Statistical analyses nonparametric

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). 

Randomness, independence and trend (upward, or downward) are fundamental concepts in a statistical analysis of observations. Distribution-free observations, or observations with unknown probability distributions, require specific nonparametric techniques, such as tests based on Spearman s D - type statistics (i.e. D, D, D, Z)k) whose application to various electrochemical data sets is herein described. The numerical illustrations include surface phenomena, technology, production time-horizons, corrosion inhibition and standard cell characteristics. The subject matter also demonstrates cross fertilization of two major disciplines. 

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). 

Statistical analysis for these assays can be performed using many statistical programs including a twotailed student t-test or a KruskalWallis nonparametric analysis of variance on averaged triplicate values. 

Statistical analysis of the data is performed by means of the Student s t-test for paired or unpaired data, or by means of analysis of variance followed by the Tukey test. Statistical analysis of nonparametric data is made by the chi square test. 

Crawford, C.G., Slack, J.R. and Hirsch, R.M. (1983) Nonparametric Tests for Trends in Water Quality Data using the Statistical Analysis System. United States Geological Survey Report, 83-550. 

Data analyses included data visualization, nonparametric statistical analysis on observations (data from study 1 only), and parametric analysis with nonlinear mixed effects modeling. 

A study is considered valid if the results obtained with positive and negative controls are consistent with the laboratory s historical data and with the literature. Statistical analysis is usually applied to compare treated and negative control groups. Both pairwise and linear trend tests can be used. Because of the low background and Poisson distribution, data transformation (e.g., log transformation) is sometimes needed before using tests applicable to normally distributed data. Otherwise, nonparametric analyses should be preferred. 

The statistical analysis was performed using the Statistical Package for Social Sciences (SPSS) for Windows 22.0 of Microsoft . Because it is a small sample (N = 13) in the evaluation of sEMG and dynamometry, nonparametric tests (Wilcoxon test) were used. The significance level used was 5% (p = 0.05). 

Another statistic often calculated is an overall index of acceptability of the form (T- Q/(r + C), where T and C are defined as above. This statistic has an expected value of 0 when there is no difference in preference of the test and control treatments and a range from - 1 (when the test treatment is never chosen) to + 1 where the test treatment is completely preferred over the control. While this statistic allows the expression of the relative attraction and deterrence of a range of compounds, its statistical properties are not well understood. The statistic is not normally distributed, and the occurrence of negative values precludes the use of common transformations (e.g., log, arcsine, square root) to remove some deviations from normality. Thus, if this statistic is used, it should be analyzed only by nonparametric, distribution-free statistical tests based on ranks. Further discussion of procedures and criteria for selecting statistical tests can be found in many standard texts and manuals for a number of statistic analysis packages for use with computers (e.g.. Steel Torrie 1980 SAS Institute 1989 Sokal Rohlf 1995). 

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. 

If the two-factor cases considered here were known to originate (at least approximately) from a normal population, the standard randomized block experiment approach would be admissible for testing the significance of the block effect. A detailed discussion of this technique, widely documented in the statistical textbook literature, is omitted. Table 10 indicates the possibility of drawing qualitatively identical inferences from nonparametric and conventional analysis of variance, even if only one of the two is correct, in principle. 

Dependencies may be detected using statistical tests and graphical analysis. Scatter plots may be particularly helpful. Some software for statistical graphics will plot scatter plots for all pairs of variables in a data set in the form of a scatter-plot matrix. For tests of independence, nonparametric tests such as Kendall s x are available, as well as tests based on the normal distribution. However, with limited data, there will be low power for tests of independence, so an assumption of independence should be scientifically plausible. 

A basic assumption underlying r-tests and ANOVA (which are parametric tests) is that cost data are normally distributed. Given that the distribution of these data often violates this assumption, a number of analysts have begun using nonparametric tests, such as the Wilcoxon rank-sum test (a test of median costs) and the Kolmogorov-Smirnov test (a test for differences in cost distributions), which make no assumptions about the underlying distribution of costs. The principal problem with these nonparametric approaches is that statistical conclusions about the mean need not translate into statistical conclusions about the median (e.g., the means could differ yet the medians could be identical), nor do conclusions about the median necessarily translate into conclusions about the mean. Similar difficulties arise when - to avoid the problems of nonnormal distribution - one analyzes cost data that have been transformed to be more normal in their distribution (e.g., the log transformation of the square root of costs). The sample mean remains the estimator of choice for the analysis of cost data in economic evaluation. If one is concerned about nonnormal distribution, one should use statistical procedures that do not depend on the assumption of normal distribution of costs (e.g., nonparametric tests of means). 

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. 

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 term parametric test, or analysis, is introduced in Section 6.2.4. In broad terms, statistical analyses can be placed into one of two categories, parametric tests and nonparametric tests. This book almost exclusively discusses parametric tests, but it should be noted here that nonparametric tests are also very valuable analyses in appropriate circumstances. As with the terms experimental design and nonexperimental design (recall Section 5.5), the term nonparametric is not a relative quality judgment compared with parametric. This nomenclature simply differentiates statistical approaches. In circumstances where nonparametric analyses are appropriate, they are powerful tests. 

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. 

A key factor in modeling is parameter estimation. One usually needs to fit the established model to experimental data in order to estimate the parameters of the model both for simulation and control. However, a task so common in a classical system is quite difficult in a chaotic one. The sensitivity of the system s behavior to the initial conditions and the control parameters makes it very hard to assess the parameters using tools such as least squares fitting. However, efforts have been made to deal with this problem [38]. For nonlinear data analysis, a combination of statistical and mathematical tests on the data to discern inner relationships among the data points (determinism vs. randomness), periodicity, quasiperiodicity, and chaos are used. These tests are in fact nonparametric indices. They do not reveal functional relationships, but rather directly calculate process features from time-series records. For example, the calculation of the dimensionality of a time series, which results from the phase space reconstruction procedure, as well as the Lyapunov exponent are such nonparametric indices. Some others are also commonly used ... 

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). 

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... 

Analysis of the data for correlation, separability, etc. using techniques drawn from statistics, nonparametric statistics, and pattern recognition. Testing of discriminants for predictive ability. 

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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