Scale measures with fuzzy data: analyzing their robustness and application to fuzzy rating scale-based questionnaires

Abstract

The Likert-type scales are frequently used in designing questionnaires to rate characteristics or attributes that cannot be numerically measured (like satisfaction, perceived quality, perception...). Although they are easy to answer and they do not require a special training to use them, the available statistical methodology to analyze the data from these questionnaires is rather limited. This is mainly due to the fact that Likert scales are discrete with a very small number of responses to choose for each item (often 4 or 5). To overcome this concern, some alternatives have been suggested in the literature, such as the visual analogue scales, the fuzzy linguistic scales or the fuzzy rating scales. The questionnaires based on this last scale have a free-response format, allowing the rater to draw the fuzzy number that better expresses his/her response to the given item, within a reference bounded interval. In this way, the values can cope (to a full extent) with the intrinsic imprecision associated with the ratings. On the other hand, the diversity and subjectivity of the responses are not lost because it is a continuous scale, and there is a substantial gain of information and accuracy in the conclusions. Moreover, along the last years a statistical methodology is being developed to analyze fuzzy-based data, allowing us to treat them in a similar way to the numerical data. A substantial part of the thesis work is devoted to summarize the statistical information involved in datasets that are based on a fuzzy rating by analyzing their scale estimate, where it is intended as a representative measure of the dispersion of the imprecise-valued attributes supplying the datasets. The Fréchet-type variance (or the corresponding standard deviation) is the best known and used dispersion measure in the fuzzy context. It preserves the main valuable properties of the variance (standard deviation) for real-valued data, but it also inherits its high sensitivity to the presence of outliers or atypical/extreme observations in the data. Therefore, it is desirable to introduce and study other scale measures and to analyze their robust behaviour. Aiming to achieve this goal, several measures of scale to deal with real-valued data are extended to deal with fuzzy-valued data. After proving some of their most relevant properties, such as the invariance by translation, the (absolute) equivariance by the product by scalars or the strong consistency, their robust behaviour is analyzed deeply from a theoretical point of view and also by means of simulations. This robustness is measured in terms of the finite sample breakdown point and the sensitivity curves. The other main contribution of the work is to compare the mentioned different rating scales to model/deal with imprecise-valued data through some studies based on simulations and also on real-life questionnaires. In particular, the Likert-type scale, through both its numerical and fuzzy linguistic encodings, and the fuzzy rating scale are compared. The first comparative tool to be analyzed will be the diversity, and it will be proved that it is always higher, under quite general reasonable conditions, for the questionnaires based on the fuzzy rating scale. This supports the idea that this scale allows us to capture better the subjectivity and diversity of responses. Then, an inferential approach based on a bootstrapped test about the equality of variances/standard deviations for independent populations is presented and applied to two questionnaires-based case studies. Finally, the three scales are compared by means of some descriptive analyses mainly based on the dispersion/scale measures studied in the first main part of the thesis.