Distribution free tests

The test presented in the previous section is useful when a smaller probability of false detection is needed than is provided by the distribution-free test. However, the test in the previous section is no panacea. Reduction of the skewness through proper choice of sampling and subsampling procedures is an alternative that may have much more potential for improving the study. 

Exact and distribution-free tests are easier to compute. 

Introduce non-parametric methods where data is converted to rankings, so they become distribution-free tests ... 

An alternative approach is to use a non-parametric test that does the same job as one of the parametric tests. These tests convert the data to rankings and the distribution of the data (largely) ceases to be an issue. They are sometimes called distribution-free tests . 

Nonparametric (distribution free) tests do not require assumptions about the distribution of the population of the features to be tested. An example is the Wilcoxon test. 

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). 

The distribution-free multiple comparison test should be used to compare three or more groups of nonparametric data. These groups are then analyzed two at a time for any significant differences (Hollander and Wolfe, 1973, pp. 124-129). The test can be used for data similar to those compared by the rank-sum test. We often employ this test for reproduction and mutagenicity studies (such as comparing survival rates of offspring of rats fed various amounts of test materials in the diet). 

When we find a significant difference, we do not know which groups are different. It is not correct to then perform a Mann-Whitney U Test on all possible combinations rather, a multiple comparison method must be used, such as the distribution-free multiple comparisons. 

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. 

Bradley, J. V. (1968)7 Distribution Free Statistical Tests, Prentice-Hall, Inc., Englewood Cliffs, New Jersey. 

Distribution free method a method for testing a hypothesis or setting up a confidence interval, which does not depend on the form of the underlying distribution. 

The only other effect possibly attributed solely to humic acid involved 2,4-dichlorophenol, the most broadly distributed compound tested in this study. In columns 108 and 109, the humic acid apparently decreased the ease of elution of the chlorophenols because lower overall recoveries were obtained from column 109, which included the humate, than from column 108, which was humate-free. Also, recovery was detected in F6 of column 109 but not in F6 of column 108. This result suggests that humate enhanced binding of the phenol to the column. The reproducibility of 2,4-dichlorophenol recovery among the various parfait fractions was poor, as illustrated by the results from replicate columns 117, 118, and 120. Because of the variability, the differences in... 

The data must be tested in order to establish the nature of error variation to enable decision on whether the transformation of the data, the utilization of distribution-free statistical analysis procedures, or the test against simulated zero-hypothesis data is necessary. 

Jonckheere, A. R. 1954. A distribution-free fc-sample test against ordered alternatives. 

Pairwise comparisons among 80 adult E. rufescens using a distribution-free multiple comparison test based upon the Kruskal-Wallis ranked sums test (Hollander Wolfe, 1973) indicate that the active surface area of paired males is significantly larger than that of single males and those of both paired and single females (P < 0.05 for each comparison)... 

The data from the pilot survey(s) should be tested and analyzed to determine whether the error variation is homogeneous, normally distributed, and independent of the mean. If this is not the case (a situation which is highly probable with biological parameters), then the data could be appropriately transformed, or distribution-free (nonparametric) procedures could be used. 

This chapter introduces two groups of statistical tests for handling data that may not be normally distributed. Methods which make no assumptions about the shape of the distribution from which the data are taken are called non-parametric or distribution-free methods. Many of them have the further advantage of greatly simplified calculations with small data sets some of the tests can be performed mentally. The second group of methods, which has grown rapidly in use in recent years, is based... 

What is the significance of these different scales of measurement As was mentioned in Section 1.5, many of the well-known statistical methods are parametric, that is, they rely on assumptions concerning the distribution of the data. The computation of parametric tests involves arithmetic manipulation such as addition, multiplication, and division, and this should only be carried out on data measured on interval or ratio scales. When these procedures are used on data measured on other scales they introduce distortions into the data and thus cast doubt on any conclusions which may be drawn from the tests. Non-parametric or distribution-free methods, on the other hand, concentrate on an order or ranking of data and thus can be used with ordinal data. Some of the non-parametric techniques are also designed to operate with classified (nominal) data. Since interval and ratio scales of measurement have all the properties of ordinal scales it is possible to use non-parametric methods for data measured on these scales. Thus, the distribution-free techniques are the safest to use since they can be applied to most types of data. If, however, the data does conform to the distributional assumptions of the parametric techniques, these methods may well extract more information from the data. 

As the results were not distributed normally, median values were used for the descriptive statistics, while parameter-free test procedures were used for the analytical statistics (Wilcoxon test for paired differences by Wilcoxon/Mann and Whitney) (ClauC and Ebner 1992). 

The FOEX ranges from -50% to +50%, with an optimum value of 0. The EAn ranges from an optimum value of 100% to 0. The factor of 2 (FA2) is most often referred to in dense gas evaluation studies (Hanna et al, 1991). The sign tests have the advantage of being distribution-free. 

The data in the CAMPUS database has been obtained with uniform, standardized test methods as descriribed in ISO 10350, ISO 11403-1, and ISO 11403-2. CAMPUS is distributed free of charge to customers directly from the resin manufacturers. In fact, CAMPUS data from a number of resin suppliers can be downloaded from their websites at no cost. CAMPUS is available in five languages English, German, French, Spanish, and Italian [102]. Some of the data in this chapter are actually from this database. 

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