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.

[1]  Beryl Hesketh,et al.  Work Adjustment Theory: An empirical test using a fuzzy rating scale☆ , 1992 .

[2]  Przemysław Grzegorzewski,et al.  Algorithms for Trapezoidal Approximations of Fuzzy Numbers Preserving the Expected Interval , 2010 .

[3]  Przemyslaw Grzegorzewski,et al.  Fuzzy number approximation via shadowed sets , 2013, Inf. Sci..

[4]  Peter Filzmoser,et al.  Benefits from Using Continuous Rating Scales in Online Survey Research , 2011, ICIS.

[5]  Ana Colubi,et al.  Asymptotic and Bootstrap techniques for testing the expected value of a fuzzy random variable , 2004 .

[6]  Didier Dubois,et al.  Gradualness, uncertainty and bipolarity: Making sense of fuzzy sets , 2012, Fuzzy Sets Syst..

[7]  María Asunción Lubiano,et al.  Fuzzy Rating vs. Fuzzy Conversion Scales: An Empirical Comparison through the MSE , 2012, SMPS.

[8]  Maria Ferraro,et al.  A multiple linear regression model for imprecise information , 2012 .

[9]  Tim Hesketh,et al.  Computerized fuzzy ratings: The concept of a fuzzy class , 1994 .

[10]  María Angeles Gil,et al.  The fuzzy hyperbolic inequality index associated with fuzzy random variables , 1998, Eur. J. Oper. Res..

[11]  M. Puri,et al.  Fuzzy Random Variables , 1986 .

[12]  Ana Colubi,et al.  Estimation of a simple linear regression model for fuzzy random variables , 2009, Fuzzy Sets Syst..

[13]  Yin-Feng Xu,et al.  Computing the Numerical Scale of the Linguistic Term Set for the 2-Tuple Fuzzy Linguistic Representation Model , 2009, IEEE Transactions on Fuzzy Systems.

[14]  Thierry Denoeux,et al.  Nonparametric rank-based statistics and significance tests for fuzzy data , 2005, Fuzzy Sets Syst..

[15]  Ana Colubi,et al.  Interval arithmetic-based simple linear regression between interval data: Discussion and sensitivity analysis on the choice of the metric , 2012, Inf. Sci..

[16]  Ana Colubi,et al.  Bootstrap approach to the multi-sample test of means with imprecise data , 2006, Comput. Stat. Data Anal..

[17]  Ulf-Dietrich Reips,et al.  Interval-level measurement with visual analogue scales in Internet-based research: VAS Generator , 2008, Behavior research methods.

[18]  Qing Li,et al.  A novel Likert scale based on fuzzy sets theory , 2013, Expert Syst. Appl..

[19]  Pedro Terán,et al.  Random fuzzy sets: why, when, how , 2014 .

[20]  José L. Verdegay,et al.  Linguistic decision‐making models , 1992, Int. J. Intell. Syst..

[21]  Phil Diamond Higher level fuzzy numbers arising from fuzzy regression , 1990 .

[22]  Witold Pedrycz,et al.  A fuzzy cognitive structure for pattern recognition , 1989, Pattern Recognit. Lett..

[23]  Stefan Van Aelst,et al.  The median of a random fuzzy number. The 1-norm distance approach , 2012, Fuzzy Sets Syst..

[24]  Francisco Herrera,et al.  A Fuzzy Linguistic Methodology to Deal With Unbalanced Linguistic Term Sets , 2008, IEEE Transactions on Fuzzy Systems.

[25]  Ana Colubi,et al.  Bootstrap techniques and fuzzy random variables: Synergy in hypothesis testing with fuzzy data , 2006, Fuzzy Sets Syst..

[26]  P. Giordani,et al.  Component Models for Fuzzy Data , 2006 .

[27]  Phil Diamond,et al.  Fuzzy least squares , 1988, Inf. Sci..

[28]  Didier Dubois,et al.  Statistical reasoning with set-valued information: Ontic vs. epistemic views , 2014, Int. J. Approx. Reason..

[29]  Didier Dubois,et al.  On the Variability of the Concept of Variance for Fuzzy Random Variables , 2009, IEEE Transactions on Fuzzy Systems.

[30]  Ernesto Damiani,et al.  Concept of Linguistic Variable-Based Fuzzy Ensemble Approach: Application to Interlaced HDTV Sequences , 2009, IEEE Transactions on Fuzzy Systems.

[31]  Jeng-Ming Yih,et al.  An evaluation of airline service quality using the fuzzy weighted SERVQUAL method , 2011, Appl. Soft Comput..

[32]  W. Bossert,et al.  The Measurement of Diversity , 2001 .

[33]  Yu-Cheng Lee,et al.  Service quality gaps analysis based on Fuzzy linguistic SERVQUAL with a case study in hospital out‐patient services , 2010 .

[34]  David García,et al.  Estimating the expected value of fuzzy random variables in the stratified random sampling from finite populations , 2001, Inf. Sci..

[35]  Frank Proske,et al.  Central limit theorem for Banach space valued fuzzy random variables , 2001 .

[36]  Przemyslaw Grzegorzewski,et al.  Trapezoidal approximations of fuzzy numbers with restrictions on the support and core , 2011, EUSFLAT Conf..

[37]  R. Coppi,et al.  A determination coefficient for a linear regression model with imprecise response , 2011 .

[38]  Ana Colubi,et al.  Rejoinder on "A distance-based statistical analysis of fuzzy number-valued data" , 2014, Int. J. Approx. Reason..

[39]  W. Pedrycz Why triangular membership functions , 1994 .

[40]  Witold Pedrycz,et al.  Granular computing with shadowed sets , 2002, Int. J. Intell. Syst..

[41]  Ulf-Dietrich Reips,et al.  Why Semantic Differentials in Web-Based Research Should Be Made from Visual Analogue Scales and Not from 5-Point Scales , 2012 .

[42]  Witold Pedrycz,et al.  From fuzzy sets to shadowed sets: Interpretation and computing , 2009, Int. J. Intell. Syst..

[43]  Humberto Bustince,et al.  Interval Type-2 Fuzzy Sets Constructed From Several Membership Functions: Application to the Fuzzy Thresholding Algorithm , 2013, IEEE Transactions on Fuzzy Systems.

[44]  Zeshui Xu,et al.  Preference Relations Based on Intuitionistic Multiplicative Information , 2013, IEEE Transactions on Fuzzy Systems.

[45]  María Asunción Lubiano,et al.  f-inequality indices for fuzzy random variables , 2002 .

[46]  Kazuhisa Takemura,et al.  A Fuzzy Linear Regression Analysis for Fuzzy Input-Output Data Using the Least Squares Method under Linear Constraints and Its Application to Fuzzy Rating Data , 1999, J. Adv. Comput. Intell. Intell. Informatics.

[47]  Wolfgang Näther Linear Statistical Inference for Random Fuzzy Data , 1997 .

[48]  Ralf Körner An asymptotic α-test for the expectation of random fuzzy variables , 2000 .

[49]  Piotr Prokopowicz,et al.  Defuzzification Functionals of Ordered Fuzzy Numbers , 2013, IEEE Transactions on Fuzzy Systems.

[50]  D. Dubois,et al.  Additions of interactive fuzzy numbers , 1981 .

[51]  Ana Colubi,et al.  SMIRE Research Group at the University of Oviedo: A distance-based statistical analysis of fuzzy number-valued data , 2014, Int. J. Approx. Reason..

[52]  N. Pidgeon,et al.  Fuzzy set analysis for behavioral and social sciences , 1988 .

[53]  Inés Couso,et al.  Approximations of upper and lower probabilities by measurable selections , 2010, Inf. Sci..

[54]  S. Jamieson Likert scales: how to (ab)use them , 2004, Medical education.

[55]  Paolo Giordani,et al.  A Proposal of Robust Regression for Random Fuzzy Sets , 2012, SMPS.

[56]  Pierpaolo D'Urso,et al.  Arithmetic and distance-based approach to the statistical analysis of imprecisely valued data , 2013, SOCO 2013.

[57]  D. Ralescu,et al.  Statistical Modeling, Analysis and Management of Fuzzy Data , 2001 .

[58]  Robert Pryor,et al.  An Application of a Computerized Fuzzy Graphic Rating Scale to the Psychological Measurement of Individual Differences , 1988, Int. J. Man Mach. Stud..

[59]  Janusz Kacprzyk,et al.  "Softer" optimization and control models via fuzzy linguistic quantifiers , 1984, Inf. Sci..

[60]  I. Burhan Turksen,et al.  A fuzzy set preference model for consumer choice , 1994 .

[61]  María Asunción Lubiano,et al.  The λ-mean squared dispersion associated with a fuzzy random variable , 2000, Fuzzy Sets Syst..

[62]  Yin-Feng Xu,et al.  Selecting the Individual Numerical Scale and Prioritization Method in the Analytic Hierarchy Process: A 2-Tuple Fuzzy Linguistic Approach , 2011, IEEE Transactions on Fuzzy Systems.

[63]  E. H. Simpson Measurement of Diversity , 1949, Nature.

[64]  M. Fréchet Les éléments aléatoires de nature quelconque dans un espace distancié , 1948 .

[65]  M. Gil,et al.  RANDOM FUZZY SETS: A MATHEMATICAL TOOL TO DEVELOP STATISTICAL FUZZY DATA ANALYSIS , 2013 .

[66]  M. Puri,et al.  The Concept of Normality for Fuzzy Random Variables , 1985 .

[67]  Rocco J. Perla,et al.  Ten Common Misunderstandings, Misconceptions, Persistent Myths and Urban Legends about Likert Scales and Likert Response Formats and their Antidotes , 2007 .

[68]  Ana Colubi,et al.  A new family of metrics for compact, convex (fuzzy) sets based on a generalized concept of mid and spread , 2009, Inf. Sci..

[69]  Carlo Bertoluzza,et al.  A generalized real-valued measure of the inequality associated with a fuzzy random variable , 2001, Int. J. Approx. Reason..

[70]  Francis Galton,et al.  English men of science : their nature and nurture , 1874 .

[71]  M. Gil,et al.  One-sample tests for a generalized Fréchet variance of a fuzzy random variable , 2010 .

[72]  Przemyslaw Grzegorzewski Trapezoidal approximations of fuzzy numbers preserving the expected interval - Algorithms and properties , 2008, Fuzzy Sets Syst..

[73]  M. Asunción Lubiano,et al.  Estimating the expected value of fuzzy random variables in random samplings from finite populations , 1999 .

[74]  Tim Hesketh,et al.  Use of fuzzy variables in developing new scales from the Strong Interest Inventory , 1995 .

[75]  J. Gerich,et al.  Visual analogue scales for mode-independent measurement in self-administered questionnaires , 2007, Behavior research methods.

[76]  Jeng-Shyang Pan,et al.  Fuzzy Rules Interpolation for Sparse Fuzzy Rule-Based Systems Based on Interval Type-2 Gaussian Fuzzy Sets and Genetic Algorithms , 2013, IEEE Transactions on Fuzzy Systems.

[77]  Przemyslaw Grzegorzewski,et al.  Trapezoidal approximation and aggregation , 2011, Fuzzy Sets Syst..

[78]  Kazuhisa Takemura,et al.  Ambiguity and Social Judgment: Fuzzy Set Model and Data Analysis , 2012 .

[79]  Wolfgang Näther,et al.  Linear Regression with Random Fuzzy Variables: Extended Classical Estimates, Best Linear Estimates, Least Squares Estimates , 1998, Inf. Sci..

[80]  Yukio Ogura,et al.  Central limit theorems for generalized set-valued random variables , 2004 .

[81]  Witold Pedrycz,et al.  Induction of Shadowed Sets Based on the Gradual Grade of Fuzziness , 2013, IEEE Transactions on Fuzzy Systems.

[82]  Wolfgang Näther Regression with fuzzy random data , 2006, Comput. Stat. Data Anal..

[83]  Ana Colubi,et al.  A generalized strong law of large numbers , 1999 .

[84]  María Asunción Lubiano,et al.  K-sample tests for equality of variances of random fuzzy sets , 2012, Comput. Stat. Data Anal..

[85]  José Muñiz,et al.  Effect of the Number of Response Categories on the Reliability and Validity of Rating Scales , 2008 .

[86]  Ana Colubi,et al.  A fuzzy representation of random variables: An operational tool in exploratory analysis and hypothesis testing , 2006, Comput. Stat. Data Anal..

[87]  R. Aitken Measurement of feelings using visual analogue scales. , 1969, Proceedings of the Royal Society of Medicine.

[88]  Vanessa Loh,et al.  A Future-Oriented Retirement Transition Adjustment Framework , 2011 .

[89]  Francisco Herrera,et al.  Hesitant Fuzzy Linguistic Term Sets for Decision Making , 2012, IEEE Transactions on Fuzzy Systems.

[90]  Carlo Bertoluzza,et al.  On a new class of distances between fuzzy numbers , 1995 .

[91]  Bernard De Baets,et al.  Only Smooth Rule Bases Can Generate Monotone Mamdani-Assilian Models Under Center-of-Gravity Defuzzification , 2009, IEEE Trans. Fuzzy Syst..

[92]  Ana Colubi,et al.  Nonparametric criteria for supervised classification of fuzzy data , 2011, Int. J. Approx. Reason..

[93]  María Asunción Lubiano,et al.  SAFD — An R Package for Statistical Analysis of Fuzzy Data , 2013 .

[94]  María Asunción Lubiano,et al.  Two-sample hypothesis tests of means of a fuzzy random variable , 2001, Inf. Sci..

[95]  Witold Pedrycz,et al.  Shadowed sets: representing and processing fuzzy sets , 1998, IEEE Trans. Syst. Man Cybern. Part B.

[96]  Volker Krätschmer,et al.  Limit theorems for fuzzy-random variables , 2002, Fuzzy Sets Syst..

[97]  Dimitrios K. Iakovidis,et al.  Intuitionistic Fuzzy Cognitive Maps for Medical Decision Making , 2011, IEEE Transactions on Information Technology in Biomedicine.

[98]  Ana Colubi,et al.  Computational Statistics and Data Analysis Fuzzy Data Treated as Functional Data: a One-way Anova Test Approach , 2022 .

[99]  Dan Meng,et al.  On weighted unbalanced linguistic aggregation operators in group decision making , 2013, Inf. Sci..

[100]  Pedro Terán Probabilistic foundations for measurement modelling with fuzzy random variables , 2007, Fuzzy Sets Syst..

[101]  E. Giné,et al.  Bootstrapping General Empirical Measures , 1990 .

[102]  Lotfi A. Zadeh,et al.  Outline of a New Approach to the Analysis of Complex Systems and Decision Processes , 1973, IEEE Trans. Syst. Man Cybern..

[103]  María Angeles Gil,et al.  Estimating the fuzzy inequality associated with a fuzzy random variable in random samplings from finite populations , 1998, Kybernetika.

[104]  Ana Colubi,et al.  Statistical inference about the means of fuzzy random variables: Applications to the analysis of fuzzy- and real-valued data , 2009, Fuzzy Sets Syst..

[105]  Kazuhisa Takemura,et al.  Ambiguous comparative judgment: Fuzzy set model and data analysis , 2007 .

[106]  María Angeles Gil,et al.  A generalized L1-type metric between fuzzy numbers for an approach to central tendency of fuzzy data , 2013, Inf. Sci..