Technology & Engineering
Non Parametric Statistics
Non-parametric statistics refers to statistical methods that do not make assumptions about the underlying probability distribution of the data. These methods are used when the data does not meet the requirements of parametric statistics, such as normal distribution or homogeneity of variance. Non-parametric statistics are valuable in situations where the data is skewed, contains outliers, or is measured on an ordinal scale.
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12 Key excerpts on "Non Parametric Statistics"
eBook - PDF- Carlos Cortinhas, Ken Black(Authors)
- 2014(Publication Date)
- Wiley(Publisher)
The name parametric statistics refers to the fact that an assumption (here, normally dis- tributed data) is being made about the data used to test or estimate the parameter (in this case, the population mean). In addition, the use of parametric statistics requires quantitative measurements that yield interval- or ratio-level data. For data that do not meet the assumptions made about the population, or when the level of data being measured is qualitative, statistical techniques called non-parametric, or distribution-free, techniques are used. Non- parametric statistics are based on fewer assumptions about the population and the parameters than are parametric statistics. Sometimes they are referred to as distribution-free statistics because many of them can be used regardless of the shape of the population distribution. A variety of non-parametric statistics are available for use with nominal or ordinal data. Some require at least ordinal-level data, but others can be specifically targeted for use with nominal- level data. Non-parametric techniques have the following advantages: 1. Sometimes there is no parametric alternative to the use of non-parametric statistics. 2. Certain non-parametric tests can be used to analyse nominal data. 3. Certain non-parametric tests can be used to analyse ordinal data. 4. The computations on non-parametric statistics are usually less complicated than those for parametric statis- tics, particularly for small samples. 5. Probability statements obtained from most non-parametric tests are exact probabilities. Using non-parametric statistics also has some disadvantages: 1. Non-parametric tests can be wasteful of data if parametric tests are available for use with the data. 2. Non-parametric tests are usually not as widely available and well known as parametric tests. 3. For large samples, the calculations for many non-parametric statistics can be tedious. Entire courses and texts are dedicated to the study of non-parametric statistics.
- M. Kraska-MIller(Author)
- 2013(Publication Date)
- Chapman and Hall/CRC(Publisher)
2.2 Overview of Nonparametric and Parametric Statistics Statistical procedures may be grouped into two major classifications: para-metric and nonparametric. Specific statistical tests are available within each category to perform various analyses; however, data for each category should meet unique requirements prior to analysis. Parametric statistics require assumptions to be more specific and more stringent than the assumptions for nonparametric statistics. The general-ity of conclusions drawn from statistical results are tied to the strength of the assumptions that the data satisfy. In other words, the more rigorous the assumptions, the more trustworthy the conclusions. Conversely, fewer or weaker assumptions indicate weaker or more general conclusions. For this reason, statisticians prefer to use parametric tests whenever the data meet the assumptions. Unlike parametric statistics, nonparametric statistics make no assumptions about the properties of the distribution from which the data are drawn, except that the distribution is continuous. The properties of a distribution are defined by its shape (normality), midpoint (mean), and spread of scores (variance and standard deviation). These measures represent the parameters of a population. Before conducting statistical tests, researchers generally check the assump-tions of the data to verify whether they are appropriate for a parametric test. Parametric tests are more powerful than nonparametric tests in that they have a better chance of rejecting the null hypothesis when the null hypothesis is false. For example, if one were to test the null hypothesis for 35 Introduction to Nonparametric Statistics mathematical achievement of students who are taught beginning algebra using three different instructional methods, one would want to use the most powerful statistical test available.
Available until 25 Jan |Learn more- Peter Sprent, Nigel C. Smeeton(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
CHAPTER 1 SOME BASIC CONCEPTS 1.1 Basic Statistics We assume most readers are familiar with the basic statistical notions met in introductory or service courses in statistics of some 20 hours duration. Nevertheless, those with no formal statistical training should be able to use this book in parallel with an introductory statistical text. Rees (2000) adopts a straightforward approach. Some may prefer a more advanced treatment, or an introduction that emphasizes applications in a discipline in which they are working. Readers trained in general statistics, but who are new to nonparametric methods will be familiar with some of the background material in this chapter. However, we urge them at least to skim through it to see where we depart from conventional treatments, and to learn how nonparametric procedures relate to other approaches. We explain the difference between parametric and nonparametric methods and survey some general statistical notions that are relevant to nonparametric methods. We also comment on good practice in applied statistics. In Chapter 2 we use simple examples to illustrate some basic nonparametric ideas and introduce some statistical notions and tools that are widely used in this field. Their application to a range of problems is covered in the remaining chapters. 1.1.1 Parametric and Nonparametric Methods The word statistics has several meanings. It is used to describe a collection of data, and also to designate operations that may be performed with that primary data. The simplest of these is to form descriptive statistics. These include the mean, range, or other quantities to summarize primary data, as well as preparing tables or pictorial representations (e.g., graphs) to exhibit specific facets of the data. The scientific discipline called statistics , or statistical inference , uses observed data — in this context called a sample — to make inferences about a larger potentially observable collection of data called a population .
- Aliakbar Montazer Haghighi, Indika Wickramasinghe(Authors)
- 2020(Publication Date)
- CRC Press(Publisher)
427 6 Nonparametric Statistics 6.1 WHY NONPARAMETRIC STATISTICS? It is believed that the idea about nonparametric is as nearly old as statistics, as early works in this theory go back to fifteenth and sixteenth centuries. Arbuthnot, a math-ematician, investigated whether the number of male births is higher than the number of female births, during the period of 1629–1710. This is considered as the first sign test. Among statisticians, it is believed that the first work in nonparametric statistics belongs to the paper written by Hotelling (1895–1973) and Pabst (1936), who dis-cussed about the rank correlation. The origin of the term “nonparametric” goes back to 1942 in a paper by Wolfowitz (1910–1981). Introduction of nonparametric in the literature moved the theory of statistics beyond the parametric setting. In inferential statistics, we considered some types of distribution regarding the underlining data. Statistical procedures based on distribution regarding the popula-tion are referred to as the parametric methods . For example, when calculating the confidence intervals or when conducting hypothesis test, we assume that the data comes from a normal distribution (or from t -distribution) and we use parameters such as the mean and the standard deviation. However, not all distributions contain parameters as normal distribution does. Hence, in those cases, we cannot use para-metric methods; instead, we turn to use statistical procedures that are not based on distributional assumptions. Such methods are referred to as the distribution free-techniques or nonparametric techniques . In parametric methods, we usually make assumptions that the population is nor-mally distributed and is symmetric about the mean. In such cases, all calculations are made based on these critical assumptions. But what if the normality feature is not present? Naturally, the use of mean as a measure of central tendency will not be available any longer.
eBook - PDFPharmaceutical Statistics
Practical and Clinical Applications, Fifth Edition
- Sanford Bolton, Charles Bon(Authors)
- 2009(Publication Date)
- CRC Press(Publisher)
15 Nonparametric Methods Nonparametric statistics, also known as distribution-free statistics, may be applicable when the nature of the distributions are unknown, and we are not willing to accept the assumptions necessary for the application of the usual statistical procedures. For most of the statistical tests described in this book, we have assumed that data are normally distributed. This assumption, although never exactly realized, is bolstered by the central limit theorem (sect. 3.4.2) when we are testing hypotheses concerning the means of distributions. However, occasions arise in which data are clearly too far from normal to accept the assumption of normality. The data may deviate so much from that expected for a normal distribution that to assume normality, even when dealing with means, would be incorrect. In these situations, a data transformation may be used, chapter 10, or nonparametric methods may be applied for statistical tests. As we shall see, many of the nonparametric tests are easy to compute, and can be used for a quick preliminary approximation of the level of significance when parametric tests may be more appropriate. Although some people believe that any kind of data, no matter what the distribution, can be correctly analyzed using nonparametric methods, a kind of panacea, this is not true. Many if not most nonparametric methods require that the distributions be continuous and symmetrical, and that data be independent, for example. These are among the assumptions underlying parametric analyses, as exemplified by the normal t , and F tests. 15.1 DATA CHARACTERISTICS AND AN INTRODUCTION TO NONPARAMETRIC PROCEDURES Before proceeding, a review of the different kinds of data that are usually encountered in scientific experiments will be useful for the understanding of the applications of nonparametric methods. 1. Perhaps the most elementary kinds of data are categorical or attribute measurements.
eBook - PDF- Myles Hollander, Douglas A. Wolfe, Eric Chicken(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
Chapter 1 Introduction 1.1 ADVANTAGES OF NONPARAMETRIC METHODS Roughly speaking, a nonparametric procedure is a statistical procedure that has certain desirable properties that hold under relatively mild assumptions regarding the underlying populations from which the data are obtained. The rapid and continuous development of nonparametric statistical procedures over the past 7 1 2 decades is due to the following advantages enjoyed by nonparametric techniques: 1. Nonparametric methods require few assumptions about the underlying populations from which the data are obtained. In particular, nonparametric procedures forgo the traditional assumption that the underlying populations are normal. 2. Nonparametric procedures enable the user to obtain exact P -values for tests, exact coverage probabilities for confidence intervals, exact experimentwise error rates for multiple comparison procedures, and exact coverage probabilities for confidence bands without relying on assumptions that the underlying populations are normal. 3. Nonparametric techniques are often (although not always) easier to apply than their normal theory counterparts. 4. Nonparametric procedures are often quite easy to understand. 5. Although at first glance most nonparametric procedures seem to sacrifice too much of the basic information in the samples, theoretical efficiency investigations have shown that this is not the case. Usually, the nonparametric procedures are only slightly less efficient than their normal theory competitors when the underlying populations are normal (the home court of normal theory methods), and they can be mildly or wildly more efficient than these competitors when the underlying populations are not normal. 6. Nonparametric methods are relatively insensitive to outlying observations. 7. Nonparametric procedures are applicable in many situations where normal theory procedures cannot be utilized.
eBook - PDFStochastic Modeling and Mathematical Statistics
A Text for Statisticians and Quantitative Scientists
- Francisco J. Samaniego(Author)
- 2014(Publication Date)
- Chapman and Hall/CRC(Publisher)
12 Nonparametric Statistical Methods In this chapter, we will discuss approaches to statistical inference in which minimal as-sumptions are made about the underlying distribution F from which data is drawn. We will continue to assume that our inferences are based on one or more random samples from the distributions of interest. In one-sample problems, we will assume that the data may be represented as X 1 ,..., X n iid ∼ F , where F is a distribution that, in some applications, will be completely general (that is, can be any distribution, continuous or discrete) and in other settings, might be assumed to obey some modest restrictions like continuity and/or sym-metry. In any case, the functional form of F is not assumed to be known. In particular, the distribution F is not indexed by a finite number of parameters as are the discrete and continuous models we discussed in Chapters 2 and 3. The term “nonparametric” is thus seen as a natural way to describe the models and methods utilized in this quite general framework. Before embarking on our tour of “nonparametric statistics,” it seems worthwhile dis-cussing, albeit briefly, why this subfield of statistics is important and, in some instances, essential. The process of parametric modeling and inference is appropriate and useful in many problems to which Statistics is applied, but it does have its difficulties and limita-tions. While the physical circumstances of an experiment, or logical reasoning about the experimental process, can often suggest a particular parametric model as appropriate, there are many situations in which the modeling of experimental data is not easily accomplished. Since goodness-of-fit tests are more reliable in discrediting a proposed model than in con-firming that a model holds, deciding on the appropriate parametric model can be a chal-lenging proposition. Using a nonparametric approach to modeling and inference represents an obvious and potentially quite useful alternative.
No longer available |Learn more- Anthony Hayter(Author)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
C H A P T E R F I F T E E N Nonparametric Statistical Analysis This chapter provides a discussion and illustration of the implementation of nonparametric or distribution-free statistical inference methodologies. The general distinction between these types of inference methods and the alternative parametric inference methods discussed in pre-vious chapters is that a parametric inference approach is based on a distributional assumption for data observations, whereas a nonparametric inference approach provides answers that are not based on any distributional assumptions. Most of the statistical methodologies discussed in the other chapters of this book would be considered parametric inference methods. For instance, the analysis of Example 14 concern-ing metal cylinder diameters in Chapters 7 and 8 is based upon the distributional assumption that the diameters are normally distributed with some unknown mean parameter μ and some unknown variance parameter σ 2 . The parametric analysis is then based on obtaining esti-mates ˆ μ and ˆ σ 2 for these two unknown parameters. For other data sets other distributional assumptions may be appropriate. For example, measurements of the failure times of certain machine parts may be modeled with the Weibull or the gamma distribution. However, in all cases the parametric inference approach operates within the framework of an assumed distributional model, and it is based on the estimation of the unknown parameters of that model. Obviously, the results of a parametric analysis are only as good as the validity of the assumptions on which they are based. If the assumption of a normal distribution is made, whereas in fact the unknown true distribution is skewed and asymmetric, then the correspond-ing inference results may in turn be misleading. Nonparametric inference methods have been developed with the objective of providing statistical inference methods that are free from any distributional assumptions.
eBook - PDFRandom Phenomena
Fundamentals of Probability and Statistics for Engineers
- Babatunde A. Ogunnaike(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
But none of this is an acceptable option for the well-trained engineer or scientist. The objective in this chapter is to present some viable alternatives to con-sider when distributional assumptions are invalid. These techniques, which make little or no demand on the specific distributional structure of the popu-lation from whence the data came, are sometimes known as “distribution-free” methods. And precisely because they do not involve population parameters, in contrast to the distribution-based techniques, they are also known as non-parametric methods. Inasmuch as entire textbooks have been written on the subject of nonparametric statistics—complete treatises on statistical analysis without the support (and, some would say, encumbrance) of hard probability distribution models—the discussion here will necessarily be limited to only the few most commonly used techniques. And to put the techniques in proper context, we will compare and contrast these nonparametric alternatives with the corresponding parametric methods, where possible. 18.1 Introduction There are at least two broad classes of problems for which the classical hypothesis tests discussed in Chapter 15 are unsuitable: 1. When the underlying distributional assumptions (especially the Gaus-sian assumptions) are seriously violated; 2. When the data in question is ordinal only, not measured on a quantita-tive scale in which the distance between succeeding entities is uniform or even meaningful (see Chapter 12). In each of these cases, even in the absence of any knowledge of the mathemati-cal characteristics of the underlying distributions, the sample data can always be rank ordered by magnitude. The data ranks can then be used to analyze such data with the little or no assumptions about the probability distributions of the populations.
eBook - PDF- Maria Dolores Ugarte, Ana F. Militino, Alan T. Arnholt(Authors)
- 2015(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 10 Nonparametric Methods 10.1 Introduction The statistical inference techniques presented in Chapter 8 and Chapter 9 are based on complete satisfaction of all of the assumptions made in the derivations of their sampling dis-tributions. Indeed, many of the techniques in Chapters 8 and 9 are commonly referred to as parametric techniques since not only was the form of the underlying population (generally normal) stated, but so was one or more of the underlying distribution’s parameters. This chapter introduces both distribution-free tests as well as nonparametric tests. The collection of inferential techniques known as distribution-free is based on functions of the sample observations whose corresponding random variable has a distribution that is independent of the specific distribution function from which the sample was drawn. Consequently, as-sumptions with respect to the underlying population are not required. Nonparametric tests involve tests of a hypothesis where there is no statement about the distribution’s parameters; however, it is common practice to refer collectively to both distribution-free tests and nonparametric tests simply as nonparametric methods . When there are analogous parametric and nonparametric tests, comparisons between tests can be made based on power. The power e ffi ciency of a test A relative to a test B is the ratio of n b /n a , where n a is the number of observations required by test A for the power of test A to equal the power of test B when n b observations are used. Since the power value is conditional on the type of alternative hypothesis and on the significance level, power e ffi ciency can be di ffi cult to interpret. One way to avoid this problem is to use the asymptotic relative e ffi ciency ( ARE ) (a limiting power e ffi ciency) of consistent tests. A test is consistent for a specified alternative if the power of the test when that alternative is true approaches 1 as the sample size approaches infinity (Gibbons, 1997).
eBook - PDFStatistics Using Stata
An Integrative Approach
- Sharon Lawner Weinberg, Sarah Knapp Abramowitz(Authors)
- 2016(Publication Date)
- Cambridge University Press(Publisher)
573 CHAPTER SEVENTEEN Nonparametric Methods In the situations of statistical inference that we have discussed regarding the mean and the variance, the dependent variable Y was tacitly assumed to be at the interval level of measurement. In addition, the statistical tests (e.g., t and F) required the assumptions of normality and homogeneity of variance of the parent population distributions. In the situ- ations of statistical inference regarding correlation and regression, both variables X and Y were tacitly assumed to be at the interval level of measurement and the statistical tests required the assumption of homogeneous variances and bivariate normality of the parent population distribution. When one or more of these assumptions regarding the population distributions and parameters are not reasonable, alternative methods of statistical infer- ence must be employed. The focus of this chapter is on the development and use of such alternative methods. PARAMETRIC VERSUS NONPARAMETRIC METHODS Because the validity of the inferential tests we have discussed so far in this book rely on explicit assumptions about population distributions and parameters, they are called “para- metric” or “distribution-tied” methods. There are occasions, however, when the use of parametric methods is not warranted. One such instance occurs when the dependent vari- able is not at the interval level of measurement, but rather at the nominal or ordinal levels of measurement as in the case of categorical or ranked data. Another instance occurs when one or more of the required assumptions regarding the parent population distributions are not reasonable, as in the case of studying the mean of a nonnormal population using a small sample of data. In situations where the use of parametric or distribution-tied methods is inap- propriate, alternative methods called “nonparametric” or “distribution-free” may be employed.
eBook - PDFStatistics Using R
An Integrative Approach
- Sharon Lawner Weinberg, Daphna Harel, Sarah Knapp Abramowitz(Authors)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Another instance occurs when one or more of the required assumptions regarding the parent population distributions are not reasonable, as in the case of studying the mean of a non-normal population using a small sample of data. In situations where the use of parametric or distribution-tied methods is inappropriate, alternative methods called “nonparametric” or “distribution-free” may be employed. In this chapter, we first present nonparametric techniques that can be used for nominal- leveled variables, and then we present techniques that can be used for ordinal-leveled variables. Although nonparametric methods have the advantage of being relatively free from assumptions about population distributions and parameters, they have the disadvantage of having lower power generally than comparable parametric methods when these para- metric methods do apply. It should also be pointed out that the hypotheses tested using nonparametric methods are not exactly the same as the hypotheses tested using parametric methods. For example, although the parametric two-group t-test specifically tests a hypothesis about the equality of two population means, the comparable nonparametric 593 alternative (to be discussed later in this chapter) tests a hypothesis about the equality of two population distributions, providing information about the equality of the two medians as a by-product. NONPARAMETRIC METHODS WHEN THE OUTCOME VARIABLE IS AT THE NOMINAL LEVEL In this section, we present two commonly used nonparametric techniques of statistical inference based on the chi-square distribution, which are applicable when the outcome variable is at the nominal level of measurement. The first method, the chi-square goodness-of-fit test, can be used to analyze the distribution of subjects among the categories of a nominal-leveled variable.
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