Distribution-free methods

The statistical methods discussed up to now have required certain assumptions about the populations from which the samples were obtained. Among these was that the population could be approximated by a normal distribution and that, when dealing with several populations, these have the same variance. There are many situations where these assumptions cannot be met, and methods have been developed that are not concerned with specific population parameters or the distribution of the population. These are referred to as non-parametric or distribution-free methods. They are the appropriate methods for ordinal data and for interval data where the requirements of normality cannot be assumed. A disadvantage of these methods is that they are less efficient than parametric methods. By less efficient is meant... 

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 adequate number of data points needed to construct an SSD depends on the method used (Assumption 8). Generally distribution-free methods need more data points (30 or more) than distribution-based methods (Newman et al. 2000). Crane et al. (2003) stated that, for chlorpyrifos, species sensitivity followed a log-normal... 

Marasciulo, L., and McSweeney, M. (1977), Nonparametric and Distribution-Free Methods for the Social Sciences, Brooks/Cole, Monterey, CA. 

Bortz, J., Lienert, G. A., Boehnke, K. (1990). Verteilungsfreie Methoden in derBiostatistik [Distribution free methods in bioslatistics]. New York Springer-Verlag. 

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. 

Volume of vessel (free volume V) Shape of vessel (area and aspect ratio) Type of dust cloud distribution (ISO method/pneumatic-loading method) Dust explosihility characteristics Maximum explosion overpressure P ax Maximum explosion constant K ax Minimum ignition temperature MIT Type of explosion suppressant and its suppression efficiency Type of HRD suppressors number and free volume of HRD suppressors and the outlet diameter and valve opening time Suppressant charge and propelling agent pressure Fittings elbow and/or stub pipe and type of nozzle Type of explosion detector(s) dynamic or threshold pressure, UV or IR radiation, effective system activation overpressure Hardware deployment location of HRD suppressor(s) on vessel... 

Exact permutation and distribution-free techniques are not as widely developed as in the cases of one-way methods. 

As mentioned earlier, older (and some newer) literature in large animal toxicology is full of two-sample, one-way parametric, and distribution-free techniques. Some of the newer works use repeated-measures and even multivariate techniques. The following is a brief expose of various methods used in the field. 

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). 

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. 

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

Ultrafiltration has been used to determine the protein bound fraction of many drags, such as methadone (Wilkins et al. 1997), phenylacetate and phenylbu-tyrate (Boudoulas et al. 1996), etoposide (Robieux et al. 1997), doxorubicin and vincristine (Mayer and St-Onge 1995), disopyramide (Echize et al. 1995), and ketamine and its active metabolites (Hijazi and Boulieu 2002). Schumacher et al. (2000) have shown the applicability for the determination of erythro-cyte/plasma distribution. The method of UF has been applied in the measurement of free unaltered thyroxin or after displacement by salicylate as well after displacement by heparin in healthy people and in patients with non-thyroidal somatic illness (Faber et al. 1993). The protein binding of tritium labeled, antidiabetic repaglinide and its displacement by warfarin, furosemide, tolbutamide, diazepam, glibenclamide and nicardipine were determined by ultrafiltration (Plumetal. 2000). 

Methods for identifying and handling of possible outlier data should be specified in the protocol. Medical or pharmacokinetic explanations for such observations should be sought and discussed. As outliers may be indicative of product failure, post hoc deletion of outlier values is generally discouraged. An approach to dealing with data containing outliers is to apply distribution-free (non-parametric), statistical methods (72). 

R382 V, I. Irzhak, Solution-Free Methods for the Determination of the Molecular Mass Distribution of Polymers , Russ. Chem. Rev., 2000, 69, in... 

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