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Fundamental characteristics and statistical analysis of ordinal variables: a review

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Abstract

The measurement of several concepts used in social sciences generates an ordinal variable, which is characterized by rawness of the output values and presents some much debated problems in data analysis. In fact, the need for effective analysis is easily satisfied with parametric models that deal with quantitative variables. However, the peculiarities of the ordinal scales, and the crude values produced by them, limit the use of parametric models, which has generated conflicting favourable and unfavourable views of the parametric approach. The main distinctive features of ordinal scales, some of which are critical points and nodal issues, are illustrated here along with the construction processes. Among the traditional procedures, the most common ordinal scales are described, including the Likert, semantic differential, feeling thermometers, and the Stapel scale. A relative new method, based on fuzzy sets, can be used to handle and generate ordinal variables. Therefore, the structure of a fuzzy inference system is exemplified in synthetic terms to show the treatment of ordinal variables to obtain one or more response variables. The nature of ordinal variables influences the interpretation and selection of many strategies used for their analysis. Four approaches are illustrated (nonparametric, parametric, latent variables, and fuzzy inference system), highlighting their potential and drawbacks. The modelling of an ordinal dependent variable (loglinear models, ordinary parametric models or logit and probit ordinal models, latent class models and hybrid models) is affected by the various approaches.

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References

  • Agresti, A.: Categorical Data Analysis. John Wiley & Sons, New York (1990)

    Google Scholar 

  • Aiello, F., Attanasio, M.: How to transform a batch of simple indicators to make up a unique one?. In: Atti della XLII Riunione Scientifica: Sessioni Plenarie e Specializzate, pp. 327–338. SIS, 9–11 giugno. University of Bari, Bari, IT (2004)

  • Aitchison, J., Silvey, S.D.: The generalization of probit analysis to the case of multiple responses. Biometrika 44(1/2), 131–140 (1957)

    Article  Google Scholar 

  • Alvarez, R.M., Bailey, D., Katz, J.N.: An empirical bayes approach to estimating ordinal treatment effects. Polit. Anal. 19(1 Winter), 20–31 (2011)

    Article  Google Scholar 

  • Amemiya, T.: Qualitative response models: a survey. J. Econ. Lit. 19(4), 1483–1538 (1981)

    Google Scholar 

  • ANES, American National Election Studies: Pre- post- election study. Survey Research Center (S473) (1964) http://www.electionstudies.org/studypages/1964prepost/int1964.txt

  • Babbie, E.: The Practice of Social Research. Cengage Learning, 12th edn. Wadsworth, Belmont (2010)

    Google Scholar 

  • Bernardi, L.: Misurazione e valutazione: le difficoltà di una coppia alle prime esperienze in comune. In: Bertin, G. (ed.) Valutazione e sapere sociologico. Metodi e tecniche di gestione dei processi decisionali, pp. 69–82. Franco Angeli, Milano (1995)

    Google Scholar 

  • Bernardi, L., Capursi, V., Librizzi, L.: Measurement awareness: the use of indicators between expectations and opportunities. In: Atti della XLII Riunione Scientifica: Sessioni Plenarie e Specializzate, pp. 315–326. SIS, 9–11 giugno. University of Bari, Bari, IT (2004)

  • Bernberg, R.E.: Socio-psychological factors in individual morale: I. The prediction of specific indicators. J. Soc. Psychol. 36(1), 73–82 (1952)

    Article  Google Scholar 

  • Bollen, K.A.: Structural Equations with Latent Variables. John Wiley & Sons, New York (1989)

    Book  Google Scholar 

  • Cacciola, S., Marradi, A.: Contributo al dibattito sulle scale Likert basato sull’analisi di interviste registrate. In: Marradi, A. (ed.) Costruire il dato. Sulle tecniche di raccolta delle informazioni nelle scienze sociali, pp. 63–102. Franco Angeli, Milano (1988)

    Google Scholar 

  • Campostrini, S., Ghellini, G., Tuzzi, A.: Con senso di misura. Riflessi statistici da alcuni allievi di Lorenzo Bernardi. CLEUP, Padova (2015)

    Google Scholar 

  • Cantril, H., Free, L.A.: Hopes and fears for self and country: the self-anchoring striving scale in cross-cultural research. Am. Behav. Sci. 6(2, Supplement: Oct.), 1–30 (1962)

    Google Scholar 

  • Conover, W.J.: Practical Nonparametric Statistics, 3rd edn. John Wiley & Sons, New York (1999)

    Google Scholar 

  • Coombs, C.H.: Psychological scaling without a unit of measurement. Psychol. Rev. 57(3), 145–158 (1950)

    Article  Google Scholar 

  • Coombs, C.H.: Theory and method of social measurement. In: Festinger, L., Katz, D. (eds.) Research Methods in the Behavioral Sciences, pp. 471–535. Dryden, New York (1953)

    Google Scholar 

  • Crespi, I.: Use of a scaling technique in surveys. J. Market. 25(July), 69–72 (1961)

    Article  Google Scholar 

  • Crespi, L.P.: Public opinion toward conscientious objectors: II. Measurement of national approval-disapproval. J. Psychol. 19(2), 209–250 (1945a)

    Article  Google Scholar 

  • Crespi, L.P.: Public opinion toward conscientious objectors: III. Intensity of social rejection in stereotype and attitude. J. Psychol. 19(2), 251–276 (1945b)

    Article  Google Scholar 

  • Das, S.: Quantifying fuzziness due to the scale of measurement in response systems. Fuzzy Sets Syst. 132(3), 317–333 (2002)

    Article  Google Scholar 

  • Das, S.: On measuring imprecision in human response due to respondent and attribute and its utility in questionnaire design. Int. J. Uncertain. Fuzz. 14(2), 155–173 (2006)

    Article  Google Scholar 

  • Domingo-Ferrer, J., Torra, V.: Extending microaggregation procedures using defuzzification methods for categorical variables. Proceedings of first International IEEE symposium on intelligent systems, Verna, Sept. pp. 44–49 (2002)

  • Dubois, D., Prade, H. (eds.): Fundamentals of Fuzzy Sets. Kluwer Academic Publ, Boston (2000)

    Google Scholar 

  • Dubois, D., Prade, H., Gil, M.A., Grzegorzewsky, P., Hryniewicz, O. (eds.): Soft Methods for Handling Variability and Imprecision. Springer-Verlag, Heidelberg (2008)

    Google Scholar 

  • Farebrother, R.W.: A class of shrinkage estimators. J. R. Stat. Soc. B 40(1), 47–49 (1977)

    Google Scholar 

  • Gilula, Z., Krieger, A.M., Ritov, Y.: Ordinal association in contingency tables: some interpretive aspects. J. Am. Stat. Assoc. 83(402), 540–545 (1988)

    Article  Google Scholar 

  • Goodman, L.A.: Exploratory latent structure analysis using both identifiable and unidentifiable models. Biometrika 61(2), 215–231 (1974)

    Article  Google Scholar 

  • Goodman, L.A.: Simple models for the analysis of association in cross-classifications having ordered categories. J. Am. Stat. Assoc. 74(367), 537–552 (1979)

    Article  Google Scholar 

  • Greene, W.H.: Econometric Analysis, 5th edn. Prentice Hall, Upper Saddle River (2003)

    Google Scholar 

  • Guttman, L.A.: The basis for scalogram analysis. In: Stouffer, S. (ed.) Measurement and Prediction, pp. 60–90. Princeton University Press, New York (1950)

    Google Scholar 

  • Guttman, L.A.: A general nonmetric technique for finding the smallest coordinate space for a configuration of points. Psychometrika 33(4), 469–506 (1968)

    Article  Google Scholar 

  • Hagenaars, J.A., McCutcheon, A.L.: Applied Latent Class Analysis. Kluwer, Dordrecht (2002)

    Book  Google Scholar 

  • Hand, D.J.: Measurement theory and practice. The world through quantification. Arnold, London (2004)

    Google Scholar 

  • Hoaglin, D.C., Mosteller, F., Tukey, J.W.: Understanding Robust and Exploratory Data Analysis. John Wiley & Sons, New York (1983)

    Google Scholar 

  • Hofacker, C.F.: Categorical judgment scaling with ordinal assumptions. Multivar. Behav. Res. 19(1), 91–106 (1984)

    Article  Google Scholar 

  • Hofmans, J., Theuns, P., Van Acker, F.: Combining quality and quantity. A psychometric evaluation of the self-anchoring scale. Qual. Quant. 43(5), 703–716 (2009)

    Article  Google Scholar 

  • Jöreskog, K.G.: A general method for estimating a linear structural equation system. In: Goldberger, A.S., Duncan, O.D. (eds.) Structural Equation Models in the Social Sciences, pp. 85–112. Seminar Press, New York (1973)

    Google Scholar 

  • Jöreskog, K.G.: New developments in LISREL: analysis of ordinal variables using polychoric correlations and weighted least squares. Qual. Quant. 24(4), 387–404 (1990)

    Article  Google Scholar 

  • Jöreskog, K.G.: On the estimation of polychoric correlations and their asymptotic covariance matrix. Psychometrika 59(3), 381–389 (1994)

    Article  Google Scholar 

  • Jöreskog, K.G., Sörbom, D.: Advances in Factor Analysis and Structural Equation Models. Abt books, Cambridge (1979)

    Google Scholar 

  • Juster, F.T.: Prediction and consumer buying intentions. Am. Econ. Rev. 50(2), 604–617 (1960)

    Google Scholar 

  • Juster, F.T.: Consumer buying intentions and purchase probability: an experiment in survey design. J. Am. Stat. Assoc. 61(315), 658–696 (1966)

    Article  Google Scholar 

  • Kampen, J., Swyngedouw, M.: The ordinal controversy revisited. Qual. Quant. 34(1), 87–102 (2000)

    Article  Google Scholar 

  • Khurshid, A., Sahai, H.: Scales of measurement: an introduction and selected bibliography. Qual. Quant. 27(3), 303–323 (1993)

    Article  Google Scholar 

  • Kilpatrick, F.P., Cantril, H.: Self-anchoring scaling: a measure of individuals’ unique reality worlds. J. Individ. Psychol. 16(2), 158–173 (1960)

    Google Scholar 

  • Lalla, M.: Le scale ordinali e i relativi problemi operativi. In: Campostrini, S., Ghellini, G., Tuzzi, A. (eds.) Con senso di misura Riflessi statistici da alcuni allievi di Lorenzo Bernardi, pp. 35–52. CLEUP, Padova (2015)

    Google Scholar 

  • Lalla, M., Facchinetti, G., Mastroleo, G.: Ordinal scales and fuzzy set systems to measure agreement: an application to the evaluation of teaching activity. Qual. Quant. 38(5), 577–601 (2004)

    Article  Google Scholar 

  • Lalla, M., Ferrari, D., Pirotti, T.: Fuzzy inference systems to analyze ordinal variables—the case of evaluating teaching activity. In: Proceedings of the international conference on fuzzy computation theory and applications. SciTePress—Science and Technology Publications Digital Library, Setubal, pp. 25–36 (2014)

  • Landenna, G., Marasini, D.: Metodi statistici non parametrici. il Mulino, Bologna (1990)

    Google Scholar 

  • Likert, R.: A technique for the measurement of attitudes. Archives of Psychology 140, 1–55 (1932)

    Google Scholar 

  • Lord, F.M.: On the statistical treatment of football members. Am. Psychol. 8(12), 750–751 (1953)

    Article  Google Scholar 

  • Marradi, A.: Termometri con vincolo di ordinabilità: il «gioco della torre» consente di aggirare la tendenza alla desiderabilità sociale? Sociologia e ricerca sociale 57, 49–59 (1998)

    Google Scholar 

  • Marradi, A.: Metodologia delle scienze sociali. il Mulino, Bologna (2007)

    Google Scholar 

  • Mantel, N.: Chi square tests with one degree of freedom: extensions of the Mantel-Haenszel procedure. J. Am. Stat. Assoc. 58(303), 690–700 (1963)

    Google Scholar 

  • McCullagh, P.: Regression models for ordinal data (with discussion). J. R. Stat. Soc. B 42(2), 109–142 (1980)

    Google Scholar 

  • McKelvey, R.D., Zavoina, W.: A statistical model for the analysis of ordinal level dependent variables. J. Math. Sociol. 4(1), 103–120 (1975)

    Article  Google Scholar 

  • Niederée, R.: There is more to measurement than just measurement: measurement theory, symmetry, and substantive theorizing. J. Math. Psychol. 38(4), 527–594 (1994)

    Article  Google Scholar 

  • Nowakowska, M.: Methodological problem of measurement of fuzzy concepts in the social sciences. Behav. Sci. 22(2), 107–115 (1977)

    Article  Google Scholar 

  • Osgood, C.E.: The nature of measurement and meaning. Psychol. Bull. 49(3), 197–237 (1952)

    Article  Google Scholar 

  • Osgood, C.E., Suci, G.J., Tannenbaum, R.H.: The Measurement of Meaning. University of Illinois Press, Urbana (1957)

    Google Scholar 

  • Pearson, K.: On a new method of determining correlation between a measured character A, and a character B, of which only the percentage of cases wherein B exceeds (or falls short of) a given intensity is recorded for each grades of A. Biometrika 7(1/2), 96–105 (1909)

    Article  Google Scholar 

  • Pesarin, F.: Multivariate Permutation Tests: With Application in Biostatistics. John Wiley & Sons, Chichester (2001)

    Google Scholar 

  • Pesarin, F., Salmaso, L.: Permutation Tests for Complex Data: Theory, Applications and Software. John Wiley & Sons, Chichester (2010)

    Book  Google Scholar 

  • Prytulac, L.S.: A critique of S. S. Stevens’ theory of measurement scale classification. Percept. Motor Skills 41(1), 3–28 (1975)

    Article  Google Scholar 

  • Ricolfi, L.: Operazioni di ricerca e scale. Rassegna italiana di sociologia XXVI(2), 189–227 (1985)

    Google Scholar 

  • Savage, I.R.: Nonparametric statistics. J. Am. Stat. Assoc. 52(279), 331–344 (1957)

    Article  Google Scholar 

  • Smithson, M.J.: Fuzzy Set Analysis for Behavioral and social sciences. Springer, Heidelberg (1987)

    Book  Google Scholar 

  • Smithson, M.J.: Fuzzy set theory and the social sciences: the scope for applications. Fuzzy Sets Syst. 26(1), 1–21 (1988)

    Article  Google Scholar 

  • Siegel, S., Castellan Jr, N.J.: Nonparametric Statistics for the Behavioral Science. McGraw-Hill, New York (1988)

    Google Scholar 

  • Stevens, S.S.: On the theory of scale measurement. Science 103(2684), 677–680 (1946)

    Article  Google Scholar 

  • Stevens, S.S.: Mathematics, measurement, and psychophysics. In: Stevens, S.S. (ed.) Handbook of Experimental Psychology, pp. 1–49. Wiley & Sons, New York (1951)

    Google Scholar 

  • Symeonaki, M., Michalopoulou, C., Kazani, A.: A fuzzy set theory solution to combining likert items into a single overall scale (or subscales). Qual. Quant. 49(2), 739–762 (2015)

    Article  Google Scholar 

  • Tanaka, H.: Fuzzy data analysis by possibilistic linear models. Fuzzy Sets Syst. 24(3), 363–375 (1987)

    Article  Google Scholar 

  • Thurstone, L.L.: A law of comparative judgment. Psychol. Rev. 34(4), 273–286 (1927a)

    Article  Google Scholar 

  • Thurstone, L.L.: The method of paired comparison for social values. J. Abnorm. Soc. Psychol. 21(4), 384–397 (1927b)

    Article  Google Scholar 

  • Thurstone, L.L.: Attitudes can be measured. Am. J. Sociol. 33(4), 529–554 (1928)

    Article  Google Scholar 

  • Tukey, J.W.: Exploratory Data Analysis. Addison-Wesley, Reading (1977)

    Google Scholar 

  • Van Leekwijck, W., Kerre, E.E.: Defuzzification: criteria and classification. Fuzzy Sets Syst. 108(1), 159–178 (1999)

    Article  Google Scholar 

  • Velleman, P.F., Wilkinson, L.: Ordinal, interval, and ratio typologies are misleading. Am. Stat. 47(1), 65–72 (1993)

    Google Scholar 

  • Viertl, R.: Statistical Methods for Fuzzy Data. John Wiley and Sons, New Delhi (2011)

    Book  Google Scholar 

  • Von Altrock, C.: Fuzzy logic and neurofuzzy applications in business and finance. Prentice Hall PTR, Upper Saddle River (1997)

    Google Scholar 

  • Weisberg, H.F., Rusk, J.G.: Dimensions of candidate evaluation. Am. Polit. Sci. Rev. 64(4), 1167–1185 (1970)

    Article  Google Scholar 

  • Westermann, R.: Interval-scale measurement of attitudes: some theoretical conditions and empirical testing methods. Br. J. Math. Stat. Psychol. 36(2), 228–239 (1983)

    Article  Google Scholar 

  • White, M.: Psychological technique and social problems. Southwest. Polit. Soc. Sci. Q. 7, 58–73 (1926)

    Google Scholar 

  • Wilson, T.P.: Critique of ordinal variables. Soc. Forces 49(3), 432–444 (1971)

    Article  Google Scholar 

  • Yang, M., Lin, T.: Fuzzy least-squares linear regression analysis for fuzzy input–output data. Fuzzy Sets Syst. 126(3), 389–399 (2002)

    Article  Google Scholar 

  • Yu, J.H., Albaum, G., Swenson, M.: Is a central tendency error inherent in the use of semantic differential scales in different cultures? Int. J. Market Res. 45(2), 213–228 (2003)

    Google Scholar 

  • Zadeh, L.A.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)

    Article  Google Scholar 

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Acknowledgments

Part of this paper, specifically Sect. 2, has been previously published (Lalla 2015) in the volume edited by Sefano Campostrini, Giulio Ghellini, and Arjuna Tuzzi (2015), a collection of papers written by pupils and colleagues in honour of Professor Lorenzo Bernardi, who was a fine, versatile, and brilliant academic and social statistician. Overall, he was a master, a mentor and a friend for many of us. This paper is dedicated to his memory.

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  1. Department of Economics “Marco Biagi”, CAPP (Centre for the Analysis of Public Policies), University of Modena and Reggio Emilia, Via Jacopo Berengario 51, 41121, Modena, Italy

    Michele Lalla

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Lalla, M. Fundamental characteristics and statistical analysis of ordinal variables: a review. Qual Quant 51, 435–458 (2017). https://doi.org/10.1007/s11135-016-0314-5

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