Antonia Salas

A hypothesis testing-based discussion on the sensitivity of means of fuzzy data with respect to data shape

Abstract In assessing fuzzy numbers to model imprecise data associated with random experiments, trapezoidal fuzzy numbers are often considered. Such an assessment is mainly due to easing both interpretation and computation. This becomes especially noticeable when those assessing fuzzy numbers to data have a weak knowledge, low background and little or no expertise in using fuzzy sets (as it happens when questionnaires whose responses involve a free fuzzy rating are conducted), since the required training to explain the meaning and use of trapezoidal fuzzy numbers is definitely lower than that associated with other shapes. Nevertheless, a question that constantly arises in connection with this trapezoidal assessment is whether it can importantly affect the conclusions of the study involving such data. This paper aims to answer the last question from a statistical perspective. More concretely, the analysis of the influence of the choice of trapezoidal fuzzy numbers to model data is to be based on the conclusions from statistical hypothesis testing about the mean values of the involved fuzzy datasets. For this purpose, the p -values of tests have been compared for trapezoidal assessment vs. other frequently used ones, like some LU assessments. The analysis is first performed by developing simulation-based pairwise comparisons, and it is later illustrated and corroborated to some extent with a real-life example. The analysis indicates that the shape of the fuzzy assessment scarcely affects statistical conclusions.

Hypothesis testing-based comparative analysis between rating scales for intrinsically imprecise data

Abstract In previous papers, it has been empirically proved that descriptive (summary measures) and inferential conclusions (in particular, tests about means p -values) with imprecise-valued data are often affected by the scale considered to model such data. More concretely, conclusions from the numerical and fuzzy linguistic encodings of Likert-type data have been compared with those for fuzzy data obtained by using a totally free fuzzy assessment: the so-called fuzzy rating scale. These previous comparisons have been performed separately for each of the scales. This paper aims to perform a joint comparison in such a way that means of linked data (one associated with the fuzzy rating and the other one with the encoded Likert scale) are to be tested for equality. Two real-life examples, as well as several simulation-based synthetic ones, have unequivocally shown that the fuzzy rating scale means are significantly different from those for the encoded Likert scales.

An Empirical Analysis of the Coherence Between Fuzzy Rating Scale- and Likert Scale-Based Responses to Questionnaires

In dealing with questionnaires concerning satisfaction , quality perception , attitude, judgement , etc., the fuzzy rating scale has been introduced as a flexible way to respond to questionnaires’ items. Designs for this type of questionnaires are often based on Likert scales. This paper aims to examine three different real-life examples in which respondents have been allowed to doubly answer: in accordance with either a fuzzy rating scale or a Likert one. By considering a minimum distance-based criterion, each of the fuzzy rating scale answers is associated with one of the Likert scale labels. The percentages of coincidences between the two responses in the double answer are computed by the criterion-based association. Some empirical conclusions are drawn from the computation of such percentages.

Sequential Bayesian test from fuzzy experimental information

This paper is devoted to the sequential problem of testing hypotheses about an experiment, when its outcomes do not provide exact but rather fuzzy information.

An Alternative to the Variation Coefficient

The aim of this paper is to introduce an invariant by translation coefficient different from the variation one (widely used in literature but not fulfilling that property) that allows us to study whether the mean is a good representation of the distribution or not. The value of this new coefficient for a normally distributed random variable is obtained in order to establish a criterion, similar to the one used in the symmetry or kurtosis coefficients, to decide the grade of representation of the mean.

Linear regression in a fuzzy context. The least square method

This paper deals with the statistical linear regression. Its aim is to extend the classical least square method to the case where the observations are not crisp, but fuzzy numbers. In order to attain our purpose, we introduce a suitable and very general squared distance between fuzzy numbers which substitute the classical (a−b)2 on the real line. Then we use a well known theorem of functional theory to prove that, under suitable and reasonable conditions, there exists a unique set of fuzzy coefficients which minimize the sum of the squared distances between the previsions and the observations. Nevertheless the classical variational methods cannot be used to find the regression coefficient due to the particular structure of the domain of the functional to be minimized (its interior is empty). So we complete the paper by describing a numerical solution based on active constraint method.