Statistics nonparametric method

Quantitative methodology uses large or relatively large samples of subjects (as a rule students) and tests or questionnaires to which the subjects answer. Results are treated by statistical analysis, by means of a variety of parametric methods (when we have continuous data at the interval or at the ratio scale) or nonparametric methods (when we have categorical data at the nominal or at the ordinal scale) (30). Data are usually treated by standard commercial statistical packages. Tests and questionnaires have to satisfy the criteria for content and construct validity (this is analogous to lack of systematic errors in measurement), and for reliability (this controls for random errors) (31). 

Current methods for supervised pattern recognition are numerous. Typical linear methods are linear discriminant analysis (LDA) based on distance calculation, soft independent modeling of class analogy (SIMCA), which emphasizes similarities within a class, and PLS discriminant analysis (PLS-DA), which performs regression between spectra and class memberships. More advanced methods are based on nonlinear techniques, such as neural networks. Parametric versus nonparametric computations is a further distinction. In parametric techniques such as LDA, statistical parameters of normal sample distribution are used in the decision rules. Such restrictions do not influence nonparametric methods such as SIMCA, which perform more efficiently on NIR data collections. 

The statistical software systems used for analysis of clitucal trial data can range from custom programs for specific statistical techniques to COTS packages. Such packages (e g, the SAS system, SPSS, S-Plus) provide the user with a library of statistical procedures (e.g., analysis of variance, regression, generahzed linear modelling, nonparametric methods) which can be accessed either by... 

The presently used statistical method is the confidence interval approach. The main concern is to rule out the possibility that the test product is inferior to the comparator pharmaceutical product by more than the specified amount. Hence a one-sided confidence interval (for efficacy and/or safety) may be appropriate. The confidence intervals can be derived from either parametric or nonparametric methods. 

Statistical methods are based on specific assumptions. Parametric statistics, those most familiar to the majority of scientists, have more stringent underlying assumptions than do nonparametric statistics. Among the underlying assumptions for many parametric statistical methods (such as the analysis of variance) is that the data are continuous. The nature of the data associated with a variable (as described previously) imparts a value to that data, the value being the power of the statistical tests which can be employed. 

The analysis of rank data, what is generally called nonparametric statistical analysis, is an exact parallel of the more traditional (and familiar) parametric methods. There are methods for the single comparison case (just as Student s t-test is used) and for the multiple comparison case (just as analysis of variance is used) with appropriate post hoc tests for exact identification of the significance with a set of groups. Four tests are presented for evaluating statistical significance in rank data the Wilcoxon Rank Sum Test, distribution-free multiple comparisons, Mann-Whitney U Test, and the Kruskall-Wallis nonparametric analysis of variance. For each of these tests, tables of distribution values for the evaluations of results can be found in any of a number of reference volumes (Gad, 1998). 

Hollander, M. and Wolfe, D.A. (1973). Nonparametric Statistical Methods. Wiley, New York, pp. 124-129. 

Like other statistical methods, the user has to be careful with the requirements of a statistical test. For many statistical tests the data have to follow a normal distribution. If this data requirement is not fulfilled, the outcome of the test can be biased and misleading. A possible solution to this problem are nonparametric tests that are much less restrictive with respect to the data distribution. There is a rich literature on... 

Hollander, M., Wolfe, D. Nonparametric Statistical Methods Wiley New York, NY, 1973. 

Lehman, E. Nonparametrics Statistical Methods Based on Ranks Holden Day San Francisco, CA, 1975. 

Optimization techniques may be classified as parametric statistical methods and nonparametric search methods. Parametric statistical methods, usually employed for optimization, are full factorial designs, half factorial designs, simplex designs, and Lagrangian multiple regression analysis [21]. Parametric methods are best suited for formula optimization in the early stages of product development. Constraint analysis, described previously, is used to simplify the testing protocol and the analysis of experimental results. 

The online statistical calculations can be performed at http //members.aol.com/ johnp71/javastat.html. To carry out linear regression analysis as an example, select Regression, correlation, least squares curve-fitting, nonparametric correlation, and then select any one of the methods (e.g., Least squares regression line, Least squares straight line). Enter number of data points to be analyzed, then data, x and y . Click the Calculate Now button. The analytical results, a (intercept), b (slope), f (degrees of freedom), and r (correlation coefficient) are returned. 

The results are expressed as the number of cells per high power field for each 10 pm distance. Statistical analysis for thick filter assays can be performed using many methods including a twotailed student to test or a KruskalWallis nonparametric analysis of variance on averaged triplicate values. 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

In general, a regularization parameter should be chosen for each voxel. Since there may be thousands of voxels, the use of graphical or other methods requiring intervention is prohibitive. In the present work, an automatic, data-driven method is utilized to obtain a reliable estimate of the regularization parameter for each voxel. It is based on nonparametric statistical theory, which can incorporate a number of performance criteria, including unbiased prediction risk (UBPR),9 cross-validation (CV),15 and generalized cross-validation (GCV).16... 

The verification of transferred reference values or intervals is both important and problematic. The comparison of a locally produced, small subset of values with the large set produced elsewhere using traditional statistical tests often is not appropriate because the underlying statistical assumptions are not fulfilled and because of the unbalanced sample sizes. Alternative methods using nonparametric tests or Monte Carlo sampling have been described. 

There are a number of nonparametric analysis methods dealing with continuous data. The last statistical method included in this chapter is to be used when the continuous outcome is time to an event. 

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