Sara de la Rosa de Sáa

Fuzzy Rating Scale-Based Questionnaires and Their Statistical Analysis

The fuzzy rating method has been introduced in psychometric studies as a tool, which allows the capture of and accurate reflection of the diversity, subjectivity, and imprecision inherent in human responses to many questionnaires. The lack of statistical techniques for in-depth analysis of these responses has been, for years, the appearance of an important barrier. At present, this barrier is being overcome thanks to new statistical techniques. In this way, the information from fuzzy rating method-based responses can be suitably explored and exploited. This paper aims to formally endorse some of the main statistical benefits of using free-response format fuzzy rating scale-based questionnaires instead of using the closed-response format involving fuzzy linguistic representations.

A generalized L1-type metric between fuzzy numbers for an approach to central tendency of fuzzy data☆

Abstract In dealing with data generated from a random experiment, L2 metrics are suitable for many statistical approaches and developments. To analyze fuzzy-valued experimental data a generalized L2 metric based on the mid/spread representation of fuzzy values has been stated, and a related methodology to conduct statistics with fuzzy data has been carried out. Most of the developed methods concern either explicitly or implicitly the mean values of the involved random mechanisms producing fuzzy data. Other statistical approaches and studies with experimental data consider L1 metrics, especially in dealing with errors or in looking for a more robust solution and intuitive interpretation. This paper aims to introduce a generalized L1 metric between fuzzy numbers based on a new characterization for them that will be referred to as the mid β / β - leftdev / β -rightdev characterization. More precisely, the metric will take into account both absolute differences in ‘location’ and absolute differences in ‘shape/imprecision’ of fuzzy numbers; moreover one can choose the weight of the influence of the second differences in contrast to the first one. After introducing the new characterization for fuzzy numbers, as well as the associated L1 metric, we will examine some properties. Finally, as an immediate application, the problem of minimizing the mean distance between a fuzzy number and the distribution of a random mechanism producing fuzzy number-valued data will be given, discussed and illustrated.

Descriptive analysis of responses to items in questionnaires. Why not using a fuzzy rating scale

In evaluating aspects like quality perception, satisfaction or attitude which are intrinsically imprecise, the fuzzy rating scale has been introduced as a psychometric tool that allows evaluators to give flexible and quite accurate, albeit non numerical, ratings. The fuzzy rating scale integrates the skills associated with the visual analogue scale, because of the total freedom in assessing ratings, with the ability of fuzzy linguistic variables to capture the natural imprecision in evaluating such aspects.Thanks to a recent methodology, the descriptive analysis of the responses to a fuzzy rating scale-based questionnaire can be now carried out. This paper aims to illustrate such an analysis through a real-life example, as well as to show that statistical conclusions can often be rather different from the conclusions one could get from either Likert scale-based responses or their fuzzy linguistic encoding. This difference encourages the use of the fuzzy rating scale when statistical conclusions are important, similarly to the use of exact real-valued data instead of grouping them.

Hypothesis testing for means in connection with fuzzy rating scale-based data: algorithms and applications

The fuzzy rating scale was introduced as a tool to measure intrinsically ill-defined/ imprecisely-valued attributes in a free way. Thus, users do not have to choose a value from a class of prefixed ones (like it happens when a fuzzy semantic representation of a linguistic term set is considered), but just to draw the fuzzy number that better represents their valuation or measurement. The freedom inherent to the fuzzy rating scale process allows users to collect data with a high level of richness, accuracy, expressiveness, diversity and subjectivity, what is especially valuable for statistical purposes.

Analyzing data from a fuzzy rating scale-based questionnaire. A case study

BACKGROUND The fuzzy rating scale was introduced to cope with the imprecision of human thought and experience in measuring attitudes in many fields of Psychology. The flexibility and expressiveness of this scale allow us to properly describe the answers to many questions involving psychological measurement. METHOD Analyzing the responses to a fuzzy rating scale-based questionnaire is indeed a critical problem. Nevertheless, over the last years, a methodology is being developed to analyze statistically fuzzy data in such a way that the information they contain is fully exploited. In this paper, a summary review of the main procedures is given. RESULTS The methods are illustrated by their application on the dataset obtained from a case study with nine-year-old children. In this study, children replied to some questions from the well-known TIMSS/PIRLS questionnaire by using a fuzzy rating scale. The form could be filled in either on the computer or by hand. CONCLUSIONS The study indicates that the requirements of background and training underlying the fuzzy rating scale are not too demanding. Moreover, it is clearly shown that statistical conclusions substantially often differ depending on the responses being given in accordance with either a Likert scale or a fuzzy rating scale.

Robust scale estimators for fuzzy data

Observations distant from the majority or deviating from the general pattern often appear in datasets. Classical estimates such as the sample mean or the sample variance can be substantially affected by these observations (outliers). Even a single outlier can have huge distorting influence. However, when one deals with real-valued data there exist robust measures/estimates of location and scale (dispersion) which reduce the influence of these atypical values and provide approximately the same results as the classical estimates applied to the typical data without outliers. In real-life, data to be analyzed and interpreted are not always precisely defined and they cannot be properly expressed by using a numerical scale of measurement. Frequently, some of these imprecise data could be suitably described and modelled by considering a fuzzy rating scale of measurement. In this paper, several well-known scale (dispersion) estimators in the real-valued case are extended for random fuzzy numbers (i.e., random mechanisms generating fuzzy-valued data), and some of their properties as estimators for dispersion are examined. Furthermore, their robust behaviour is analyzed using two powerful tools, namely, the finite sample breakdown point and the sensitivity curves. Simulations, including empirical bias curves, are performed to complete the study.

Fuzzy Rating vs. Fuzzy Conversion Scales: An Empirical Comparison through the MSE

The scale of fuzzy numbers have been used in the literature to measurement of many ratings/perceptions/valuations, expectations, and so on. Among the most common uses one can point out: the so-called ‘fuzzy rating’, which is based on a free fuzzy numbered response scheme, and the ‘fuzzy conversion’, which corresponds to the conversion of linguistic (often Likert-type) labels into fuzzy numbers. This paper aims to present an empirical comparison of the two scales. This comparison has been carried out by considering the following steps: fuzzy responses have been first freely simulated; these responses have been ‘Likertized’ in accordance with a five-point measurement and a plausible criterion; each of the five Likert class has been transformed into a fuzzy number (two fuzzification procedures will be examined); the mean squared error (MSE) has been employed to perform the comparison. On the basis of the simulations we will conclude that for most of the simulated samples the Aumann-type mean is more representative for the fuzzy rating than for the fuzzy conversion scale.

Empirical Sensitivity Analysis on the Influence of the Shape of Fuzzy Data on the Estimation of Some Statistical Measures

This paper means an introduction to analyze whether the choice of the shape for fuzzy data in their statistical analysis can or cannot affect the conclusions of such an analysis. More concretely, samples of fuzzy data are simulated in accordance with different assumptions (distributions) concerning four relevant points (namely, those determining their core and support), and later, by preserving core and support, the ‘arms’ are changed by considering trapezoidal, Π-curves, and some LR fuzzy numbers. For the simulations obtained with each of the considered shapes, several characteristics have been estimated: Aumann-type mean, 1-norm and wabl/ldev/rdev medians and Frechet’s variance. A comparative analysis with the bias, mean squared distance and variance of the estimates is finally included.

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 Overview on the Statistical Central Tendency for Fuzzy Data Sets

When Statistics deals with data which cannot be expressed in a numerical scale, the scale of fuzzy values (in particular, the scale of fuzzy numbers) often becomes a suitable tool to express such data. In this way, many ratings, opinions, judgements, etc. mostly coming from human valuations can be appropriately described in terms of fuzzy data. To summarize the central tendency of a fuzzy dataset, some measures have been suggested in the literature. This paper aims to review some of the main ones, and examine their properties in a comparative way. A real-life example illustrates their application. Furthermore, the paper shows the statistical robustness (both through the finite sample breakdown point and a simulation study) and the empirical “precision” of the fuzzy number-valued sample measures. Finally, some related developments and future directions are pointed out.