Statistics method comparison data model

In this paper, we discuss studies based on comparison with background measurements that may have a skew distribution. We discuss below the design of such a study. The design is intended to insure that the model for the comparison is valid and that the amount of skewness is minimized. Subsequently, we present a statistical method for the comparison of the background measurements with the largest of the measurements from the suspected region. This method, which is based on the use of power transformations to achieve normality, is original in that it takes into account estimation of the transformation from the data. 

A couple of years ago a workshop was organized to compare the performance of the various statistical methods applied for receptor model (12). To create an objective basis for the comparison of the different analyses, a synthetic data set was generated according to the following equation ... 

Strictly speaking, a lifetime prediction should be done with a specified failure probability and confidence level, see for instance Ronold and Echtermeyer [30], or Sutherland and Veers [31], but for the sake of simplicity in comparing the mean values of different models to the experimentally determined mean lifetime, these statistics and the associated myriad of possibilities for the choice of distribution types and statistical methods have been omitted. Moreover, for the comparison of models, comparing the mean of the data is sufficient. 

The purpose of statistical evaluation of sample data is to extrapolate from a collection of individual events (e.g., 30 min of process time) to the entire population of events (e.g., 8-h shift). Because microbial monitoring data usually measure the impact of human activity, which is not reproducible exactly from one event to the next, results usually do not fit standard statistical models for normal distributions. In spite of this limitation, it is necessary to summarize the data for comparison to limits. The best statistical methods of evaluation are determined by the nature of the data. Wilson suggests that microbial monitoring data histograms generally resemble Poisson or negative... 

The derivation of functional relationships between independent variables, i.e., a concentration or amount proportional quantity and dependent variables - the response - belongs to the daily work of an analytical chemist. The functional relation has to be established in the calibration step and the concentration of an unknown sample can be estimated by its inverse application. Really both the dependent and independent variables are superimposed by error. Statistical methods accounting for errors in both responses (y) and concentrations (x) can hardly be applied if only a small sample size is available because the estimates become poor. Furthermore, in comparison to the Bayesian approach the incorporation of prior knowledge or subjective aspects with respect to the uncertainty of the data is carried out more easily by fuzzy methods. Results relying on the Bayesian approach can be doubtful if standard model assumptions do not hold. ... 

This comparison is performed on the basis of an optimality criterion, which allows one to adapt the model to the data by changing the values of the adjustable parameters. Thus, the optimality criteria and the objective functions of maximum likelihood and of weighted least squares are derived from the concept of conditioned probability. Then, optimization techniques are discussed in the cases of both linear and nonlinear explicit models and of nonlinear implicit models, which are very often encountered in chemical kinetics. Finally, a short account of the methods of statistical analysis of the results is given. 

It is noteworthy that comparisons of existing assessment schemes reveal dissimilarities in the use of extrapolation methods and their input data between different jurisdictions and between prospective and retrospective assessment schemes. This is clearly apparent from, for example, a set of scientific comparisons of 5% hazardous concentration (HC5) values for different substances. Absolute HC5 values and their lower confidence values were different among the different statistical models that can be used to describe a species sensitivity distribution (SSD Wheeler et al. 2002a). As different countries have made different choices in the prescribed modeling by SSDs (regarding data quality, preferred model, etc.), it is clear that different jurisdictions may have different environmental quality criteria for the same substance. Considering the science, the absolute values could be the same in view of the fact that the assessment problem, the available extrapolation methods, and the possible set of input data are (scientifically) similar across jurisdictions. When it is possible, however, to look at the confidence intervals, the numerical differences resulting from different details in method choice become smaller because confidence intervals show overlap. 

Fig. 2. Method overview Ligand sets derived from existing databases (a) are used in set-wise comparisons (b) against a query set, the result of which is quantified by the statistical model inferred from that reference database (c).The generated probabilistic data can be used to construct chemical mappings of the ligand sets and correspondingly the biological targets (d).

Roe, D.J. Comparison of population pharmacokinetic modeling methods using simulated data Results from the population modeling workgroup. Statistics in Medicine 1997 16 1241-1262. 

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