observed fluorescence decay match in a least-squares sense. The goodness-of-fit parameter is defined as... 

Equally important is to check the descriptor set for multicolinearity. Correlation between descriptor values results in unreliable MLR with overestimate goodness-of-fit parameters and poor predictive capability. Crosscorrelation matrices provide information on descriptor multicolinearity. It is worth mentioning that when two descriptors, X and Z, are statistically correlated, it does not necessarily mean that physicochemically they are also redundant. Principal component regression or partial least squares regression can be used to address multicolinearity. Alternatively, the impact of descriptors X and Z on the QSRR should be inspected separately. It should be possible to select from the solute set those solutes with varying X values and constant Z values and vice versa. 

Dyar MD (1986) Practical application of Mossbauer goodness-of-fit parameters for evaluation of real experimental results A reply. Am Mineral 71 1266-1267 Dyar (1987) A review of Mossbauer data on trioctahedral micas evidence for tetrahedral Fe and cation ordering. Am Mineral 72 102-112... 

Dyar MD (1989) Applications of Mossbauer goodness-of-fit parameters to experimental spectra Further discussion. Am Mineral 74 688... 

Some endpoints cannot be measured on a continuous scale, or for scientific or regulatory purposes can only be divided into two or more qualitative classes, for instance active/inactive. Goodness of fit can be described by the Cooper statistics, introduced to assess the significance of carcinogenicity tests.The goodness of fit parameters for a 2-level classification are shown in Table 9.3. This table is often, and very aptly, entitled a confusion matrix. 

The parameters (oti and v) are varied to yield the best fit between the data and the calculated values, as indicated by a minimum value for the goodness-of-fit parameters xi. 

The simulations in Figure 14.22 and data for a flexible and difltising D-A pair in Figure 14.24 provide a good example of how the ability to determine a parameter value depends on the infixmation content of the data. Recall that parameter values are often correlated, so that the values of the goodness-of-fit parameter xi depends on the entire set of parameters, not just the value of a single parameter. 

For extracting the frequency content of a TF spectrum, the maximiun entropy method has been described by Rainford and Daniell (1994) and Alves et al. (1994) as an unproved technique compared to the usually employed Fourier transform. Aspects of the treatment of transverse field (J.SR data (with emphasis on pulsed beam measurements) have been discussed recently by Rainford (1999b). In the experimental spectra shown in this article, the solid line through the data points is the least-squares fitted theoretical function least-squares fitting. 

Nonlinear least squares method allows to compare the measured data N(tk) with values predicted Ndtk) from a model and the parameters of the model to be varied to yield the minimum deviation from the data through minimization of the goodness-of-fit parameter calculated from... 

Cp = 2.83 (0.01) mF, and /fad = 0, where the numbers in parentheses denote the standard deviations of the parameters. The fit was carried out in the frequency range 100 kHz to 0.4 Hz and the goodness-of-fit parameter is = 7.5 x 10. At lower frequencies deviations appear. The obtained results indicate that both models might be indistinguishable in practice, although the CPE model contains fewer adjustable parameters. 

Instead of dl, the R factors similar to those commonly applied in X-ray structure analysis, may be used as goodness-of-fit parameters. The conventional and weighted R factors are defined as... 

The output from the modeling tools is often not used to the extent possible. Often, the FITEQL results used are the numerical values of the optimized parameters and the overall goodness of fit. Sometimes, also the standard deviations are considered. The numerical value of the goodness-of-fit parameter and the standard deviations are dependent on the defined experimental error estimates. The values for these, which are most frequently used, are the standard values, which may not at all be reasonable for the actual equilibrium problem treated [36]. 

---