Goodness-of-fit test

The chi-square distribution can be applied to other types of apph-catlon which are of an entirely different nature. These include apph-cations which are discussed under Goodness-of-Fit Test and Two-Way Test for Independence of Count Data. In these applications, the mathematical formulation and context are entirely different, but they do result in the same table of values. 

The value of the eorrelation eoeffieient using the least squares teehnique and the use of goodness-of-fit tests (in the non-linear domain) together probably provide the means to determine whieh distribution is the most appropriate (Keeeeioglu, 1991). However, a more intuitive assessment about the nature of the data must also be made when seleeting the eorreet type of distribution, for example when there is likely to be a zero threshold. 

As regards the x-ray data, the conclusion in the text was reached by applying the chi-square goodness-of-fit test. See C. A. Bennett and N. L. Franklin, Statistical Analysis, page 620, for details of the test. 

Numerical solutions of such problems cause some difficulties during the course of many methods in connection with nonlinearity and tendency to infinity of the partial derivatives at the front of the temperature way. It is hoped that the exact solutions obtained in such a way help motivate what is done and could serve in practical implementations as goodness-of-fit tests. 

Two issues present themselves when the question of PB-PK model validation is raised. The first issue is the accuracy with which the model predicts actual drug concentrations. The actual concentration-time data have most likely been used to estimate certain total parameters. Quantitative assessment, via goodness-of-fit tests, should be done to assess the accuracy of the model predictions. Too often, model acceptance is based on subjective evaluation of graphical comparisons of observed and predicted concentration values. 

Goodness-of-fit tests may be a simple calculation of the sum of squared residuals for each organ in the model [26] or calculation of a log likelihood function [60], In the former case,... 

Frequency domain performance has been analyzed with goodness-of-fit tests such as the Chi-square, Kolmogorov-Smirnov, and Wilcoxon Rank Sum tests. The studies by Young and Alward (14) and Hartigan et. al. (J 3) demonstrate the use of these tests for pesticide runoff and large-scale river basin modeling efforts, respectively, in conjunction with the paired-data tests. James and Burges ( 1 6 ) discuss the use of the above statistics and some additional tests in both the calibration and verification phases of model validation. They also discuss methods of data analysis for detection of errors this last topic needs additional research in order to consider uncertainties in the data which provide both the model input and the output to which model predictions are compared. 

The existence of a normal distribution can only be confirmed by a goodness-of-fit test (e.g., yf, according to Kolmogoroff [1941] and Smirnoff [1948]). 

The Statistical Model. The residue levels of the individual specimens in a particular subpopulation (e.g., a given Census Division and age, sex, race category) are assumed to follow a lognormal distribution. Previous studies on NHATS data have found the lognormal distribution to be appropriate and goodness of fit tests performed on the collected data verified that the assumption is still reasonable. The lognormal model assumes only non-negative values and allows the variances of the different subpopulation distributions to increase with the mean levels. This distribution is commonly used to model pollutant levels in the environment. 

The BMC approach can provide a more refined assessment of the prediction of the empirical NOAEL. It must be emphasized that even the empirical NOAEL may represent a response level that is not detected. When 5 to 10 animals are used in an experiment, a 10-20% response can be missed (Leisenring and Ryan 1992) and even a BMCio is similar to a LOAEL with dichotomized data (Gaylor 1996). It is expected that the BMC is less than the empirical LOAEL. In the Fowles et al. (1999) analysis of the data, the BMCqs and BMCoi values were always below the empirical LOAEL for the studies analyzed. The probit analysis of the data by Fowles et al. (1999) provided a better fit with the data as measured by the chi-squared goodness-of-fit test, mean width of confidence intervals, and number of data sets amenable to analysis by the model. ... 

In the future, the BMCqs and MLEqi for lethality will be determined, presented, and discussed. Results from the above models will be compared with the log probit EPA (2000) benchmark dose software (http //www.epa.gov/ncea/ bmds.htm). In all cases, the MLE and BMC at specific response levels will be considered. Other statistical models such as the Weibull may also be considered. Since goodness-of fit-tests consider an average fit, they may not be valid predictors of the fit in the low-exposure region of interest. In this case, the output of the different models will be plotted and compared visually with the experimental data to determine the most appropriate model. The method that results in values consistent with the experimental data and the shape of the exposure-response curve will be selected for AEGL derivations. 

Two examples are treated here one on multicomponent diffusion and one on reaction kinetics. Parameter estimation and model discrimination are demonstrated, along with goodness-of-fit testing when replicates are available. 

The numerical results, summarized below, show the fourth model to be the most probable one according to the data, and also show that this model does better than the others on the goodness-of-fit test. These results are consistent with those of Stewart, Henson, and Box (1996), who found this model to be the most probable a posteriori of the 18 models considered by Tschernitz et al. (1946). The linearized model forms used by Tscher-nitz et al. yield the same conclusion if one uses the appropriate variable weighting for each linearized model form. 

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