Distribution-free statistics

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

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. 

Parametric results are numerical, whereas nonparametric results often aim for yes-or-no answers and often require no assumption about the underlying distribution of observations. The use of a median instead of the mean to estimate the location of a distribution is an example of the use of distribution-free statistics. The median is more robust than the mean in that it is more distribution-free, but the median is less efficient (requires more observations to achieve the same precision) than the mean for... 

The statistical methods based on assumption that the lifetime were drawn from known distributed populations such as normal etc., and below there are techniques that do not make such assumptions. The methods are recognized as distribution-free statistics or nonparametric statistics. In situations where the known assumption holds, the nonparametric tests are less efficient than parametric methods. 

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). 

Fig. 25. Reverse osmosis, ultrafiltration, microfiltration, and conventional filtration are related processes differing principally in the average pore diameter of the membrane filter. Reverse osmosis membranes are so dense that discrete pores do not exist transport occurs via statistically distributed free volume areas. The relative size of different solutes removed by each class of membrane is illustrated in this schematic.

Flow cytometer cell counts are much more precise and more accurate than hemocytometer counts. Hemocytometer cell counts are subject both to distributional (13) and sampling (14—16) errors. The distribution of cells across the surface of a hemocytometer is sensitive to the technique used to charge the hemocytometer, and nonuniform cell distribution causes counting errors. In contrast, flow cytometer counts are free of distributional errors. Statistically, count precision improves as the square root of the number of cells counted increases. Flow cytometer counts usually involve 100 times as many cells per sample as hemocytometer counts. Therefore, flow cytometry sampling imprecision is one-tenth that of hemocytometry. 

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. 

Nonparametric technique A statistical technique that does not depend for its validity upon the assumption that the data were drawn from a specific distribution, such as the normal or lognormal. A distribution-free technique. 

Corrected for isomer distribution and statistical factor. b 0, free base is reacting species. 

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... 

Iman RL, Conover WJ (1982) A distribution free approach to inducing rank correlation among input variables. Communications in Statistics, B11(3) 311-334. 

Corrected for isomer distribution and statistical factor. b 0, free base is reacting species. u +, conjugate acid is reacting species. d Free base nitration is assumed. 

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). 

Hauschke D, Steinijans VW, Diletti E. A distribution-free procedure for the statistical analysis of bioequivalence studies. InternationalJournal of Clinical Pharmacology, Therapy and Toxicology, 1990, 28 72-78. 

Moreover, the equation can only be accurate for small strains, since considerable change in the end-to-end distance of the cords would distort the Gaussian distribution of statistical chain elements. This happens more readily for a smaller value of It also implies that at increasing strain, the chemical bonds in the primary chain become increasingly distorted. Consequently, the increase in elastic free energy is due not merely to a decrease in conformational entropy but also to an increase in bond enthalpy. If the value of is quite small, even a small strain will cause an increase in enthalpy. (In a crystalline solid, only the increase in bond enthalpy contributes to the elastic modulus.)... 

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. 

There are many other statistical techniques in addition to those presented in the preceding sections of this paper which can be used to advantage by the food research worker. Many of these are concerned with enumeration data (i.e., data which arise by counting), and others are recently developed methods for dealing with measurement data. Examples of the latter are control chart techniques, sequential analysis, procedures involving the sample range in place of the sample standard deviation, and nonparametric and distribution-free techniques. Since these methods have as yet received little attention by food research workers, published examples are difficult if not impossible to find. However, we have mentioned these methods so that interested persons may consult appropriate references (Ostle, 1954 Snedecor, 1948 Goulden, 1939 and Dixon and Massey, 1951) for the details of operation of particular techniques. 

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