The class of cub models: statistical foundations, inferential issues and empirical evidence. (English) Zbl 1435.62242
Author’s summary: This paper discusses a general framework for the analysis of rating and preference data that is rooted on a class of mixtures of discrete random variables. These models have been extensively studied and applied in the last 15 years thanks to a flexible and parsimonious parametrization of data generating process and to prompt interpretation of results. The approach considers the final response as the combination of feeling and uncertainty, by allowing for finer model specifications to include refuge options, response styles and possible overdispersion, also in relation to subjects’ and objects’ covariates. The article establishes the state of art of the research inherent to this paradigm, in terms of methodology, inferential procedures and fitting measures, by emphasizing capabilities and limitations yet establishing new findings. In particular, explicative power and predictive performances of cub statistical models for ordinal data are examined and new topics that could boost and support the modelling of uncertainty in this framework are provided. Possible developments are outlined throughout the whole presentation and final comments conclude the paper.
From the introduction: This paper is devoted to a critical investigation of the class ofcubstatistical models (Combination of a Uniform and a Binomial random variable) that is grounded on the data generating process of the discrete response choice.
From the introduction: This paper is devoted to a critical investigation of the class ofcubstatistical models (Combination of a Uniform and a Binomial random variable) that is grounded on the data generating process of the discrete response choice.
MSC:
| 62H30 | Classification and discrimination; cluster analysis (statistical aspects) |
| 62D20 | Causal inference from observational studies |
Keywords:
ordinal data; ratings; data generating process; cub models; explicative power; predictabilityReferences:
| [1] | Agresti A (1986) Applying \(R^2\)-type measures to ordered categorical data. Technometrics 28(2):133-138 |
| [2] | Agresti A (2010) Analysis of ordinal categorical data, 2nd edn. Wiley, Hoboken · Zbl 1263.62007 |
| [3] | Agresti A, Kateri M (2017) Ordinal probability effect measures for group comparisons in multinomial cumulative link models. Biometrics 73:214-219 · Zbl 1366.62201 |
| [4] | Agresti A, Lang JB (1993) A proportional odds model with subject-specific effects for repeated ordered categorical responses. Biometrika 80:527-534 · Zbl 0800.62750 |
| [5] | Agresti A, Natarajan R (2001) Modeling clustered ordered categorical data: a survey. Int Stat Rev 69:345-371 · Zbl 1213.62106 |
| [6] | Agresti A, Tarantola C (2018) Simple ways to interpret effects in modeling ordinal categorical data. Statistica Neerlandica 72(3):210-223 · Zbl 1541.62176 |
| [7] | Allik J (2014) A mixed-binomial model for Likert-type personality measure. Front Psychol 5:1-13 |
| [8] | Anderson JA (1984) Regression and ordered categorical variables. J R Stat Soc Ser B 46:1-30 · Zbl 0578.62064 |
| [9] | Anderson JA, Philips PR (1981) Regression, discrimination and measurement models for ordered categorical variables. J R Stat Soc Ser C 30:22-31 · Zbl 0459.62045 |
| [10] | Andreis F, Ferrari PA (2013) On a copula model with cub margins. Quaderni di Statistica. J Methodol Appl Stat 15:33-51 |
| [11] | Andrich (1978) A binomial latent trait model for the study of Likert-style attitude questionnaires. Br J Math Stat Psychol 31:84-98 · Zbl 0387.92019 |
| [12] | Banfield JD, Raftery AE (1993) Model-based Gaussian and non-Gaussian clustering. Biometrics 49:803-821 · Zbl 0794.62034 |
| [13] | Bartolucci F, Farcomeni A, Pennoni F (2012) Latent Markov models for longitudinal data. Chapman and Hall/CRC, Boca Raton · Zbl 1341.62002 |
| [14] | Baum CF, Cerulli C, Di Iorio F, Piccolo D, Simone R (2018) The Stata module cub for fitting mixture models for ordinal data. XV Italian Meeting of STATA Users, Bologna, Nov 15-16 |
| [15] | Baumgartner H, Steenback JB (2001) Response styles in marketing research: across-national investigation. J Market Res 38:143-156 |
| [16] | Bianconcini S, Mignani S (2008) Latent variable models for longitudinal data in educational studies. In: Proceedings of the XLIV scientific meeting of SIS, CLEUP, Padua, pp 225-232 |
| [17] | Birnbaum, A.; Lord, FM (ed.); Novick, MR (ed.), Some latent trait models and their use in inferring an examinee’s ability, 397-472 (1968), Reading |
| [18] | Bonnini S, Piccolo D, Salmaso L, Solmi F (2012) Permutation inference for a class of mixture models. Commun Stat Theory Methods 41(16-17):2879-2895 · Zbl 1319.62092 |
| [19] | Box GEP, Draper NR (1987) Empirical model building and response surfaces. Wiley, New York · Zbl 0614.62104 |
| [20] | Bradbum NM, Sudman S, Blair E (1979) Improving interview method and questionnaire design. Jossey-Bass Publishers, San Francisco |
| [21] | Breen R, Luijkx R (2010) Mixture models for ordinal data. Sociol Methods Res 39:3-24 |
| [22] | Brentari E, Manisera M, Zuccolotto P (2018) Modelling preceived variety in a choice process with nonlinear cub. In: Capecchi S, Di Iorio F, Simone R (eds.), Proceedings of the international conference ASMOD 2018. Federico II University Press, Naples, pp 69-76. ISBN 978-88-6887-042-3 |
| [23] | Brier GW (1950) Verification of forecasts expressed in terms of probability. Month Weather Rev 78(1):1-3 |
| [24] | Cagnone, S.; Mignani, S.; Moustaki, S.; Bini, M. (ed.); Monari, P. (ed.); Piccolo, D. (ed.); Salmaso, L. (ed.), Latent variable models for ordinal data, 17-28 (2009), Berlin |
| [25] | Cagnone S, Moustaki I, Vasdekis V (2009) Latent variable models for multivariate longitudinal ordinal responses. Br J Math Stat Psychol 62(2):401-415 · Zbl 1272.62136 |
| [26] | Capecchi S (2015) Modelling the perception of conflict in working conditions. Electron J Appl Stat 8(3):298-311 |
| [27] | Capecchi, S.; Brulé, G. (ed.); Maggino, F. (ed.), Measuring indecision in happiness studies, 133-153 (2017), Dordrecht |
| [28] | Capecchi S, Endrizzi I, Gasperi F, Piccolo D (2016) A multi-product approach for detecting subjects’ and objects’ covariates in consumer preferences. Br Food J 118(3):515-526 |
| [29] | Capecchi S, Iannario M (2016) Gini heterogeneity index for detecting uncertainty in ordinal data surveys. Metron 74:223-232 · Zbl 1365.62116 |
| [30] | Capecchi S, Iannario M, Simone R (2018) Well-being and relational goods: a model-based approach to detect significant relationships. Soc Indic Res 135(2):729-750 |
| [31] | Capecchi, S.; Piccolo, D.; Perna, C. (ed.); Sibillo, M. (ed.), Modelling the latent components of personal happiness, 49-52 (2014), Springer |
| [32] | Capecchi S, Piccolo D (2016) Investigating the determinants of job satisfaction of Italian graduates: a model-based approach. J Appl Stat 43(1):169-179 · Zbl 1514.62457 |
| [33] | Capecchi S, Piccolo D (2017) Dealing with heterogeneity in ordinal responses. Qual Quant 51:2375-2393 |
| [34] | Capecchi S, Simone R (2019) A proposal for a model-based composite indicators: experience on perceived discrimination in Europe. Soc Indic Res 141(1):95-110 |
| [35] | Cappelli C, Simone R, Di Iorio F (2019) cubremot: a tool for building model-based trees for ordinal responses. Expert Syst Appl 124:39-49 |
| [36] | Carpita M, Ciavolino E, Nitti M (2018) The MIMIC-CUB model for the prediction of the economic public opinions in Europe. Soc Indic Res. https://doi.org/10.1007/s11205-018-1885-4 · doi:10.1007/s11205-018-1885-4 |
| [37] | Colombi R, Giordano S (2016) A class of mixture models for multidimensional ordinal data. Stat Model 16(4):322-340 · Zbl 07289466 |
| [38] | Colombi R, Giordano S (2018) A flexible distribution to handle responses styles when modelling rating scale data. In: Capecchi S, Di Iorio F, Simone R (eds) Proceedings of the international conference ASMOD2018. Federico II University Press, Naples, pp 77-84. ISBN 978-88-6887-042-3 |
| [39] | Colombi R, Giordano S, Gottard A, Iannario M (2018) Hierarchical marginal models with latent uncertainty. Scand J Stat. https://doi.org/10.1111/sjos.12366 · Zbl 1418.62552 · doi:10.1111/sjos.12366 |
| [40] | Corduas M (2008a) Clustering cub models by Kullback-Liebler divergence. In: Proceedings of SCF-CLAFAG Meeting, ESI, Napoli, pp 245-248 |
| [41] | Corduas M (2008b) A statistical procedure for clustering ordinal data. Quaderni di Statistica 10:177-189 |
| [42] | Corduas, M.; Attanasio, M. (ed.); Capursi, V. (ed.), A study on University students’ opinions about teaching quality: a model based approach to clustering ordinal data, 67-78 (2011), Berlin |
| [43] | Corduas, M.; Fichet, A. (ed.); etal., Assessing similarity of rating distributions by Kullback-Liebler divergence, 221-228 (2011), Berlin |
| [44] | Corduas M (2011c) Modelling correlated bivariate ordinal data with cub marginals. Quaderni di Statistica 13:109-119 |
| [45] | Corduas M (2015a) Analyzing bivariate ordinal data with cub margins. Stat Model 15(5):411-432 · Zbl 07258996 |
| [46] | Corduas, M.; Carpita, M. (ed.); Brentari, E. (ed.); Qannari, El Mostafa (ed.), Modelling correlated consumer preferences, 27-36 (2015), Berlin |
| [47] | Corduas M (2015c) Modelling correlated ordinal data by a copula approach. In: Proceedings of the 30th international workshop on statistical modelling, Johannes Kepler Universität , Linz, 2:71-74 |
| [48] | Corduas M (2018) Joint modelling of ordinal data: a copula-based method. In: Capecchi S, Di Iorio F, Simone R (eds) Proceedings of the international conference ASMOD 2018. Federico II University Press, Naples, pp 84-92. ISBN 978-88-6887-042-3 |
| [49] | Corduas, M.; Iannario, M.; Piccolo, D.; Bini, M. (ed.); etal., A class of statistical models for evaluating services and performances, 99-117 (2009), Berlin Heidelberg |
| [50] | Cugnata F, Salini S (2017) Comparison of alternative imputation methods for ordinal data. Communications in Statistics. Simul Comput 46(1):315-330 · Zbl 1360.62018 |
| [51] | Dayton CM, Macready GB (1988) Concomitant-variable latent-class models. J Am Stat Assoc 83:173-178 |
| [52] | Deldossi L, Paroli R (2015) Bayesian variable selection in a class of mixture models for ordinal data: a comparative study. J Stat Comput Simul 85(10):1926-1944 · Zbl 1457.62084 |
| [53] | D’Elia A (2000a) The mechanism of paired comparisons in rank modelling: statistical issues and critical considerations (in Italian). Quaderni di Statistica 2:173-203 |
| [54] | D’Elia A (2000b) A shifted Binomial model for rankings. In: Nunez-Anton V, Ferreira E (eds) Statistical modelling. In: 15th International workshop on statistical modelling. Servicio Editorial de la Universidad del Pais Vasco, pp 412-416 |
| [55] | D’Elia A (2003a) Finite sample performance of the E-M algorithm for ranks data modelling. Statistica 63(1):41-51 · Zbl 1187.62043 |
| [56] | D’Elia A (2003b) Modelling ranks using the inverse hypergeometric distribution. Stat Model 3(1):65-78 · Zbl 1195.62014 |
| [57] | D’Elia A, Piccolo D (2005a) A mixture model for preference data analysis. Comput Stat Data Anal 49:917-934 · Zbl 1429.62077 |
| [58] | D’Elia A, Piccolo D (2005b) The moment estimator for the IHG distribution. In: Provasi C (ed) Proceedings of the IV S.Co. 2005 Meeting, CLEUP, Padova, pp 245-250 |
| [59] | D’Elia A, Piccolo D (2005c) A model based approach for testing homogeneity among evaluation data. In: Zani S, Cerioli A (eds) Proceedings of the CLADAG-2005 Meeting, Parma, pp 83-86 |
| [60] | D’Elia A, Piccolo D (2005d) Uno studio sulla percezione delle emergenze metropolitane: un approccio modellistico. Quaderni di Statistica 7:121-161 |
| [61] | D’Elia A, Piccolo D (2006) Analyzing evaluation data: modelling and testing for homogeneity. In: Zani S et al (eds) Data analysis, classification and the forward search, Springer, Berlin, pp 299-307 |
| [62] | Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm (with discussion). J Royal Stat Soc Ser B 39:1-38 · Zbl 0364.62022 |
| [63] | Di Iorio F, Iannario M (2012) Residual diagnostics for assessing the fit of cub models. Statistica 72:163-172 · Zbl 1453.62761 |
| [64] | Di Iorio F, Piccolo D (2009) Generalized residuals in cub models. Quaderni di Statistica 11:73-88 |
| [65] | Di Nardo E, Simone R (2018) A model-based fuzzy analysis of questionnaires. Stat Meth Appl. https://doi.org/10.1007/s10260-018-00443-9 · Zbl 1427.62068 · doi:10.1007/s10260-018-00443-9 |
| [66] | Easterlin, RA; David, PA (ed.); Reder, MW (ed.), Does economic growth improve the human lot? (1974), New York |
| [67] | Easterlin RA, McVey LA, Switek M, Sawangfa O, Zweig JS (2010) The happiness-income paradox revisited. Proc Natl Acad Sci USA 107(52):22463-22468 |
| [68] | Epstein ES (1969) A scoring system for probability forecasts of ranked categories. J Appl Meteorol Climatol 8(6):985-987 |
| [69] | Everitt BS (1988) A finite mixture for the clustering of mixed-mode data. Stat Prob Lett 6(5):305-309 |
| [70] | Fasola, S.; Sciandra, M.; Morlini, I. (ed.); Minerva, T. (ed.); Vichi, M. (ed.), New flexible probability distributions for ranking data, 117-124 (2015), Berlin |
| [71] | Ferrari S, Cribari-Neto F (2004) Beta regression for modelling rates and proportions. J Appl Stat 31:799-815 · Zbl 1121.62367 |
| [72] | Fin F, Iannario M, Simone R, Piccolo D (2017) The effect of uncertainty on the assessment of individual performance: empirical evidence from professional soccer. Electron J Appl Stat Anal 10(3):677-692 |
| [73] | Forcina A, Dardanoni V (2008) Regression models for multivariate ordered responses via the Plackett distribution. J Multivar Anal 99:2472-2478 · Zbl 1152.62031 |
| [74] | Fraley C, Raftery AE (1998) How many clusters? Which clustering method?—answers via model-based cluster analysis. Comput J 41:578-588 · Zbl 0920.68038 |
| [75] | Fraley C, Raftery AE (2002) Model-based clustering, discriminant analysis, and density estimation. J Am Stat Assoc 97:611-631 · Zbl 1073.62545 |
| [76] | Gelman A, Hill J (2007) Data analysis using regression and multilevel/hierarchical models. Cambridge University Press, New York |
| [77] | Gini C (1912) Variabilità e mutabilità. Studi economico-giuridici, Facoltà di Giurisprudenza, Università di Cagliari, A, III, parte II |
| [78] | Gneiting T, Raftery AE (2007) Strictly proper scoring rules, prediction, and estimation. J Am Stat Assoc 102(477):359-378 · Zbl 1284.62093 |
| [79] | Golia S (2015) On the interpretation of the uncertainty parameter in cub models. Electron J Appl Stat Anal 8(3):312-328 |
| [80] | Gormley IC, Murphy TB (2006) Analysis of Irish third-level college application data. J R Stat Soc Ser A 169:361-379 |
| [81] | Gormley IC, Murphy TB (2008) Exploring voting blocs within the Irish electorate: a mixture modeling approach. J Am Stat Assoc 103(483):1014-1027 · Zbl 1205.62198 |
| [82] | Gottard A, Iannario M, Piccolo D (2016) Varying uncertainty in cub models. Adv Data Anal Classif 10(2):225-244 · Zbl 1414.62327 |
| [83] | Granger CJ (1969) Investigating Causal Relationships by Econometrics Models and Cross Spectral Methods. Econometrica 37:425-435 |
| [84] | Greene WH, Hensher DA (2003) A latent class model for discrete choice analysis: contrasts with mixed logit. Transp Res Part B 37:681-689 |
| [85] | Grilli L, Iannario M, Piccolo D, Rampichini C (2014) Latent class cub models. Adv Data Anal Classif 8:105-119 |
| [86] | Grilli, L.; Rampichini, C.; Kenett, RS (ed.); Salini, S. (ed.), Multilevel models for ordinal data, 391-411 (2012), New York |
| [87] | Grilli L, Rampichini C, Varriale R (2015) Binomial mixture modelling of university credits. Communications in statistics. Theory Methods 44(22):4866-4879 · Zbl 1365.62476 |
| [88] | Grün B, Leisch F (2008) Identifiability of finite mixtures of multinomial logit models with varying and fixed effects. J Classif 25:225-247 · Zbl 1276.62021 |
| [89] | Grün B, Leisch F (2009) Dealing with label switching in mixture models under genuine multimodality. J Multivar Anal 100:851-861 · Zbl 1157.62040 |
| [90] | Han KT (2012) Fixing the \(c\) parameter in the three-parameter logistic model. Pratic Assess Res Eval 17(1):1-24 |
| [91] | Hedeker, D.; Leeuw, J. (ed.); Meijer, E. (ed.), Multilevel models for ordinal and nominal variables, 237-274 (2008), New York |
| [92] | Hedeker D, Gibbons RD (1994) A random-effects ordinal regression model for multilevel analysis. Biometrics 50:933-944 · Zbl 0826.62049 |
| [93] | Hox JJ (2002) Multilevel analysis: techniques and applications. Erlbaum, Mahwah · Zbl 1226.62001 |
| [94] | Hox JJ, Roberts JK (2010) Handbook of advanced multilevel analysis. Routledge, New York |
| [95] | Iannario M (2010) On the identifiability of a mixture model for ordinal data. Metron 68:87-94 · Zbl 1301.62017 |
| [96] | Iannario M (2012a) Modelling shelter choices in a class of mixture models for ordinal responses. Stat Methods Appl 21:1-22 · Zbl 1333.62181 |
| [97] | Iannario M (2012b) cube models for interpreting ordered categorical data with overdispersion. Quaderni di Statistica 14:137-140 |
| [98] | Iannario M (2012c) Preliminary estimators for a mixture model of ordinal data. Adv Data Anal Classif 6:163-184 · Zbl 1254.62004 |
| [99] | Iannario M (2012d) Hierarchical cub models for ordinal variables. Commun Stat Theory Meth 41(16-17):3110-3125 · Zbl 1296.62067 |
| [100] | Iannario M (2014) Modelling uncertainty and overdispersion in ordinal data. Commun Stat Theory Meth 43:771-786 · Zbl 1287.62001 |
| [101] | Iannario M (2015) Modelling scale effects and uncertainty in rating surveys. Electron J Appl Stat Anal 8(3):329-345 |
| [102] | Iannario M, Manisera M, Piccolo D, Zuccolotto P (2018) Ordinal data models for No-opinion responses in attitude surveys. Sociol Methods Res 6(4):1-27 |
| [103] | Iannario M, Monti AC, Piccolo D (2016) Robustness issues in cub models. TEST 25(4):731-750 · Zbl 1422.62117 |
| [104] | Iannario M, Monti AC, Piccolo D, Ronchetti E (2017) Robust inference for ordinal response models. Electron J Stat 11:3407-3445 · Zbl 1390.62041 |
| [105] | Iannario M, Monti AC, Scalera P (2018) Why the number of response categories in rating scales should be large. In: Capecchi S, Di Iorio F, Simone R (eds) Proceedings of the international conference ASMOD 2018. Federico II University Press, Naples, pp 139-146. ISBN 978-88-6887-042-3 |
| [106] | Iannario M, Piccolo D (2012) A framework for modelling ordinal data in rating surveys. In: Proceedings of Joint statistical meetings, section on statistics in marketing, San Diego, California, pp 3308-3322 |
| [107] | Iannario M, Piccolo D (2016a) A generalized framework for modelling ordinal data. Stat Methods Appl 25:163-189 · Zbl 1405.62101 |
| [108] | Iannario M, Piccolo D (2016b) A comprehensive framework of regression models for ordinal data. Metron 74:233-252 · Zbl 1365.62295 |
| [109] | Iannario M, Piccolo D, Simone R (2018) CUB: a class of mixture models for ordinal data. R package version 1(1):3. http://CRAN.R-project.org/package=CUB |
| [110] | Iannario M, Simone R (2017a) Mixture models for rating data: the method of moments via Gröbner basis. J Algebr Stat 8(2):1-28 · Zbl 06834676 |
| [111] | Iannario M, Simone R (2017b) Zero inflated cub models for the evaluation of leisure time activities. In: CLADAG 2017 Book of Short Papers, pp 1-6. ISBN: 9788899459710 |
| [112] | Jasberg K, Sizov S (2017) The Magic barrier revisited: accessing natural limitations of recommender assessment. In: Proceedings of the 11th ACM conference on recommender systems, pp 56-64 |
| [113] | Jonung L (1986) Uncertainty about inflationary perceptions and expectations. J Econ Psychol 7:315-325 |
| [114] | Kateri M (2014) Contingency table analysis: methods and implementations using R. Birkäuser, Springer, New York · Zbl 1291.62012 |
| [115] | Kenett RS, Salini S (2011) Modern analysis of customer satisfaction surveys: comparison of models and integrated analysis. Appl Stoch Models Bus Ind 27(5):465-475 |
| [116] | Kenett RS, Salini S (eds) (2012) Modern Analysis of Customer Surveys: with Applications using R. Wiley, Chivhester |
| [117] | Kleyner A, Bhagath S, Gasparini M, Robinson J, Bender M (1997) Bayesian techniques to reduce the sample size in automotive electronics attribute testing. Microelectron Reliabil 37(6):879-883 |
| [118] | Köster EP (2009) Diversity in the determinants of food choice: a psychological perspective. Food Qual Prefer 20:70-82 |
| [119] | Krosnick JA (1991) Response strategies for coping with the cognitive demands of attitude measures in surveys. Appl Cogn Psychol 5:213-236 |
| [120] | Krosnick JA (1999) Surveys research. Ann Rev Psychol 50:537-567 |
| [121] | Laakso M, Taagepera R (1989) Effective number of parties: a measure with application to West Europe. Compar Polit Stud 12:3-27 |
| [122] | Lambert D (1992) Zero-inflated poisson regression, with an application to defects in manufacturing. Technometrics 34(1):1-14 · Zbl 0850.62756 |
| [123] | Lang JB, Agresti JW (1994) Simultaneously modelling joint and marginal distribution of multivariate categorical responses. J Am Stat Assoc 89:625-632 · Zbl 0799.62063 |
| [124] | Lee PM (2012) Bayesian statistics: an introduction, 4th edn. Wiley, New York · Zbl 1258.62028 |
| [125] | Li C, Shepherd BE (2012) A new residual for ordinal outcomes. Biometrika 99(2):473-480 · Zbl 1239.62042 |
| [126] | Liu D, Zhang H (2018) Residuals and diagnostics for ordinal regression models: a surrogate approach. J Am Stat Soc 113(522):845-854 · Zbl 1398.62195 |
| [127] | Lord FM (1980) Applications of item response theory to practical testing problems. Erlbaum, Hillsdale |
| [128] | Louis TA (1982) Finding the observed information matrix when using the EM algorithm. J R Stat Soc Ser B 44:226-233 · Zbl 0488.62018 |
| [129] | Luchini S, Watson V (2013) Uncertainty and framing in a valuation task. J Econ Psychol 39:204-214 |
| [130] | Magee L \((1990) R^2\) measures based on wald and likelihood ratio joint significance tests. Technometrics 44(3):250-253 |
| [131] | Manisera M, Zuccolotto P (2014a) Modeling “don’t know” responses in rating scales. Pattern Recognit Lett 45:226-234 · Zbl 1506.62123 |
| [132] | Manisera M, Zuccolotto P (2014b) Modeling rating data with nonlinear cub models. Comput Stat Data Anal 78:100-118 · Zbl 1506.62123 |
| [133] | Manisera M, Zuccolotto P (2015) On the identifiability of nonlinear cub models. J Multivar Anal 140:302-316 · Zbl 1360.62112 |
| [134] | Marasini D, Quatto P, Ripamonti E (2015) Intuitionistic fuzzy sets in questionnaire analysis. Qual Quant 50:767-790 |
| [135] | McFadden K (1978) Modeling the choice of residential location. In: Karlqvist A et al (ed) Spatial interaction theory and residential location. Amsterdam, North-Holland, pp 75-76 |
| [136] | McCullagh P (1980) Regression models for ordinal data (with discussion). J R Stat Soc Ser B 42:109-142 · Zbl 0483.62056 |
| [137] | McCullagh P, Nelder JA (1989) Generalized linear models, 2nd edn. Chapman and Hall, London · Zbl 0744.62098 |
| [138] | McLachlan G, Krishnan T (2008) The EM algorithm and extensions, 2nd edn. Wiley, New York · Zbl 1165.62019 |
| [139] | McLachlan G, Peel GJ (2000) Finite mixture models. Wiley, New York · Zbl 0963.62061 |
| [140] | Molenberghs G, Lesaffre E (1994) Marginal modelling of correlated ordinal data using multivariate Plackett distribution. J Am Stat Assoc 89:633-644 · Zbl 0802.62063 |
| [141] | Molenberghs G, Verbeke G (2007) Likelihood ratio, score, and Wald tests in a constrained parameter space. Am Stat 61:22-27 |
| [142] | Morrison DG (1979) Purchase intentions and purchase behavior. J Market 43:65-74 |
| [143] | Moustaki I, Knott M (2000) Generalized latent trait models. Psychometrika 65:391-411 · Zbl 1291.62236 |
| [144] | Mullahy J (1986) Specification and testing of some modified count data models. J Econom 33:341-365 |
| [145] | Murphy AH (1970) The ranked probability score and the probability score: a comparison. Month Weather Rev 98(12):917-924 |
| [146] | Murphy TB, Martin D (2002) Mixtures of distances-based models for ranking data. Comput Stat Data Anal 41:645-655 · Zbl 1429.62258 |
| [147] | Muschelli, J.; Betz, J.; Varadhan, R.; Rao, MB (ed.); Rao, CR (ed.), Binomial regression in R (chapter 7), No. 32, 257-308 (2014), Amsterdam |
| [148] | Oberski DL, Vermunt JK (2015) The relationship between cub and loglinear models with latent variables. Electron J Appl Stat Anal 8(3):374-383 |
| [149] | Peryam DR, Pilgrim FJ (1957) Hedonic scale method of measuring food preferences. Food Technol 11:9-14 |
| [150] | Piccolo D (2003) On the moments of a mixture of uniform and shifted binomial random variables. Quad Stat 5:85-104 |
| [151] | Piccolo D (2006) Observed information matrix for MUB models. Quad Stat 8:33-78 |
| [152] | Piccolo D (2015) Inferential issues on cube models with covariates. Communications in statistics. Theory Methods 44:5023-5036 · Zbl 1381.62036 |
| [153] | Piccolo, D.; Abbruzzo, A. (ed.); Brentari, E. (ed.); Chiodi, M. (ed.); Piacentino, D. (ed.), A new paradigm for rating data models, 1-12 (2018), New York |
| [154] | Piccolo D, D’Elia A (2008) A new approach for modelling consumers’ preferences. Food Qual Pref 19:247-259 |
| [155] | Piccolo D, Simone R, Iannario M (2018) Cumulative and cub models for rating data: a comparative analysis. Int Stat Rev. https://doi.org/10.1111/insr.12282 · Zbl 07763644 · doi:10.1111/insr.12282 |
| [156] | Pinto da Costa JF, Alonso H, Cardoso JS (2008) The unimodal modal for the classification of ordinal data. Neur Netw 21:78-91 Corrigendum in: (2014). Neural Networks 59:73-75 · Zbl 1254.68220 |
| [157] | Powers DA, Xie Y (2000) Statistical methods for categorical data analysis. Academic Press, San Diego · Zbl 0967.62101 |
| [158] | Rao CR (1973) Linear statistical inference and its applications, 2nd edn. Wiley, New York · Zbl 0256.62002 |
| [159] | Raudenbush SW, Bryk AS (2002) Hierarchical linear models. Sage, Newbury Park · Zbl 1001.62004 |
| [160] | Samejima F (1997) Graded response model, handbook of modern item response theory. Springer, Berlin, pp 85-100 · Zbl 0872.62099 |
| [161] | Schutz HG, Cardello AV (2001) A labelled affective magnitude (LAM) scale for assessing food liking/disliking. J Sens Stud 16(2):117-159 |
| [162] | Self SG, Liang KY (2003) Asymptotic properties of maximum likelihood estimators and likelihood ratio test under nonstandard conditions. J Am Stat Assoc 82:605-610 · Zbl 0639.62020 |
| [163] | Simon HA (1957) Models of man. Wiley, New York |
| [164] | Simone, R.; Abbruzzo, A. (ed.); Brentari, E. (ed.); Chiodi, M. (ed.); Piacentino, D. (ed.), A test for variable importance (2018), New York |
| [165] | Simone R (2018b) Louis’ identity and fast estimation of mixture models for rating data (under review) |
| [166] | Simone R (2018c) A note on predictability for binomial models (Technical Report) |
| [167] | Simone R, Iannario M (2018) Analysing sport data with clusters of opposite preferences. Stat Model 18(5-6):1-20 · Zbl 07289520 |
| [168] | Simone R, Tutz G (2018) Modelling uncertainty and response styles in ordinal data. Stat Neerlandica 72:224-245 · Zbl 1541.62186 |
| [169] | Simone R, Cappelli C, Di Iorio F (2019a) Modelling marginal ranking distributions: the uncertainty tree (Forthcoming) |
| [170] | Simone R, Tutz G, Iannario M (2019b) Subjective heterogeneity in response attitude for multivariate ordinal outcomes (Forthcoming) |
| [171] | Skellam JG (1948) A probability distribution derived from the Binomial distribution by regarding the probability of success as variable between the sets of trials. J R Stat Soc Ser B 10(2):257-261 · Zbl 0032.41903 |
| [172] | Tamhane A, Ankemanman B, Yang Y (2002) The Beta distribution as a latent response model for ordinal data (I): Estimation of location and dispersion parameters. J Stat Comput Simul 72(6):473-494 · Zbl 1026.62078 |
| [173] | Titterington DM, Smith AFM, Makov UE (1985) Statistical analysis of finite mixture distributions. Wiley, New York · Zbl 0646.62013 |
| [174] | Tourangeau R, Rips LJ, Rasinski K (2000) The psychology of survey response. Cambridge University Press, Cambridge |
| [175] | Train KE (2003) Discrete choice methods with simulations. Cambridge University Press, Cambridge · Zbl 1047.62098 |
| [176] | Tripathi RC, Gupta RC, Gurland J (1994) Estimation of parameters in Beta Binomial models. Ann Inst Stat Math 46(2):317-331 · Zbl 0816.62024 |
| [177] | Tutz G (2012) Regression for categorical data. Cambridge University Press, Cambridge · Zbl 1304.62021 |
| [178] | Tutz G (2018) Uncertainty, dispersion and response styles in ordinal regression. In: Capecchi S, Di Iorio F, Simone R (eds) Proceedings of the international conference ASMOD 2018, Federico II University Press, Naples, pp 33-41. ISBN 978-88-6887-042-3 |
| [179] | Tutz G, Schauberger G (2013) Visualization of categorical response models: from data glyphs to parameter glyphs. J Comput Gr Stat 22(1):156-177 |
| [180] | Tutz G, Schauberger G, Berger M (2018) Response styles in the partial credit model. Appl Psychol Measur 42(6):407-427 |
| [181] | Tutz G, Schneider M, Iannario M, Piccolo D (2017) Mixture models for ordinal responses to account for uncertainty of choice. Adv Data Anal Classif 11(2):281-305 · Zbl 1414.62019 |
| [182] | Ursino M (2014) Ordinal data: a new model with applications. Ph.D. Thesis, XXVI cycle, Polytechnic University of Turin, Turin |
| [183] | Ursino M, Gasparini M (2018) A new parsimonious model for ordinal longitudinal data with application to subjective evaluations of a gastrointestinal disease. Stat Methods Med Res 27(5):1376-1393 |
| [184] | van der Linden WJ, Hambleton RK (eds) (1996) Handbook of modern item response theory. Springer, New York |
| [185] | Vermunt JK, Magidson J (2013) Technical guide for latent gold 5.0: basic, advanced, and sintax. Statistical Innovations, Inc., Belmont |
| [186] | von Eye A, Mun E-Y (2012) Log-linear modeling: concepts, interpretation, and application. Wiley, New York · Zbl 1256.62042 |
| [187] | Vu HTV, Zhou S (1997) Generalization of likelihood ratio tests under nonstandard conditions. Ann Stat 25:897-916 · Zbl 0873.62022 |
| [188] | Wedel M, DeSarbo WS (1995) A mixture likelihood approach for generalized linear models. J Classif 12:21-55 · Zbl 0825.62611 |
| [189] | Zhou H, Lange K (2009) Rating movies and rating the raters who rate them. Am Stat 63:297-307 |
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