Chapter

# Testing Hypotheses by Fuzzy Methods: A Comparison with the Classical Approach

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## Abstract

Testing hypotheses could sometimes benefit from the fuzzy context of data or from the lack of precision in specifying the hypotheses. A fuzzy approach is therefore needed for reflecting the right decision regarding these hypotheses. Different methods of testing hypotheses in a fuzzy environment have already been presented. On the basis of the classical approach, we intend to show how to accomplish a fuzzy test. In particular, we consider that the fuzziness does not only come from data but from the hypotheses as well. We complete our review by explaining how to defuzzify the fuzzy test decision by the signed distance method in order to obtain a crisp decision. The detailed procedures are presented with numerical examples of real data. We thus present the pros and cons of both the fuzzy and classical approaches. We believe that both approaches can be used in specific conditions and contexts, and guidelines for their uses should be identified.

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... This method will be eventually useful in hypotheses testing. As such, we remember that hypotheses testing methods based on confidence intervals are widely used in different contexts in both the classical and the fuzzy environments (see Berkachy & Donzé [1], Berkachy & Donzé [2], Chachi & al. [3], Grzegorzewski [9] and others). ...
... We hereby propose an original method to construct a fuzzy confidence interval based on the likelihood ratio. The main objective of this study is to provide a general way of computing a confidence interval when fuzziness occurs in the data set, and give us the possibility to extend our former results on fuzzy hypotheses tests by confidence intervals (see Berkachy & Donzé [1] and Berkachy & Donzé [2]). Our approach is adapted to a broad spectrum of parameters using any type of statistical distributions. ...
... We highlight that this fuzzy confidence interval is constructed intentionally in the perspective of integrating it in the approach of the fuzzy inference tests by confidence intervals described in Berkachy & Donzé [1] and [2]. ...
... A subset of J = 5 items was selected for the sake of illustration. 2 The first step of the analysis consisted in the definition of an IRTree model for the six-point rating scale. To this end, we used the tree suggested by [2] (see Figure 2 values of the scale. ...
... 2 The first step of the analysis consisted in the definition of an IRTree model for the six-point rating scale. To this end, we used the tree suggested by [2] (see Figure 2 values of the scale. ...
... A subset of J = 5 items was selected for the sake of illustration. 2 The first step of the analysis consisted in the definition of an IRTree model for the six-point rating scale. To this end, we used the tree suggested by [2] (see Figure 2). ...
... 2 The first step of the analysis consisted in the definition of an IRTree model for the six-point rating scale. To this end, we used the tree suggested by [2] (see Figure 2). It formally represents a response model where (i) the node M determines whether the respondent has a weak A w or strong A s attitude/trait toward the item, (ii) the node A s activates the strength/extremity E of the rating response, by selecting either low values or higher values of the scale. ...
Preprint
In this contribution we describe a novel procedure to represent fuzziness in rating scales in terms of fuzzy numbers. Following the rationale of fuzzy conversion scale, we adopted a two-step procedure based on a psychometric model (i.e., Item Response Theory-based tree) to represent the process of answer survey questions. This provides a coherent context where fuzzy numbers, and the related fuzziness, can be interpreted in terms of decision uncertainty that usually affects the rater's response process. We reported results from a simulation study and an empirical application to highlight the characteristics and properties of the proposed approach.
... The signed distance measure was firstly used in the context of ranking fuzzy numbers by Yao and Wu [23]. It has also served in some other contexts: Berkachy and Donzé [3] used it in the assessment of linguistic questionnaires; Berkachy and Donzé [6] used it in hypotheses testing; etc. Although this measure is considered to be simple in terms of computations, it has interested specialists because of its directionality. ...
Article
Full-text available
We propose a practical procedure of construction of fuzzy confidence intervals by the likelihood method where the observations and the hypotheses are considered to be fuzzy. We use the bootstrap technique to estimate the distribution of the likelihood ratio. The chosen bootstrap algorithm consists on randomly drawing observations by preserving the location and dispersion measures of the original fuzzy data set. A metric $$d_{SGD}^{\theta ^{\star }}$$ d SGD θ ⋆ based on the well-known signed distance measure is considered in this case. We expose a simulation study to investigate the influence of the fuzziness of the computed maximum likelihood estimator on the constructed confidence intervals. Based on these intervals, we introduce a hypothesis test for the equality of means of two groups with its corresponding decision rule. The highlight of this paper is the application of the defended approach on the Swiss SILC Surveys. We empirically investigate the influence of the fuzziness vs. the randomness of the data as well as of the maximum likelihood estimator on the confidence intervals. In addition, we perform an empirical analysis where we compare the mean of the group “Swiss nationality” to the group “Other nationalities” for the variables Satisfaction of health situation and Satisfaction of financial situation.
... In [25], the authors demonstrate how to accomplish a fuzzy test with fuzzy data and fuzzy formulated hypotheses and discuss the defuzzification of fuzzy test decisions by means of the signed distance method. In [26], the author reviews and compares the R packages "FPV" and "Fuzzy.p.value" for hypothesis testing in fuzzy environments by using the fuzzy p-value for decision making. ...
Article
Full-text available
In this paper, we develop fuzzy, possibilistic hypothesis tests for testing crisp hypotheses for a distribution parameter from crisp data. In these tests, fuzzy statistics are used, which are produced by the possibility distribution of the estimated parameter, constructed by the known from crisp statistics confidence intervals. The results of these tests are in much better agreement with crisp statistics than the ones produced by the respective tests of a popular book on fuzzy statistics, which uses fuzzy critical values. We also present an error that we found in the implementation of the unbiased fuzzy estimator of the variance in this book, due to a poor interpretation of its mathematical content, which leads to disagreement of some fuzzy hypotheses tests with their respective crisp ones. Implementing correctly this estimator, we produce test statistics that achieve results in hypotheses tests that are in much better agreement with the results of the respective crisp ones.
... Hryniewicz [10] presents a comprehensive review with regard to statistical properties of different approaches for calculating fuzzy p-values. Berkachy and Donze [11] demonstrate how to accomplish a fuzzy test with fuzzy data and fuzzy formulated hypotheses, and discuss the defuzzification of fuzzy test decisions by means of the signed distance method. Parchami [12] reviews and compares the R packages ''FPV'' and ''Fuzzy.p.value'' for hypothesis testing in fuzzy environments by using the fuzzy p-value for decision making. ...
Article
In statistical inference hypotheses related to different kinds of phenomena are formulated, and then data are collected and analyzed, which either confirm or falsify these hypotheses. Considering traditional statistics, in the underlying models hypotheses and sample data should be well defined. However, these models are often inadequate with regard to real-life problems, as theoretical specifications and observed information are frequently imprecise, vague, incomplete, qualitative, linguistic or noisy. To relax this rigidity, numerous researchers have proposed modifications and extensions of statistical inference approaches with the help of concepts of fuzzy statistics. In the meantime there are many papers on the topic of hypothesis testing in fuzzy environments, especially based on fuzzy hypotheses and/or by using fuzzy data. In order to structure this variety of contributions, proposals and applications, we give a comprehensive systematic review in this paper and offer a bibliography on fuzzy hypothesis testing. The paper seeks to consolidate the topic of fuzzy hypothesis testing with the purpose of supporting new researchers in this field and highlighting potential directions for future research.
Chapter
This chapter first shows the definition of a fuzzy hypothesis. We after display the construction of a given fuzzy confidence interval. One of the highlights of this chapter is a new procedure of construction of fuzzy confidence intervals by the likelihood ratio method using the bootstrap technique. Moreover, we show in detail the hypotheses testing approaches, based on these intervals, followed by the fuzzy p-values. Both the fuzzy decisions and the fuzzy p-values are afterward defuzzified. We propose to defuzzify them by the signed distance operator from one side and from the generalized signed distance given from another one. The purpose is to sort out the main differences and drawbacks that might occur when using both distances in such contexts. All these procedures are illustrated by multiple detailed examples. Applications on a financial data set are also provided. In addition, a discussion on the comparison between the classical and fuzzy approaches is given. This chapter is closed by some guidelines on the use of each one of both approaches. The choice between the classical and fuzzy hypotheses testing approaches should be well-argued.
Article
Full-text available
In testing statistical hypotheses, as in other statistical problems, we may be confronted with fuzzy concepts. This paper deals with the problem of testing hypotheses, when the hypotheses are fuzzy and the data are crisp. We first introduce the notion of fuzzy p-value, by applying the extension principle and then present an approach for testing fuzzy hypotheses by comparing a fuzzy p-value and a fuzzy significance level, based on a comparison of two fuzzy sets. Numerical examples are also provided to illustrate the approach.
Chapter
We develop a fuzzy hypothesis testing approach where we consider the fuzziness of data and the fuzziness of the hypotheses as well. We give the corresponding fuzzy p-value with its $$\alpha$$-cuts. In addition, we use the so-called “signed distance” operator to defuzzify this p-value and we provide the convenient decision rule. Getting a defuzzified p-value and being able to interpret it can be of good use in many situations. We illustrate our testing procedure by a detailed numerical example where we study a right one-sided fuzzy test and compare it with a classical one. We close the paper by an application of the method on a survey from the financial place of Zurich, Switzerland. We display the decisions related to tests on the mean made on a set of variables of the sample. Both fuzzy and classical tests are conducted. One of our main findings is that despite the fact that each of both approaches have a different decision rule in terms of interpretation, the decisions made are by far the same. In this perspective, we can state that the fuzzy testing procedure can be seen as a generalization of the classical one.
Conference Paper
We extend the classical approach of hypothesis testing to the fuzzy environment. We propose a method based on fuzziness of data and on fuzziness of hypotheses at the same time. The fuzzy p-value with its α-cuts is provided and we show how to defuzzify it by the signed distance method. We illustrate our method by numerical applications where we treat a one and a two sided test. For the one-sided test, applying our method to the same data and performing tests on the same significance level, we compare the defuzzified p-values between different cases of null and alternative hypotheses.
Book
Preliminary review / Publisher's description: Statistical data are not always precise numbers, or vectors, or categories. Real data are frequently what is called fuzzy. Examples where this fuzziness is obvious are quality of life data, environmental, biological, medical, sociological and economics data. Also the results of measurements can be best described by using fuzzy numbers and fuzzy vectors respectively. Statistical analysis methods have to be adapted for the analysis of fuzzy data. In this book, the foundations of the description of fuzzy data are explained, including methods on how to obtain the characterizing function of fuzzy measurement results. Furthermore, statistical methods are then generalized to the analysis of fuzzy data and fuzzy a-priori information. Key Features: * Provides basic methods for the mathematical description of fuzzy data, as well as statistical methods that can be used to analyze fuzzy data. * Describes methods of increasing importance with applications in areas such as environmental statistics and social science. * Complements the theory with exercises and solutions and is illustrated throughout with diagrams and examples. * Explores areas such quantitative description of data uncertainty and mathematical description of fuzzy data. This work is aimed at statisticians working with fuzzy logic, engineering statisticians, finance researchers, and environmental statisticians. It is written for readers who are familiar with elementary stochastic models and basic statistical methods.
Article
A definition of fuzzy test for testing statistical hypotheses with vague data is proposed. Then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one-sided or two-sided alternative hypotheses is shown. This fuzzy test, contrary to the classical approach, leads not to the binary decision: to reject or to accept given null hypothesis, but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypothesis, respectively. However, it is a natural generalization of the traditional test, i.e. if the data are precise, not vague, we get a classical statistical test with the binary decision. A measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed.
Article
By using the decomposition principle and the crisp ranking system on R, we construct a new ranking system for fuzzy numbers which is very realistic and also matching our intuition as in R.
Article
Statistical hypothesis testing is very important for finding decisions in practical problems. Usually, the underlying data are assumed to be precise numbers, but it is much more realistic in general to consider fuzzy values which are non-precise numbers. In this case the test statistic will also yield a non-precise number. This article presents an approach for statistical testing at the basis of fuzzy values by introducing the fuzzy p-value. It turns out that clear decisions can be made outside a certain interval which is determined by the characterizing function of the fuzzy p-values. Copyright Springer-Verlag 2004
A new approach of testing fuzzy hypotheses by confidence intervals and defuzzification of the fuzzy decision by the signed distance
• R Berkachy
• L Donzé