{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,4]],"date-time":"2026-08-04T15:46:28Z","timestamp":1785858388262,"version":"3.56.0"},"reference-count":72,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T00:00:00Z","timestamp":1721952000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T00:00:00Z","timestamp":1721952000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["PID2019-104486GB-I00"],"award-info":[{"award-number":["PID2019-104486GB-I00"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100011941","name":"Gobierno del Principado de Asturias","doi-asserted-by":"publisher","award":["AYUD\/2021\/50897"],"award-info":[{"award-number":["AYUD\/2021\/50897"]}],"id":[{"id":"10.13039\/100011941","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Gobierno del Principado de Asturias - Programa Severo Ochoa","award":["BP22-173"],"award-info":[{"award-number":["BP22-173"]}]},{"DOI":"10.13039\/501100006382","name":"Universidad de Oviedo","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006382","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Data Anal Classif"],"published-print":{"date-parts":[[2025,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Along recent years, interval-valued rating scales have been considered as an alternative to traditional single-point psychometric tools for human evaluations, such as Likert-type or visual analogue scales. More concretely, in answering to intrinsically imprecise items in a questionnaire, interval-valued scales seem to allow capturing a richer information than conventional ones. When analyzing data from given performances of questionnaires, one of the main targets is that of ensuring the internal consistency of the items in a construct or latent variable. The most popular indicator of internal consistency, whenever answers to items are given in accordance with a numerically based\/encoded scale, is the well-known Cronbach <jats:italic> \u03b1<\/jats:italic> coefficient. This paper aims to extend such a coefficient to the case of interval-valued answers and to analyze some of its main statistical properties. For this purpose, after presenting some formal preliminaries for interval-valued data, firstly Cronbach\u2019s <jats:italic> \u03b1<\/jats:italic> coefficient is extended to the case in which the constructs of a questionnaire allow interval-valued answers to their items. The range of the potential values of the extended coefficient is then discussed. Furthermore, the asymptotic distribution of the sample Cronbach <jats:italic> \u03b1<\/jats:italic> coefficient along with its bias and consistency properties, are examined from a theoretical perspective. Finally, the preceding asymptotic distribution of the sample coefficient as well as the influence of the number of respondents to the questionnaire and the number of items in the constructs are empirically illustrated through simulation-based studies.<\/jats:p>","DOI":"10.1007\/s11634-024-00601-w","type":"journal-article","created":{"date-parts":[[2024,7,26]],"date-time":"2024-07-26T20:02:04Z","timestamp":1722024124000},"page":"831-854","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["On some properties of Cronbach\u2019s \u03b1 coefficient for interval-valued data in questionnaires"],"prefix":"10.1007","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0991-1866","authenticated-orcid":false,"given":"Jos\u00e9","family":"Garc\u00eda-Garc\u00eda","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0096-104X","authenticated-orcid":false,"given":"Mar\u00eda \u00c1ngeles","family":"Gil","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9847-5164","authenticated-orcid":false,"given":"Mar\u00eda Asunci\u00f3n","family":"Lubiano","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,7,26]]},"reference":[{"key":"601_CR1","doi-asserted-by":"publisher","unstructured":"Alcacer A, Mart\u00ednez-Garc\u00eda M, Epifanio I (2024) Ordinal classification for interval-valued data and interval-valued functional data. Expert Syst Appl 238(Part F):122277. https:\/\/doi.org\/10.1016\/j.eswa.2023.122277","DOI":"10.1016\/j.eswa.2023.122277"},{"issue":"1","key":"601_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0022-247X(65)90049-1","volume":"12","author":"RJ Aumann","year":"1965","unstructured":"Aumann RJ (1965) Integrals of set-valued functions. J Math Anal Appl 12(1):1\u201312. https:\/\/doi.org\/10.1016\/0022-247X(65)90049-1","journal-title":"J Math Anal Appl"},{"issue":"2","key":"601_CR3","first-page":"71","volume":"2","author":"C Bertoluzza","year":"1995","unstructured":"Bertoluzza C, Corral N, Salas A (1995) On a new class of distances between fuzzy numbers. Mathw Soft Comput 2(2):71\u201384","journal-title":"Mathw Soft Comput"},{"issue":"462","key":"601_CR4","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1198\/016214503000242","volume":"98","author":"L Billard","year":"2003","unstructured":"Billard L, Diday E (2003) From the statistics of data to the statistics of knowledge: symbolic data analysis. J Am Stat Assoc 98(462):470\u2013487. https:\/\/doi.org\/10.1198\/016214503000242","journal-title":"J Am Stat Assoc"},{"issue":"9","key":"601_CR5","doi-asserted-by":"publisher","first-page":"2568","DOI":"10.1016\/j.csda.2011.03.005","volume":"55","author":"\u00c1 Blanco-Fern\u00e1ndez","year":"2011","unstructured":"Blanco-Fern\u00e1ndez \u00c1, Corral N, Gonz\u00e1lez-Rodr\u00edguez G (2011) Estimation of a flexible simple linear model for interval data based on set arithmetic. Comput Stat Data Anal 55(9):2568\u20132578. https:\/\/doi.org\/10.1016\/j.csda.2011.03.005","journal-title":"Comput Stat Data Anal"},{"key":"601_CR6","series-title":"Advances in intelligent systems and computing","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/978-3-642-33042-1_12","volume-title":"Synergies of soft computing and statistics for intelligent data analysis","author":"\u00c1 Blanco-Fern\u00e1ndez","year":"2013","unstructured":"Blanco-Fern\u00e1ndez \u00c1, Colubi A, Garc\u00eda-B\u00e1rzana M, Montenegro M (2013) A linear regression model for interval-valued response based on set arithmetic. In: Kruse R, Berthold M, Moewes C, Gil M\u00c1, Grzegorzewski P, Hryniewicz O (eds) Synergies of soft computing and statistics for intelligent data analysis. Advances in intelligent systems and computing, vol 190. Springer, Berlin, Heidelberg, pp 105\u2013113, https:\/\/doi.org\/10.1007\/978-3-642-33042-1_12"},{"issue":"1","key":"601_CR7","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1080\/02664763.2011.575125","volume":"39","author":"P Brito","year":"2012","unstructured":"Brito P, Duarte Silva AP (2012) Modelling interval data with normal and skew-normal distributions. J Appl Stat 39(1):3\u201320. https:\/\/doi.org\/10.1080\/02664763.2011.575125","journal-title":"J Appl Stat"},{"key":"601_CR8","doi-asserted-by":"publisher","unstructured":"Carmines EG, Zeller RA (1979) Reliability and validity assessment, sage university paper series on quantitative applications in the social sciences, vol 17. Sage, Beverly Hills. https:\/\/doi.org\/10.4135\/9781412985642","DOI":"10.4135\/9781412985642"},{"issue":"7","key":"601_CR9","doi-asserted-by":"publisher","first-page":"1660","DOI":"10.1016\/j.jmva.2005.05.012","volume":"97","author":"A Christmann","year":"2006","unstructured":"Christmann A, Van Aelst S (2006) Robust estimation of Cronbach\u2019s alpha. J Multivar Anal 97(7):1660\u20131674. https:\/\/doi.org\/10.1016\/j.jmva.2005.05.012","journal-title":"J Multivar Anal"},{"issue":"3","key":"601_CR10","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1109\/TFUZZ.2002.1006441","volume":"10","author":"A Colubi","year":"2002","unstructured":"Colubi A, Fern\u00e1ndez-Garc\u00eda C, Gil M\u00c1 (2002) Simulation of random fuzzy variables: an empirical approach to statistical\/probabilistic studies with fuzzy experimental data. IEEE Trans Fuzzy Syst 10(3):384\u2013390. https:\/\/doi.org\/10.1109\/TFUZZ.2002.1006441","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"601_CR11","volume-title":"Introduction to Classical and Modern Test Theory","author":"LM Crocker","year":"1986","unstructured":"Crocker LM, Algina J (1986) Introduction to Classical and Modern Test Theory. Holt, Rinehart and Winston"},{"issue":"3","key":"601_CR12","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1007\/BF02310555","volume":"16","author":"LJ Cronbach","year":"1951","unstructured":"Cronbach LJ (1951) Coefficient alpha and the internal structure of tests. Psychometrika 16(3):297\u2013334. https:\/\/doi.org\/10.1007\/BF02310555","journal-title":"Psychometrika"},{"issue":"1","key":"601_CR13","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1109\/TFUZZ.2014.2307895","volume":"23","author":"S de la Rosa de S\u00e1a","year":"2015","unstructured":"de la Rosa de S\u00e1a S, Gil M\u00c1, Gonz\u00e1lez-Rodr\u00edguez G, L\u00f3pez MT, Lubiano MA (2015) Fuzzy rating scale-based questionnaires and their statistical analysis. IEEE Trans Fuzzy Syst 23(1):111\u2013126. https:\/\/doi.org\/10.1109\/TFUZZ.2014.2307895","journal-title":"IEEE Trans Fuzzy Syst"},{"key":"601_CR14","doi-asserted-by":"publisher","unstructured":"de la Rosa de S\u00e1a S, Lubiano MA, Sinova B, Filzmoser P, (2017) Robust scale estimators for fuzzy data. Adv Data Anal Classif 11(4):731\u2013758. https:\/\/doi.org\/10.1007\/s11634-015-0210-1","DOI":"10.1007\/s11634-015-0210-1"},{"issue":"6","key":"601_CR15","doi-asserted-by":"publisher","first-page":"1682","DOI":"10.1109\/TFUZZ.2020.2984203","volume":"29","author":"S de la Rosa de S\u00e1a","year":"2021","unstructured":"de la Rosa de S\u00e1a S, Lubiano MA, Sinova B, Gil M\u00c1, Filzmoser P (2021) Location-free robust scale estimates for fuzzy data. IEEE Trans Fuzzy Syst 29(6):1682\u20131694. https:\/\/doi.org\/10.1109\/TFUZZ.2020.2984203","journal-title":"IEEE Trans Fuzzy Syst"},{"issue":"3","key":"601_CR16","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1016\/j.ejor.2016.09.006","volume":"258","author":"S Dias","year":"2017","unstructured":"Dias S, Brito P (2017) Off the beaten track: a new linear model for interval data. Eur J Ope Res 258(3):1118\u20131130. https:\/\/doi.org\/10.1016\/j.ejor.2016.09.006","journal-title":"Eur J Ope Res"},{"issue":"1","key":"601_CR17","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1016\/j.ejor.2021.01.025","volume":"294","author":"S Dias","year":"2021","unstructured":"Dias S, Brito P, Amaral P (2021) Discriminant analysis of distributional data via fractional programming. Eur J Ope Res 294(1):206\u2013218. https:\/\/doi.org\/10.1016\/j.ejor.2021.01.025","journal-title":"Eur J Ope Res"},{"issue":"3","key":"601_CR18","doi-asserted-by":"publisher","first-page":"516","DOI":"10.1007\/s00357-015-9189-8","volume":"32","author":"AP Duarte Silva","year":"2015","unstructured":"Duarte Silva AP, Brito P (2015) Discriminant analysis of interval data: an assessment of parametric and distance-based approaches. J Classif 32(3):516\u2013541. https:\/\/doi.org\/10.1007\/s00357-015-9189-8","journal-title":"J Classif"},{"issue":"2","key":"601_CR19","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/j.chemolab.2003.11.005","volume":"70","author":"P D\u2019Urso","year":"2004","unstructured":"D\u2019Urso P, Giordani P (2004) A least squares approach to principal component analysis for interval valued data. Chemometr Intell Lab Syst 70(2):179\u2013192. https:\/\/doi.org\/10.1016\/j.chemolab.2003.11.005","journal-title":"Chemometr Intell Lab Syst"},{"issue":"2","key":"601_CR20","doi-asserted-by":"publisher","first-page":"251","DOI":"10.1007\/s00180-006-0262-y","volume":"21","author":"P D\u2019Urso","year":"2006","unstructured":"D\u2019Urso P, Giordani P (2006) A robust fuzzy k-means clustering model for interval valued data. Comput Stat 21(2):251\u2013269. https:\/\/doi.org\/10.1007\/s00180-006-0262-y","journal-title":"Comput Stat"},{"key":"601_CR21","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/j.patcog.2016.04.005","volume":"58","author":"P D\u2019Urso","year":"2016","unstructured":"D\u2019Urso P, Leski JM (2016) Fuzzy c-ordered medoids clustering for interval-valued data. Pattern Recognit 58:49\u201367. https:\/\/doi.org\/10.1016\/j.patcog.2016.04.005","journal-title":"Pattern Recognit"},{"issue":"1","key":"601_CR22","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s11634-014-0169-3","volume":"9","author":"P D\u2019Urso","year":"2015","unstructured":"D\u2019Urso P, De Giovanni L, Massari R (2015) Trimmed fuzzy clustering for interval-valued data. Adv Data Anal Classif 9(1):21\u201340. https:\/\/doi.org\/10.1007\/s11634-014-0169-3","journal-title":"Adv Data Anal Classif"},{"issue":"1","key":"601_CR23","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/s10700-016-9238-8","volume":"16","author":"P D\u2019Urso","year":"2017","unstructured":"D\u2019Urso P, Massari R, De Giovanni L, Cappelli C (2017) Exponential distance-based fuzzy clustering for interval-valued data. Fuzzy Optim Decis Making 16(1):51\u201370. https:\/\/doi.org\/10.1007\/s10700-016-9238-8","journal-title":"Fuzzy Optim Decis Making"},{"key":"601_CR24","doi-asserted-by":"publisher","unstructured":"Ellerby Z, Wagner C (2021) Do people prefer to give interval-valued or point estimates and why? In: 2021 IEEE international conference on fuzzy systems (FUZZ-IEEE), pp 1\u20136. https:\/\/doi.org\/10.1109\/FUZZ45933.2021.9494507","DOI":"10.1109\/FUZZ45933.2021.9494507"},{"issue":"3","key":"601_CR25","doi-asserted-by":"publisher","first-page":"1240","DOI":"10.3758\/s13428-021-01635-0","volume":"54","author":"Z Ellerby","year":"2022","unstructured":"Ellerby Z, Wagner C, Broomell SB (2022) Capturing richer information: on establishing the validity of an interval-valued survey response mode. Behav Res Methods 54(3):1240\u20131262. https:\/\/doi.org\/10.3758\/s13428-021-01635-0","journal-title":"Behav Res Methods"},{"key":"601_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-4549-5","volume-title":"A course in large sample theory","author":"TS Ferguson","year":"1996","unstructured":"Ferguson TS (1996) A course in large sample theory. Chapman and Hall, London"},{"issue":"4","key":"601_CR27","first-page":"215","volume":"10","author":"M Fr\u00e9chet","year":"1948","unstructured":"Fr\u00e9chet M (1948) Les \u00e9l\u00e9ments al\u00e9atoires de nature quelconque dans un espace distanci\u00e9. Ann Inst Henri Poincar\u00e9 10(4):215\u2013310","journal-title":"Ann Inst Henri Poincar\u00e9"},{"issue":"2","key":"601_CR28","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1037\/h0074329","volume":"14","author":"M Freyd","year":"1923","unstructured":"Freyd M (1923) The graphic rating scale. J Educ Psychol 14(2):83\u2013102. https:\/\/doi.org\/10.1037\/h0074329","journal-title":"J Educ Psychol"},{"key":"601_CR29","doi-asserted-by":"crossref","unstructured":"Garc\u00eda-Garc\u00eda J (2022) IntervalQuestionStat: tools to deal with interval-valued responses in questionnaires. https:\/\/CRAN.R-project.org\/package=IntervalQuestionStat, R package version 0.2.0","DOI":"10.32614\/CRAN.package.IntervalQuestionStat"},{"key":"601_CR30","doi-asserted-by":"publisher","unstructured":"Garc\u00eda-Garc\u00eda J, Gil M\u00c1, Lubiano MA (2024) Empirical sensitivity analysis on the choice of the metric on Cronbach\u2019s $$\\alpha$$ coefficient for interval-valued data in questionnaires. In: Ansari J, Fuchs S, Trutschnig W, Lubiano MA, Gil M\u00c1, Grzegorzewski P, Hryniewicz O (eds) Combining, modelling and analyzing imprecision, Randomness and Dependence. Advances in intelligent systems and computing series. Springer Nature Switzerland AG, Cham. https:\/\/doi.org\/10.1007\/978-3-031-65993-5_18","DOI":"10.1007\/978-3-031-65993-5_18"},{"issue":"1","key":"601_CR31","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BF02595831","volume":"10","author":"M\u00c1 Gil","year":"2001","unstructured":"Gil M\u00c1, L\u00f3pez-Garc\u00eda MT, Lubiano MA, Montenegro M (2001) Regression and correlation analyses of a linear relation between random intervals. TEST 10(1):183\u2013201. https:\/\/doi.org\/10.1007\/BF02595831","journal-title":"TEST"},{"issue":"2","key":"601_CR32","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s001840100160","volume":"56","author":"M\u00c1 Gil","year":"2002","unstructured":"Gil M\u00c1, Lubiano MA, Montenegro M, L\u00f3pez MT (2002) Least squares fitting of an affine function and strength of association for interval-valued data. Metrika 56(2):97\u2013111. https:\/\/doi.org\/10.1007\/s001840100160","journal-title":"Metrika"},{"issue":"6","key":"601_CR33","doi-asserted-by":"publisher","first-page":"3002","DOI":"10.1016\/j.csda.2006.01.015","volume":"51","author":"M\u00c1 Gil","year":"2007","unstructured":"Gil M\u00c1, Gonz\u00e1lez-Rodr\u00edguez G, Colubi A, Montenegro M (2007) Testing linear independence in linear models with interval-valued data. Comput Stat Data Anal 51(6):3002\u20133015. https:\/\/doi.org\/10.1016\/j.csda.2006.01.015","journal-title":"Comput Stat Data Anal"},{"issue":"5","key":"601_CR34","doi-asserted-by":"publisher","first-page":"642","DOI":"10.1016\/j.ins.2008.10.018","volume":"179","author":"G Gonz\u00e1lez-Rodr\u00edguez","year":"2009","unstructured":"Gonz\u00e1lez-Rodr\u00edguez G, Colubi A, Trutschnig W (2009) Simulation of fuzzy random variables. Inf Sci 179(5):642\u2013653. https:\/\/doi.org\/10.1016\/j.ins.2008.10.018","journal-title":"Inf Sci"},{"issue":"4","key":"601_CR35","doi-asserted-by":"publisher","first-page":"943","DOI":"10.1016\/j.csda.2010.06.013","volume":"56","author":"G Gonz\u00e1lez-Rodr\u00edguez","year":"2012","unstructured":"Gonz\u00e1lez-Rodr\u00edguez G, Colubi A, Gil M\u00c1 (2012) Fuzzy data treated as functional data. a one-way anova test approach. Comput Stat Data Anal 56(4):943\u2013955. https:\/\/doi.org\/10.1016\/j.csda.2010.06.013","journal-title":"Comput Stat Data Anal"},{"issue":"1","key":"601_CR36","doi-asserted-by":"publisher","first-page":"1650","DOI":"10.2991\/ijcis.d.201012.003","volume":"13","author":"P Grzegorzewski","year":"2020","unstructured":"Grzegorzewski P, Hryniewicz O, Romaniuk M (2020) Flexible bootstrap for fuzzy data based on the canonical representation. Int J Comput Intell Syst 13(1):1650\u20131662. https:\/\/doi.org\/10.2991\/ijcis.d.201012.003","journal-title":"Int J Comput Intell Syst"},{"key":"601_CR37","volume-title":"Grundz\u00fcge der Mengenleher","author":"F Hausdorff","year":"1914","unstructured":"Hausdorff F (1914) Grundz\u00fcge der Mengenleher. Verlag Von Veit & Comp, Leipzig"},{"key":"601_CR38","first-page":"98","volume":"18","author":"MHS Hayes","year":"1921","unstructured":"Hayes MHS, Patterson DG (1921) Experimental development of the graphic rating method. Psychol Bull 18:98\u201399","journal-title":"Psychol Bull"},{"issue":"2","key":"601_CR39","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1007\/s11634-016-0245-y","volume":"11","author":"K Hron","year":"2017","unstructured":"Hron K, Brito P, Filzmoser P (2017) Exploratory data analysis for interval compositional data. Adv Data Anal Classif 11(2):223\u2013241. https:\/\/doi.org\/10.1007\/s11634-016-0245-y","journal-title":"Adv Data Anal Classif"},{"issue":"2","key":"601_CR40","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1007\/s11634-014-0176-4","volume":"9","author":"A Irpino","year":"2015","unstructured":"Irpino A, Verde R (2015) Basic statistics for distributional symbolic variables: a new metric-based approach. Adv Data Anal Classif 9(2):143\u2013175. https:\/\/doi.org\/10.1007\/s11634-014-0176-4","journal-title":"Adv Data Anal Classif"},{"issue":"2","key":"601_CR41","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1016\/j.amc.2007.02.113","volume":"191","author":"GR Jahanshahloo","year":"2007","unstructured":"Jahanshahloo GR, Hosseinzadeh Lotfi F, Rezai Balf F, Zhiani Rezai H (2007) Discriminant analysis of interval data using Monte Carlo method in assessment of overlap. Appl Math Comput 191(2):521\u2013532. https:\/\/doi.org\/10.1016\/j.amc.2007.02.113","journal-title":"Appl Math Comput"},{"key":"601_CR42","doi-asserted-by":"publisher","unstructured":"Kong L, Gao X (2024) A regularized MM estimate for interval-valued regression. Expert Syst Appl 238, Part C:122044. https:\/\/doi.org\/10.1016\/j.eswa.2023.122044","DOI":"10.1016\/j.eswa.2023.122044"},{"issue":"2","key":"601_CR43","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1007\/s11634-016-0255-9","volume":"11","author":"JG Le-Rademacher","year":"2017","unstructured":"Le-Rademacher JG, Billard L (2017) Principal component analysis for histogram-valued data. Adv Data Anal Classif 11(2):327\u2013351. https:\/\/doi.org\/10.1007\/s11634-016-0255-9","journal-title":"Adv Data Anal Classif"},{"key":"601_CR44","first-page":"140","volume":"22","author":"R Likert","year":"1932","unstructured":"Likert R (1932) A technique for the measurement of attitudes. Arch Psychol 22:140\u2013155","journal-title":"Arch Psychol"},{"issue":"4","key":"601_CR45","doi-asserted-by":"publisher","first-page":"897","DOI":"10.1007\/s11634-022-00527-1","volume":"17","author":"LC Lin","year":"2023","unstructured":"Lin LC, Guo M, Lee S (2023) Monitoring photochemical pollutants based on symbolic interval-valued data analysis. Adv Data Anal Classif 17(4):897\u2013926. https:\/\/doi.org\/10.1007\/s11634-022-00527-1","journal-title":"Adv Data Anal Classif"},{"key":"601_CR46","volume-title":"Statistical theories of mental test scores","author":"FM Lord","year":"2008","unstructured":"Lord FM, Novick MR, Birnbaum A (2008) Statistical theories of mental test scores. Addison-Wesley Pub. Co., Reading"},{"key":"601_CR47","doi-asserted-by":"publisher","unstructured":"Lubiano MA, de la Rosa de S\u00e1a S, Montenegro M, Sinova B, Gil M\u00c1 (2016) Descriptive analysis of responses to items in questionnaires. Why not using a fuzzy rating scale? Inf Sci 160:131\u2013148. https:\/\/doi.org\/10.1016\/j.ins.2016.04.029","DOI":"10.1016\/j.ins.2016.04.029"},{"key":"601_CR48","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1016\/j.ins.2020.10.042","volume":"550","author":"MA Lubiano","year":"2021","unstructured":"Lubiano MA, Garc\u00eda-Izquierdo AL, Gil M\u00c1 (2021) Fuzzy rating scales: does internal consistency of a measurement scale benefit from coping with imprecision and individual differences in psychological rating? Inf Sci 550:91\u2013108. https:\/\/doi.org\/10.1016\/j.ins.2020.10.042","journal-title":"Inf Sci"},{"key":"601_CR49","doi-asserted-by":"publisher","unstructured":"Lubiano MA, Garc\u00eda-Garc\u00eda J, Garc\u00eda-Izquierdo AL, Casta\u00f1o AM (2023a) The extended version of Cohen\u2019s $$d$$ index for interval-valued data. In: Garc\u00eda-Escudero LA, Gordaliza A, Mayo A, Lubiano MA, Gil M\u00c1, Grzegorzewski P, Hryniewicz O (eds) Building Bridges between Soft and Statistical Methodologies for Data Science, Advances in Intelligent Systems and Computing, vol 1433. Springer, Cham, pp 263\u2013270, https:\/\/doi.org\/10.1007\/978-3-031-15509-3_35","DOI":"10.1007\/978-3-031-15509-3_35"},{"key":"601_CR50","doi-asserted-by":"publisher","unstructured":"Lubiano MA, Montenegro M, P\u00e9rez-Fern\u00e1ndez S, Gil M\u00c1 (2023b) Analyzing the influence of the rating scale for items in a questionnaire on Cronbach coefficient alpha. In: Balakrishnan N, Gil M\u00c1, Mart\u00edn N, Morales D, Pardo MdC (eds) Trends in mathematical, information and data sciences: a tribute to Leandro Pardo. Springer, Cham, pp 377\u2013388, https:\/\/doi.org\/10.1007\/978-3-031-04137-2_32","DOI":"10.1007\/978-3-031-04137-2_32"},{"key":"601_CR51","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/BF01445180","volume":"57","author":"H Minkowski","year":"1903","unstructured":"Minkowski H (1903) Volumen und oberf\u00e4che. Math Ann 57:447\u2013495","journal-title":"Math Ann"},{"key":"601_CR52","doi-asserted-by":"publisher","unstructured":"Montenegro M, Casals MR, Colubi A, Gil M\u00c1 (2008) Testing \u201ctwo-sided\u201d hypothesis about the mean of an interval-valued random set. In: Dubois D, Lubiano MA, Prade H, Gil M\u00c1, Grzegorzewski P, Hryniewicz O (eds) Soft methods for handling variability and imprecision, advances in soft computing, vol\u00a048. Springer, Berlin, pp 133\u2013139, https:\/\/doi.org\/10.1007\/978-3-540-85027-4_17","DOI":"10.1007\/978-3-540-85027-4_17"},{"key":"601_CR53","volume-title":"Psychometric theory","author":"JC Nunnally","year":"1967","unstructured":"Nunnally JC (1967) Psychometric theory. McGraw-Hill, New York"},{"key":"601_CR54","doi-asserted-by":"publisher","first-page":"591","DOI":"10.1016\/j.ins.2016.08.068","volume":"372","author":"AB Ramos-Guajardo","year":"2016","unstructured":"Ramos-Guajardo AB, Grzegorzewski P (2016) Distance-based linear discriminant analysis for interval-valued data. Inf Sci 372:591\u2013607. https:\/\/doi.org\/10.1016\/j.ins.2016.08.068","journal-title":"Inf Sci"},{"key":"601_CR55","unstructured":"R Core Team (2022) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, https:\/\/www.R-project.org\/"},{"issue":"4","key":"601_CR56","doi-asserted-by":"crossref","first-page":"843","DOI":"10.1007\/s11336-021-09789-8","volume":"86","author":"Revisiting Cronbach\u2019s Alpha","year":"2021","unstructured":"Revisiting Cronbach\u2019s Alpha (2021) Special section. Psychometrika 86(4):843\u2013892","journal-title":"Psychometrika"},{"issue":"14","key":"601_CR57","doi-asserted-by":"publisher","first-page":"5883","DOI":"10.1007\/s00500-018-3251-5","volume":"23","author":"M Romaniuk","year":"2019","unstructured":"Romaniuk M, Hryniewicz O (2019) Interval-based, nonparametric approach for resampling of fuzzy numbers. Soft Comput 23(14):5883\u20135903. https:\/\/doi.org\/10.1007\/s00500-018-3251-5","journal-title":"Soft Comput"},{"issue":"3","key":"601_CR58","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1109\/TFUZZ.2019.2957253","volume":"29","author":"M Romaniuk","year":"2021","unstructured":"Romaniuk M, Hryniewicz O (2021) Discrete and smoothed resampling methods for interval-valued fuzzy numbers. IEEE Trans Fuzzy Syst 29(3):599\u2013611. https:\/\/doi.org\/10.1109\/TFUZZ.2019.2957253","journal-title":"IEEE Trans Fuzzy Syst"},{"issue":"1","key":"601_CR59","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s11336-008-9101-0","volume":"74","author":"K Sijtsma","year":"2009","unstructured":"Sijtsma K (2009) On the use, the misuse, and the very limited usefulness of Cronbach\u2019s alpha. Psychometrika 74(1):107\u2013120. https:\/\/doi.org\/10.1007\/s11336-008-9101-0","journal-title":"Psychometrika"},{"key":"601_CR60","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/j.chemolab.2016.05.003","volume":"156","author":"B Sinova","year":"2016","unstructured":"Sinova B (2016) M-estimators of location for interval-valued random elements. Chemometr Intell Lab Syst 156:115\u2013127. https:\/\/doi.org\/10.1016\/j.chemolab.2016.05.003","journal-title":"Chemometr Intell Lab Syst"},{"key":"601_CR61","doi-asserted-by":"publisher","unstructured":"Sinova B, Van Aelst S (2013) Comparing the medians of a random interval defined by means of two different $${L}^{1}$$ metrics. In: Borgelt C, Gil M\u00c1, Sousa JM, Verleysen M (eds) Towards advanced data analysis by combining soft computing and statistics, studies in fuzziness and soft computing, vol 285. Springer, Berlin, pp 75\u201386. https:\/\/doi.org\/10.1007\/978-3-642-30278-7_7","DOI":"10.1007\/978-3-642-30278-7_7"},{"key":"601_CR62","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1016\/j.spl.2015.02.014","volume":"100","author":"B Sinova","year":"2015","unstructured":"Sinova B, Van Aelst S (2015) On the consistency of a spatial-type interval-valued median for random intervals. Stat Probab Lett 100:130\u2013136. https:\/\/doi.org\/10.1016\/j.spl.2015.02.014","journal-title":"Stat Probab Lett"},{"issue":"3","key":"601_CR63","doi-asserted-by":"publisher","first-page":"479","DOI":"10.1080\/02331888.2018.1435658","volume":"52","author":"B Sinova","year":"2018","unstructured":"Sinova B, Van Aelst S (2018) A spatial-type interval-valued median for random intervals. Statistics 52(3):479\u2013502. https:\/\/doi.org\/10.1080\/02331888.2018.1435658","journal-title":"Statistics"},{"key":"601_CR64","doi-asserted-by":"publisher","unstructured":"Sinova B, Colubi A, Gil M\u00c1, Gonz\u00e1lez-Rodr\u00edguez G (2012) Interval arithmetic-based simple linear regression between interval data: Discussion and sensitivity analysis on the choice of the metric. Inf Sci 199:109\u2013124. https:\/\/doi.org\/10.1016\/j.ins.2012.02.040","DOI":"10.1016\/j.ins.2012.02.040"},{"key":"601_CR65","doi-asserted-by":"publisher","unstructured":"Sinova B, Gonz\u00e1lez-Rodr\u00edguez G, Van\u00a0Aelst S (2013) An alternative approach to the median of a random interval using an $${L}^{2}$$ metric. In: Kruse R, Berthold MR, Moewes C, Gil M\u00c1, Grzegorzewski P, Hryniewicz O (eds) Synergies of Soft Computing and Statistics for Intelligent Data Analysis, Advances in Intelligent Systems and Computing, vol 190. Springer, Berlin, pp 273\u2013281. https:\/\/doi.org\/10.1007\/978-3-642-33042-1_30","DOI":"10.1007\/978-3-642-33042-1_30"},{"issue":"7","key":"601_CR66","doi-asserted-by":"publisher","first-page":"1487","DOI":"10.1016\/j.ijar.2013.09.020","volume":"55","author":"SMIRE Research Group","year":"2014","unstructured":"SMIRE Research Group (2014) A distance-based statistical analysis of fuzzy number-valued data. Int J Approx Reason 55(7):1487\u20131501. https:\/\/doi.org\/10.1016\/j.ijar.2013.09.020","journal-title":"Int J Approx Reason"},{"issue":"4","key":"601_CR67","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1177\/1470785319831227","volume":"61","author":"C Themistocleous","year":"2019","unstructured":"Themistocleous C, Pagiaslis A, Smith A, Wagner C (2019) A comparison of scale attributes between interval-valued and semantic differential scales. Int J Mark Res 61(4):394\u2013407. https:\/\/doi.org\/10.1177\/1470785319831227","journal-title":"Int J Mark Res"},{"key":"601_CR68","doi-asserted-by":"publisher","unstructured":"van der Vaart A (1998) Asymptotic statistics. cambridge series in statistical and probabilistic mathematics. Cambridge University Press, Cambridge. https:\/\/doi.org\/10.1017\/CBO9780511802256","DOI":"10.1017\/CBO9780511802256"},{"issue":"3","key":"601_CR69","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1007\/BF02296146","volume":"65","author":"JM van Zyl","year":"2000","unstructured":"van Zyl JM, Neudecker H, Nel D (2000) On the distribution of the maximum likelihood estimator of Cronbach\u2019s alpha. Psychometrika 65(3):271\u2013280. https:\/\/doi.org\/10.1007\/BF02296146","journal-title":"Psychometrika"},{"issue":"3","key":"601_CR70","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1016\/0021-9045(85)90051-6","volume":"45","author":"RA Vitale","year":"1985","unstructured":"Vitale RA (1985) $${L}^{p}$$ metrics for compact, convex sets. J Approx Theory 45(3):280\u2013287. https:\/\/doi.org\/10.1016\/0021-9045(85)90051-6","journal-title":"J Approx Theory"},{"issue":"2","key":"601_CR71","doi-asserted-by":"publisher","first-page":"248","DOI":"10.1109\/TFUZZ.2014.2310734","volume":"23","author":"C Wagner","year":"2015","unstructured":"Wagner C, Miller S, Garibaldi JM, Anderson DT, Havens TC (2015) From interval-valued data to general type-2 fuzzy sets. IEEE Trans Fuzzy Syst 23(2):248\u2013269. https:\/\/doi.org\/10.1109\/TFUZZ.2014.2310734","journal-title":"IEEE Trans Fuzzy Syst"},{"issue":"1","key":"601_CR72","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1177\/0013164402239314","volume":"63","author":"KH Yuan","year":"2003","unstructured":"Yuan KH, Guarnaccia CA, Hayslip B Jr (2003) A study of the distribution of sample coefficient alpha with the Hopkins Symptom Checklist: bootstrap versus asymptotics. Educ Pyschol Meas 63(1):5\u201323. https:\/\/doi.org\/10.1177\/0013164402239314","journal-title":"Educ Pyschol Meas"}],"container-title":["Advances in Data Analysis and Classification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11634-024-00601-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11634-024-00601-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11634-024-00601-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T06:47:50Z","timestamp":1758264470000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11634-024-00601-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,7,26]]},"references-count":72,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["601"],"URL":"https:\/\/doi.org\/10.1007\/s11634-024-00601-w","relation":{},"ISSN":["1862-5347","1862-5355"],"issn-type":[{"value":"1862-5347","type":"print"},{"value":"1862-5355","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,7,26]]},"assertion":[{"value":"9 February 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 May 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 July 2024","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 July 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}