Hierarchical CUB Models for Ordinal Variables

References

• Abramowitz , M. , Stegun , C. A. ( 1972 ). Handbook of Mathematical Functions . New York : Dover .
   Google Scholar
• Agresti , A. ( 2002 ). Categorical Data Analysis. , 2nd ed. New York : J. Wiley & Sons .
   Google Scholar
• Agresti , A. (2010). Analysis of Ordinal Categorical Data. , 2nd ed. New York : J. Wiley & Sons.
   Google Scholar
• Agresti , A. , Lang , J. B. ( 1993 ). A proportional odds model with subject-specific effects for repeated ordered categorical responses . Biometrika 80 : 527 – 534 .
   Web of Science ®Google Scholar
• Agresti , A. , Natarajan , R. ( 2001 ). Modeling clustered ordered categorical data: a survey . Int. Statist. Rev. 69 : 345 – 371 .
   Web of Science ®Google Scholar
• Aitkin , M. , Bonnett , S. N. , Hesketh , J. ( 1981 ). Teaching styles and pupil progress: a reanalysis . Brit. J. Educat. Psychol. 51 : 170 – 186 .
   Web of Science ®Google Scholar
• Bartholomew , D. J. ( 1987 ). Latent Variable Models and Factor Analysis . New York : M. Dekker .
   Google Scholar
• Berkhof , J. , Snijders , T. A. B. ( 2001 ). Variance component testing in multilevel models . J. Educat. Behav. Statist. 26 : 133 – 152 .
   Web of Science ®Google Scholar
• Bock , R. D. , Atkin , M. ( 1981 ). Marginal maximum likelihood estimation of item parameters: application of an em algorithm . Psychometrika 46 : 443 – 459 .
   Web of Science ®Google Scholar
• Corduas , M. , Iannario , M. , Piccolo , D. ( 2009 ). A class of statistical models for evaluating services and performances . In: et al. , eds. Statistical Methods for the Evaluation of Educational Services and Quality of Products . Contribution to Statistics . Heidelberg : Springer, Physica-Verlag , pp. 99 – 117 .
   Google Scholar
• Cronbach , L. J. , Webb , N. ( 1975 ). Between-class and within-class effects in a reported aptitude × treatment interaction: a reanalysis of a study by G. L. Anderson . J. Educat. Psychol. 67 : 717 – 724 .
   Web of Science ®Google Scholar
• D'Elia , A. , Piccolo , D. ( 2005 ). A mixture model for preference data analysis . Computat. Statist. Data Anal. 49 : 917 – 934 .
   Web of Science ®Google Scholar
• Dobson , A. J. , Barnett , A. G. ( 2002 ). An Introduction to Generalized Linear Models. , 3rd ed. London : Chapman & Hall/CRC .
   Google Scholar
• Everitt , B. S. ( 1984 ). An Introduction to Latent Variable Models . New York : Chapman and Hall .
   Google Scholar
• Fielding , A. ( 1997 ). On scoring ordered classification . Brit. J. Mathemat. Statist. Psychol. 50 : 285 – 307 .
   Web of Science ®Google Scholar
• Fielding , A. ( 2004 ). Scaling for residual variance components of ordered category responses in generalized linear mixed multilevel models . Qual. Quant. 38 : 425 – 433 .
   Web of Science ®Google Scholar
• Fielding , A. , Yang , M. ( 2005 ). Generalized linear mixed models for ordered responses in complex multilevel structures: effects beneath the school or college in education . J. Roy. Statist. Soc. Ser. A 168 : 159 – 183 .
   Google Scholar
• Gelman , A. , Hill , J. ( 2007 ). Data Analysis using Regression and Multilevel/Hierarchical Models . New York : Cambridge University Press .
   Google Scholar
• Grilli , L. , Rampichini , C. ( 2012 ). Multilevel models for ordinal data . In: Kenett , R. S. , Salini , S. , eds. Modern Analysis of Customer Surveys . New York : J. Wiley & Sons , pp. 391 – 411 .
   Google Scholar
• Heck , R. H. , Thomas , S. L. ( 2000 ). An Introduction to Multilevel Modeling Techniques . Mahwah , NJ : Erlbaum .
   Google Scholar
• Hedeker , D. ( 2008 ). Multilevel models for ordinal and nominal variables . In: De Leeuw , J. , Meijer , E. , eds. Handbook of Multilevel Analysis . New York : Springer , pp. 237 – 274 .
   Google Scholar
• Hedeker , D. , Gibbons , R. D. ( 1994 ). A random-effects ordinal regression model for multilevel analysis . Biometrics 50 : 933 – 944 .
   PubMed Web of Science ®Google Scholar
• Hedeker , D. , Gibbons , R. D. ( 1996 ). MIXOR: A computer program for mixed effects ordinal regression analysis . Comput. Meth. Prog. Biomed. 49 : 157 – 176 .
   PubMed Web of Science ®Google Scholar
• Hedeker , D. , Gibbons , R. D. ( 2006 ). Longitudinal Data Analysis . New Jersey : John Wiley & Sons .
   Google Scholar
• Hox , J. J. ( 2002 ). Multilevel Analysis: Techniques and Applications . Mahwah , NJ : Erlbaum .
   Google Scholar
• Hox , J. J. , Roberts , J. K. ( 2010 ). Handbook of Advanced Multilevel Analysis . New York : Routledge .
   Google Scholar
• Iannario , M. ( 2008 ). A class of models for ordinal variables with covariates effects . Quaderni di Statistica 10 : 53 – 72 .
   Google Scholar
• Iannario , M. (2010). On the identifiability of a mixture model for ordinal data. Metron LXVIII:87–94.
   Google Scholar
• Iannario , M. ( 2012 ). Modelling shelter choices in a class of mixture models for ordinal responses . Statist. Meth. Appli. 21 : 1 – 22 .
   Web of Science ®Google Scholar
• Iannario , M. , Piccolo , D. ( 2012 ). CUB Models: Statistical Methods and Empirical Evidence . In: Kenett , R. S. , Salini , S. , eds. Modern Analysis of Customer Surveys . New York : J. Wiley & Sons , pp. 231 – 254 .
   Google Scholar
• Klein , K. , Kozlowski , S. W. J. ( 2000 ). Multilevel Theory, Research, and Methods in Organizations . San Francisco : Jossey-Bass .
   Google Scholar
• Kreft , I. , De Leeuw , J. ( 2007 ). Introducing Multilevel Modeling . London : Sage Publications .
   Google Scholar
• Lewis , C. ( 2007 ). Selected topics in classical test theory . In: Rao , C. R. , Sinharay , S. , eds. Handbook of Statistics , Amsterdam : Elsevier B. V. , pp. 29 – 43 .
   Google Scholar
• Liu , L. C. , Hedeker , D. ( 2006 ). A mixed-effects regression model for longitudinal multivariate ordinal data . Biometrics 62 : 261 – 268 .
   PubMed Web of Science ®Google Scholar
• McCullagh , P. ( 1980 ) Regression models for ordinal data (with discussion) . J. Roy. Statist. Soc. Ser. B 42 : 109 – 142 .
   Google Scholar
• McCullagh , P. , Nelder , J. A. ( 1989 ). Generalized Linear Models. , 2nd ed. London : Chapman and Hall .
   Google Scholar
• Molenberghs , G. , Verbeke , G. ( 2007 ). Likelihood ratio, score, and Wald tests in a constrained parameter space . Amer. Statistician 61 : 22 – 27 .
   Web of Science ®Google Scholar
• Moustaki , I. ( 2000 ). A latent variable model for ordinal data . Appl. Psychol. Measure. 24 : 211 – 223 .
   Web of Science ®Google Scholar
• Moustaki , I. ( 2003 ). A general class of latent variable model for ordinal manifest variables with covariate effects on the manifest and latent variables . Brit. J. Mathemat. Statist. Psychol. 56 : 337 – 357 .
   PubMed Web of Science ®Google Scholar
• Moustaki , I. , Knott , M. ( 2000 ). Generalized latent trait models . Psychometrika 65 : 391 – 411 .
   Web of Science ®Google Scholar
• Pawitan , Y. ( 2001 ). In All Likelihood: Statistical Modelling and Inference Using Likelihood . Oxford : Oxford Science Publications, Clarendon Press .
   Google Scholar
• Piccolo , D. ( 2003 ). On the moments of a mixture of uniform and shifted binomial random variables . Quaderni di Statistica 5 : 85 – 104 .
   Google Scholar
• Piccolo , D. ( 2006 ). Observed information matrix for MUB models . Quaderni di Statistica 8 : 33 – 78 .
   Google Scholar
• Piccolo , D. , D'Elia , A. ( 2008 ). A new approach for modelling consumers’ preferences . Food Qual. Pref. 19 : 247 – 259 .
   Web of Science ®Google Scholar
• Rabe-Hesketh , S. , Pickles , A. , Skrondal , A. ( 2001 ). GLLAMM: A general class of multilevel models and a stata program . Multilevel Model. Newslet. 13 : 17 – 23 .
   Google Scholar
• Rabe-Hesketh , S. , Skrondal , A. ( 2008 ). Multilevel and Longitudinal Modeling Using Stata. , 2nd ed. Stata Press , TX : College Station .
   Google Scholar
• Raudenbush , S. W. , Bryk , A. S. ( 2002 ). Hierarchical Linear Models . Newbury Park , CA : Sage .
   Google Scholar
• Sammel , M. D. , Ryan , L. M. , Legler , J. M. ( 1997 ). Latent variable models for mixed discrete and continuous outcomes . J. Roy. Statist. Soc. Ser. B 59 : 667 – 678 .
   Google Scholar
• Schoenberg , R. ( 1995 ). CML Users Guide . Maple Valley , WA : Aptech Systems, Inc .
   Google Scholar
• Self , S. G. , Liang , K. Y. ( 1997 ). Generalization of likelihood ratio tests under nonstandard conditions . Ann. Statist. 82 : 605 – 610 .
   Google Scholar
• Skrondal , A. , Rabe-Hesketh , S. ( 2004 ). Generalized Latent Variable Modeling: Multilevel, Longitudinal and Structural Equation Models . London : Chapman & Hall/CRC Press .
   Google Scholar
• Snijders , T. A. B. , Bosker , R. J. ( 1999 ). Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling . Thousand Oaks , CA : Sage .
   Google Scholar
• Steele , F. , Goldstein , H. ( 2007 ). Multilevel models in psychometrics . In: Rao , C. R. , Sinharay , S. , eds. Handbook of Statistics . Amsterdam : Elsevier B.V. , pp. 401 – 420 .
   Google Scholar
• Stroud , A. H. , Secrest , D. ( 1966 ). Gaussian Quadrature Formulas . Englewood Cliffs , NJ : Prentice Hall .
   Google Scholar
• Tutz , G. , Hennevogl , W. (1996). Random effects in ordinal regression models. Computat. Statist. Data Anal. 22:537–557.
   Web of Science ®Google Scholar
• Verbeke , G. , Molenberghs , G. ( 2007 ). Likelihood ratio, score, and Wald tests in a constrained parameter space . Amer. Statistician 61 : 22 – 27 .
   Web of Science ®Google Scholar
• Vermunt , J. K. ( 2004 ). An EM algorithm for the estimation of parametric and nonparametric hierarchical nonlinear models . Statistica Neerlandica 58 : 220 – 233 .
   Web of Science ®Google Scholar
• Vu , H. T. V. , Zhou , S. ( 1987 ). Asymptotic properties of maximum likelihood estimators and likelihood ratio test under nonstandard conditions . J. Amer. Statist. Assoc. 82 : 605 – 610 .
   Web of Science ®Google Scholar