Evaluation of Linguistic Questionnaire

Abstract

Linguistic questionnaires have gained lots of attention in the last decades. They are a prominent tool used in many fields to convey the opinion of people on different subjects. In this chapter, we propose a model for the assessment of linguistic questionnaires describing the global and individual evaluations, where we suppose that the sample weights and the missingness are both allowed. For the problem of missingness, we show a method based on the readjustment of weights. We should clarify that the proposed approach is not a correction for the missingness in the sample as widely known in survey statistics. We give the expressions of the individual and global evaluations, followed by the description of the indicators of information rate related to missing answers. The model is after illustrated by a numerical application related to the Finanzplatz data set. The objective of this empirical application is to clearly see that the obtained individual evaluations can be treated similarly to any data set in the classical theory. In addition, we will empirically remark that the obtained distributions tend to be normally distributed. Afterward, we perform different analyses by simulations on the statistical measures of these distributions. We compare the individual evaluations with respect to a variety of distances, in order to see the influence of the symmetry of the modelling fuzzy numbers and the sample sizes on different statistical measures. We close the chapter by a comparison between the evaluations by the defended model, and the ones obtained through a usual fuzzy system using different defuzzification operators. Interesting findings of this chapter are that corresponding statistical measures are independent from the sample sizes, and that the use of the traditional fuzzy rule-based systems is not always the most convenient tool when non-symmetrical modelling shapes are used, contrariwise to the defended approach. Our approach by the individual evaluations is in such situations suggested.

Appendix

The first section is devoted to the description of the Finanzplatz data set.

1.1 B Description of the Finanzplatz Data Base

The questionnaire, the so-called Finanzplatz Zurich: Umfrage 2010 described in BAK (2010) is a survey of the financial place of Zurich, Switzerland. On behalf of the Department of Economic Affairs of the Canton of Zurich, the ASAM (Applied Statistics And Modelling) group was in charge of the design plan and the data analysis of this survey in 2010. The chosen sample is mainly composed of financial enterprises, i.e. banks or insurances of the canton of Zurich and the surrounding. The aim of this survey is to investigate the situation of the financial market of the canton of Zurich, by answering to a written questionnaire about their actual situation and their views for the future. As such, it intends to understand the expectations of the demand, the income, and the employment of the different firms for the foreseeable future.

This questionnaire consists of 21 questions from which 19 are linguistic ones with five Likert scale terms each. This survey is then recognized as a linguistic one. These questions are divided into 4 groups. The data base is composed of 245 observations, i.e. firms. Variables related to company size and branch are also present, as well as the sampling weights. From this questionnaire, we consider 18 variables only, including 15 linguistic ones divided into 3 groups. The Table 7.4 gives a detailed description of the considered attributes.

Table 7.4 The variables of the Finanzplatz data set—Sect. B

Table 7.5 Summary of the weighted statistical measures of the individual evaluations using the signed distance, classified by company size—Sect. 7.5

Table 7.6 Summary of the weighted statistical measures of the individual evaluations using the signed distance, classified by company branch—Sect. 7.5

Table 7.7 Summary of the weighted statistical measures of the distributions obtained from individual evaluations using the signed distance, for cases with and without NA—Sect. 7.5

Table 7.8 Summary of the statistical measures of the distribution obtained from the individual evaluations for all distances—Sect. 7.6

Table 7.8 (continued)

Note that non-response exists in this data base. We excluded the records for which a particular firm did not answer to any of the questions of this questionnaire. The size of the sample becomes then 234 observations.

1.2 C Results of the Applications of Chap. 7

This Annex provides the detailed tables of results of the studies of Chap. 7. Tables 7.5 and 7.6 show the summary of the weighted statistical measures of the individual evaluations using the signed distance, classified by company size and company branch correspondingly. Table 7.7 shows the weighted statistical measures of the distributions obtained from individual evaluations using the signed distance, for cases with and without “NA”. Finally, Table 7.8 gives the summary of the statistical measures of the distributions obtained from individual evaluations using all distances, while Table 7.9 provides the measures of the output distribution of the fuzzy rule-based system with different defuzzification operators.

Table 7.9 Summary of the statistical measures of the output distribution of the fuzzy rule-base system with different defuzzification operators—Sect. 7.6

Table 7.9 (continued)