Developing a fuzzy Likert scale for measuring xenophobia in Greece

Developing a fuzzy Likert scale for measuring xenophobia in Greece Maria Symeonaki 1, and Aggeliki Kazani 2 1 Panteion University of Political and Social Sciences Department of Social Policy 136 Syggrou Av., 17671, Athens, Greece (E-mail: msimeon@panteion.com) 2 Panteion University of Political and Social Sciences Department of Social Policy 136 Syggrou Av., 17671, Athens, Greece (E-mail: agkazani@gmail.com) Abstract. This paper presents the fuzzification of a Likert scale that was used for measuring xenophobia in Greece in a large-scale survey conducted by National Centre of Social Research of Greece. The main motivation of the paper lies in the fact that closed type units used for the development of these scales, guide the respondent towards given choices that may not fully represent his/her real opinion or attitude. In order to fuzzify the Likert scale three methodological steps were used. Firstly, the operational definitions and relations between the questions-units were developed and a fuzzy system was constructed. Secondly, the fuzzification of input variables was realized and finally, rule-blocks were constructed describing the system of xenophobia. Keywords: Likert scales, fuzzy sets and fuzzy systems, xenophobia, fuzzification. 1 Introduction The mathematical basis of fuzzy logic is founded on fuzzy set theory regarded as a generalization of the classical theory. The classical theory is based on the two-value logic (true-false), namely the classical (Aristotelian) logic. In 1951, Lukasiewiez[3] was the first to visualize the concept of fuzzy logic using an early idea of fuzzy sets. Lukasiewiez proposed a third value, the value possible, and he assigned to it a numeric value between true and false. Fuzzy logic resembles human logic and the human knowledge using abstract concepts. This approach is achieved with the introduction of fuzzy sets which are sets without clear boundaries. The present paper deals with the theory of fuzzy logic and fuzzy systems and especially with the fuzzification of Likert scales. According to the quantum principle of indeterminacy or Zadeh s uncertainty, fuzzy logic provides a satisfactory solution to the principle of incompatibility: As the complexity of a system increases, our ability to make precise and significant statements about its behaviour diminishes until a threshold is reached beyond which precision and significance become almost mutually exclusive characteristics (Zimmermann[9]). Zadeh points that everything is a matter of degree and that any logical system can be fuzzified

(Zadeh[7], Zadeh[8]). The membership in a fuzzy set is not a matter of affirmation or denial, but rather a matter of degree. More specifically, when A is a fuzzy set and x is a relevant object, the statement x is a member of A is not necessarily either true or false, but it may be true only to some degree, being usually a number in the interval [0, 1]. This degree is represented by the membership function of the fuzzy set A. Fuzzy Systems are those systems whose variables have as domain fuzzy sets. They encode structured, empirical (heuristic) or linguistic knowledge in a numerical framework. They are able to describe the operation of the system in natural language with the aid of human-like IF-THEN rules. In Likert scaling the respondent is asked to choose between several response categories, indicating various strengths of agreement or disagreement. Closedtype questionnaire requires respondents to choose only among the available options, which may not reflect to the actual views or attitudes. Likert scales are subject to bias due to various factors: respondents may avoid the acres (central tendency bias), may be inclined to agree with the suggestions as presented (acquiescence response bias) or may want to present an improved image for themselves (social desirability bias). Moreover, there is a matter of subjectivity, particularly in measuring complex social attitudes, when the respondent may not be able to assess accurately his/her attitude (Moser and Calton[5]). It is therefore important to examine an alternative approach using a fuzzy scale for measuring attitude. It is obvious that uncertainty, crucial for measuring the attitude of each person, is being created in the boundaries of the response categories. By fuzzifying these categories we try to reduce this uncertainty and define a new type for measuring attitudes. In Section 2 we focus on the definition of fuzzy scales, after studying the different possibities of fuzzy partitions of the categories. In Section 3 the fuzzy Likert scale for measuring xenophobia is described. Finally, Section 4 describes the methodological steps being followed in order to develop the fuzzy Likert scale. 2 A fuzzy Likert scale for Xenophobia The Likert scale examined is included in the questionnaire of a large-scale survey by the National Centre for Social Research 1 (Michalopoulou et al[4]). In order to examine the effects of xenophobia in the development of Northern Greece/Macedonia three different types of examination were used. The first part of the study focused on measuring xenophobia using a questionnaire through sample survey. The sample of the survey was 1200 people, aged 18-80 years, residents of Macedonia during the time of the fieldwork. In order to explore opinions and attitudes on social stereotypes between foreigners and others a Likert scale was included in the questionnaire of the survey. In this study a random sample (N = 307) of the sample examined (N = 1200) was used. In this specific scale 18 questions-units were used. 1 National Centre of Social Research of Greece: www.ekke.gr

1. Foreigners who live in our country must have equal rights with us. 2. Many of the foreigners who live in our country are responsible for the increase in the crime rate. 3. Like Greeks, foreigners can be either good or bad. 4. Foreigners must have lower wages even when they do the same job as we. 5. A foreigner s children must attend the same schools as our children. 6. The foreigners in our country increase unemployment for Greeks. 7. The local authorities must organize events so we get to know the foreigners who live and work here. 8. I would never marry a foreigner. 9. Foreigners who live in our country gave us the chance to get to know new people. 10. I would never work for a foreigner. 11. We should facilitate foreigners who want to settle in our country. 12. Foreigners who work in our country do harm to our economy. 13. The state must organise programmes of further education to help those foreigners who live in our country. 14. The more foreigners there arrive the lower the wages get. 15. We must create reception departments in our schools for the foreigners children. 16. Only as tourists should foreigners come. 17. Work permits must be given to foreigners who want to live here. 18. We must close our borders to foreigners who come to work here. The units had 5 response categories, ranging from total agreement to total disagreement. For the fuzzification of the scale three methodological steps (as developed in Lala et al[2]) were used. 1st Methodological Step: Identification of the problem The identification of the problem requires a detailed analysis of the problem in order to clarify concepts, develop operational definitions and a description of the relation between variables (Lala et al[2]). In order to construct a hypothetical fuzzy system, abbreviations for the units of the scale are provided, as shown in Table 1. Fuzzy modules are linked together at different levels of importance. Each aggregation produces intermediate variables that have a particular meaning. For example (Figure 1), the question LDP (Less Days Pay) and the question and HE (Harm Economy) create the new variable IOE (Impacts On Economy). In the same way we create a hypothetical fuzzy system for xenophobia, where the 18 units of the scale provide 4 major categories. More specificly these categories are: Rights, Impacts, General Issues and Actions. We assume that four of the units entail two subcategories which we call: SI (Social Impacts) and IOE (Impacts on Economy) (Figure 1). Notice that the separation of both levels of abstraction and fuzzy categories at each level is, in this case, intuitive and takes place with the aid of the experts of the system based and the results of the data analysis (Lala et al[2]). Figure 2 shows the relational diagram of a hypothetical fuzzy system for xenophobia which describes the specific relations of fuzzy modules. The

Table 1. Descriptives of xenophobia per cluster UNITS FUZZY MODULE UNITS FUZZY MODULE Unit 1 ER (Equal Rights) Unit 10 WF (Work Foreigner) Unit2 CR (Criminality) Unit 11 EM (Easy Move) Unit 3 GB (Good/Bad) Unit 12 HE (Harm Economy) Unit 4 LMJ (Less Money/Job) Unit 13 DTP (Develop Training Programs) Unit 5 SS (Same School) Unit 14 LDP (Less Day s Pay) Unit 6 UN (Unemployment) Unit 15 DRC (Develop Reception Classes) Unit 7 GTK (Get To Know) Unit 16 CT (Come as Tourists) Unit 8 MF (Marry Foreigner) Unit 17 PSW (Permission Stay and Work) Unit 9 MNM (Meet New Nations) Unit 18 CB (Close Borders) question CR (Criminality) and the question UN (Unemployment) create the new variable SI (Social Impacts). Moreover, the question LDP (Less Days Pay) and the question HE (Harm Economy) create the new variable, IOE (Impacts On Economy). These two new variables, HE and IOE, together with MNM (Meet New Nations), create the new variable IMPACTS. The variables IMPACTS, RIGHTS, GENERAL ISSUES and ACTIONS compose the variable of XENOPHOBIA. LDL CR IOE SI HE UN Fig. 1. Representation of the new variables IOE and SI. 2nd Methodological Step: Fuzzification of input variables This specific step concerns the definition of the membership functions of all categories for each question and for all questions. This particular scale of measuring xenophobia consists of five categories which we attempt to fuzzify. In order to fuzzify the variables the pollster method was used. This method requires a survey of a target population sample so as to identify a relative frequency distribution of the categories of all questions (Lala et al [2]). The shapes of the empirical distributions will suggest the membership functions. More specifically, the fuzzification of the input variables was carried out by examining the bar charts (Figure 3) to identify the form, the peak and the amplitude of the corresponding fuzzy number, after recoding the reversed questions.

Fig. 2. Hypothetical fuzzy system for xenophobia. Figure 3 represents the frequency bar chart of the variables and Figure 4 the fuzzy partitions of the 18 variables. 3rd Methodological Step: Construction of the rule-blocks Fuzzy knowledge consists of linguistic rules of the form IF-THEN. These rules are usually obtained with the help of the experts of the phenomenon and the system. Based on the figure of the hypothetical fuzzy system for xenophobia (Figure 5) rule-blocks for one of relations of the fuzzy variables were constructed. We define the fuzzy partitions A(1), A(2), A(3) on the domains of the three basic variables respectively Equal Rights (ER), Less Money/Job (LMJ) and Same School (SS). The fuzzy partitions are linguistic representations of their universe of discourse, therefore their elements are linguistic terms like Low, Big, Quite or Less Medium, etc. The relationship of the crisp universe of discourse is represented using linguistic rules that define a mapping of the fuzzy partitions of one level to the fuzzy partitions of another level. This mapping is said to be a fuzzy association and represents the

Fig. 3. Frequency bar charts of the 18 variables. empirical, linguistic rules (Symeonaki[6]). As far as the three variables have a linguistic meaning, heuristic or empirical linguistic rules can be used in order to describe the input-output relationship. Figure 5 shows that the variable Equal Rights (ER), the variable Less Money/ Job (LMJ) and the variable Same School (SS) create the new variable Rights. We assume that the variable RIGHTS can take the values DISBELIEVE, NEUTRAL and BELIEVE. Moreover, the variables ER, LMJ and SS can take the values DISBELIEVE, NEUTRAL and BELIEVE. So a possible realization of linguistic rules for the variable RIGHTS can be the following. 1. IF (ER, LMJ, SS) IS (D, D, D), THEN RIGHTS IS DISBELIEVE 2. IF (ER, LMJ, SS) IS (D, D, N), THEN RIGHTS IS DISBELIEVE 3. IF (ER, LMJ, SS) IS (D, D, B), THEN RIGHTS IS DISBELIEVE. 4. IF (ER, LMJ, SS) IS (D, N, D), THEN RIGHTS IS DISBELIEVE 5. IF (ER, LMJ, SS) IS (D, N, N), THEN RIGHTS IS DISBELIEVE 6. IF (ER, LMJ, SS) IS (D, N, B), THEN RIGHTS IS NEUTRAL 7. IF (ER, LMJ, SS) IS (D, B, D), THEN RIGHTS IS NEUTRAL 8. IF (ER, LMJ, SS) IS (D, B, N), THEN RIGHTS IS NEUTRAL 9. IF (ER, LMJ, SS) IS (D, B, B), THEN RIGHTS IS BELIEVE

Fig. 4. Fuzzy partitions of the 18 variables. 10. IF (ER, LMJ, SS) IS (N, D, D), THEN RIGHTS IS DISBELIEVE 11. IF (ER, LMJ, SS) IS (N, D, N), THEN RIGHTS IS NEUTRAL 12. IF (ER, LMJ, SS) IS (N, D, B), THEN RIGHTS IS NEUTRAL 13. IF (ER, LMJ, SS) IS (N, N, D), THEN RIGHTS IS NEUTRAL 14. IF (ER, LMJ, SS) IS (N, N, N), THEN RIGHTS IS NEUTRAL 15. IF (ER, LMJ, SS) IS (N, N, B), THEN RIGHTS IS NEUTRAL 16. IF (ER, LMJ, SS) IS (N, B, D), THEN RIGHTS IS NEUTRAL 17. IF (ER, LMJ, SS) IS (N, B, D), THEN RIGHTS IS NEUTRAL 18. IF (ER, LMJ, SS) IS (N, B, B), THEN RIGHTS IS BELIEVE 19. IF (ER, LMJ, SS) IS (B, D, D), THEN RIGHTS IS DISBELIEVE 20. IF (ER, LMJ, SS) IS (B, D, N), THEN RIGHTS IS NEUTRAL 21. IF (ER, LMJ, SS) IS (B, D, B), THEN RIGHTS IS BELIEVE 22. IF (ER, LMJ, SS) IS (B, N, D), THEN RIGHTS IS NEUTRAL 23. IF (ER, LMJ, SS) IS (B, N, N), THEN RIGHTS IS NEUTRAL 24. IF (ER, LMJ, SS) IS (B, N, B), THEN RIGHTS IS BELIEVE 25. IF (ER, LMJ, SS) IS (B, B, D), THEN RIGHTS IS NEUTRAL

26. IF (ER, LMJ, SS) IS (B, B, N), THEN RIGHTS IS BELIEVE 27. IF (ER, LMJ, SS) IS (B, B, B), THEN RIGHTS IS BELIEVE In the same manner the linguistic rules for variables IMPACTS, GEN- ERAL ISSUES and ACTIONS are constructed. 3 Conclusions A hypothetical fuzzy system was constructed in order to fuzzify the Likert scale of xenophobia. Fuzzy logic can provide an alternative approach for measuring attitudes in the social sciences since in fuzzy Likert scales intermediate levels of analysis are taken into account. Moreover, a fuzzy system can easily integrate the knowledge of the experts of the system in the fuzzification of the input variables and in the construction of rule-blocks. Thus, the knowledge of the experts of the system can be used not only in forming the questionnaire but also in the analysis helping the researcher to make a multiple analysis for the system he/she studies. A fuzzy scale is a flexible method which could enable the application to place the respondent between 1 and 5 choosing also intermediate values. The combination of the knowledge of experts of the system with a fuzzy system could represent a useful aspect in analyzing individual behaviour. The human action is characterized by complexity as it involves many variables, some of which are not always measurable. The techniques of soft computing, including fuzzy reasoning could be an interesting strategy for modelling social processes. References 1.G.J Klir and B. Yuan. Fuzzy sets and fuzzy logic. Theory and applications, Prentice Hall, USA, 1995. 2.M. Lalla, G. Facchinetti and G. Mastroleo. Ordinal scales and fuzzy set systems to measure agreement: An application to the evaluation of teaching activity, Quality and Quantity, 38, 577-601, 2005. 3.J. Lukasiewicz. Aristotle s Syllogistic from the Standpoint of Modern Formal Logic, Oxford University Press, 2nd Edition, 1951. 4.A. Michalopoulou, P. Tsartas, M. Giannisopoulou, P. Kafetzis and Eudokia Manologlou. Macedonia and the Balkans: Xenophobia and Development, National Centre of Social Research, Alexandria (Abridged English edition), Athens, Greece, 1999. 5.C. Moser and G. Kalton. Survey methods in social investigation, Heineman Educational Books, London, 1971. 6.M. Symeonaki, G. Stamou and S. Tzafestas. Fuzzy non-homogenous Markov systems, Applied Intelligence, 17, 2, 203-214, 2002. 7.L. Zadeh. Fuzzy sets, Information and Control, 8, 3, 1965. 8.L. Zadeh. The calculus of fuzzy if/then rules, AI Expert, 7, 3, 1992. 9.H. Zimmermann. Fuzzy set theory and its applications, Kluwer Academic Publishers, USA, 2001.