Skip to main content
SpringerLink
Log in

Multilevel Models for Ordinal and Nominal Variables

  • Chapter
Handbook of Multilevel Analysis

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. R. J. Adams, M. Wilson, and M. Wu. Multilevel item response models: An approach to errors in variable regression. Journal of Educational and Behavioral Statistics, 22:47–76, 1997.

    Google Scholar 

  2. A. Agresti. Categorical Data Analysis. Wiley, New York, 1990.

    MATH  Google Scholar 

  3. A. Agresti, J. G. Booth, J. P. Hobart, and B. Caffo. Random-effects modeling of categorical response data. Sociological Methodology, 30:27–80, 2000.

    Google Scholar 

  4. A. Agresti and J. B. Lang. A proportional odds model with subject-specific effects for repeated ordered categorical responses. Biometrika, 80:527–534, 1993.

    MATH  MathSciNet  Google Scholar 

  5. A. Agresti and R. Natarajan. Modeling clustered ordered categorical data: A survey. International Statistical Review, 69:345–371, 2001.

    MATH  Google Scholar 

  6. P. D. Allison. Discrete-time methods for the analysis of event histories. Sociological Methodology, 13:61–98, 1982.

    Google Scholar 

  7. P. D. Allison. Survival Analysis using the SAS System: A Practical Guide. SAS Institute, Cary, NC, 1995.

    Google Scholar 

  8. T. Amemiya. Qualitative response models: A survey. Journal of Economic Literature, 19:483–536, 1981.

    Google Scholar 

  9. D. A. Anderson and M. Aitkin. Variance component models with binary response: Interviewer variability. Journal of the Royal Statistical Society, Series B, 47:203–210, 1985.

    MathSciNet  Google Scholar 

  10. K. J. Arrow. Social Choice and Individual Values. Wiley, New York, 1951.

    MATH  Google Scholar 

  11. D. J. Bartholomew and M. Knott. Latent Variable Models and Factor Analysis, 2nd edition. Oxford University Press, New York, 1999.

    MATH  Google Scholar 

  12. E. K. Berndt, B. H. Hall, R. E. Hall, and J. A. Hausman. Estimation and inference in nonlinear structural models. Annals of Economic and Social Measurement, 3:653–665, 1974.

    Google Scholar 

  13. C. R. Bhat. Quasi-random maximum simulated likelihood estimation of the mixed multinomial logit model. Transportation Research, Part B, 35:677–693, 2001.

    Google Scholar 

  14. R. D. Bock. Estimating item parameters and latent ability when responses are scored in two or more nominal categories. Psychometrika, 37:29–51, 1972.

    MATH  MathSciNet  Google Scholar 

  15. R. D. Bock. Multivariate Statistical Methods in Behavioral Research. McGraw-Hill, New York, 1975.

    MATH  Google Scholar 

  16. R. D. Bock and M. Aitkin. Marginal maximum likelihood estimation of item parameters: An application of the EM algorithm. Psychometrika, 46:443–459, 1981.

    MathSciNet  Google Scholar 

  17. R. D. Bock and M. Lieberman. Fitting a response model for n dichotomously scored items. Psychometrika, 35:179–197, 1970.

    Google Scholar 

  18. R. D. Bock and S. Shilling. High-dimensional full-information item factor analysis. In M. Berkane, editor, Latent Variable Modeling and Applications to Causality, pages 163–176. Springer, New York, 1997.

    Google Scholar 

  19. N. E. Breslow and X. Lin. Bias correction in generalised linear mixed models with a single component of dispersion. Biometrika, 82:81–91, 1995.

    MATH  MathSciNet  Google Scholar 

  20. G. Camilli and L. A. Shepard. Methods for Identifying Biased Test Items. Sage, Thousand Oaks, CA, 1994.

    Google Scholar 

  21. M. R. Conaway. Analysis of repeated categorical measurements with conditional likelihood methods. Journal of the American Statistical Association, 84:53–61, 1989.

    MathSciNet  Google Scholar 

  22. C. Corcoran, B. Coull, and A. Patel. Egret for Windows User Manual. Cytel Software Corporation, Cambridge, MA, 1999.

    Google Scholar 

  23. L. A. Cupples, R. B. D'Agostino, K. Anderson, and W. B. Kannel. Comparison of baseline and repeated measure covariate techniques in the Framingham Heart Study. Statistics in Medicine, 7:205–218, 1985.

    Google Scholar 

  24. R. B. D'Agostino, M.-L. Lee, A. J. Belanger, L. A. Cupples, K. Anderson, and W. B. Kannel. Relation of pooled logistic regression to time dependent Cox regression analysis: The Framingham Heart Study. Statistics in Medicine, 9:1501–1515, 1990.

    Google Scholar 

  25. M. J. Daniels and C. Gatsonis. Hierarchical polytomous regression models with applications to health services research. Statistics in Medicine, 16:2311–2325, 1997.

    Google Scholar 

  26. K. A. Doksum and M. Gasko. On a correspondence between models in binary regression analysis and in survival analysis. International Statistical Review, 58:243–252, 1990.

    MATH  Google Scholar 

  27. D. M. Drukker. Maximum simulated likelihood: Introduction to a special issue. The Stata Journal, 6:153–155, 2006.

    Google Scholar 

  28. J. Engel. On the analysis of grouped extreme-value data with GLIM. Applied Statistics, 42:633–640, 1993.

    MATH  Google Scholar 

  29. F. Ezzet and J. Whitehead. A random effects model for ordinal responses from a crossover trial. Statistics in Medicine, 10:901–907, 1991.

    Google Scholar 

  30. A. Fielding, M. Yang, and H. Goldstein. Multilevel ordinal models for examination grades. Statistical Modelling, 3:127–153, 2003.

    MATH  MathSciNet  Google Scholar 

  31. G. M. Fitzmaurice, N. M. Laird, and A. G. Rotnitzky. Regression models for discrete longitudinal responses. Statistical Science, 8:284–309, 1993.

    MATH  MathSciNet  Google Scholar 

  32. S. A. Freels, R. B. Warnecke, T. P. Johnson, and B. R. Flay. Evaluation of the effects of a smoking cessation intervention using the multilevel thresholds of change model. Evaluation Review, 26:40–58, 2002.

    Google Scholar 

  33. S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6:721–741, 1984.

    MATH  Google Scholar 

  34. R. D. Gibbons and R. D. Bock. Trend in correlated proportions. Psychometrika, 52:113–124, 1987.

    MATH  MathSciNet  Google Scholar 

  35. R. D. Gibbons and D. Hedeker. Random-effects probit and logistic regression models for three-level data. Biometrics, 53:1527–1537, 1997.

    MATH  Google Scholar 

  36. G. Glasgow. Mixed logit models for multiparty elections. Political Analysis, 9:116–136, 2001.

    Google Scholar 

  37. H. Goldstein. Nonlinear multilevel models, with an application to discrete response data. Biometrika, 78:45–51, 1991.

    MathSciNet  Google Scholar 

  38. H. Goldstein. Multilevel Statistical Models, 3rd edition. Edward Arnold, London, 2003.

    MATH  Google Scholar 

  39. H. Goldstein and J. Rasbash. Improved approximations for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 159:505–513, 1996.

    MATH  MathSciNet  Google Scholar 

  40. W. H. Greene. LIMDEP Version 8.0 User's Manual, 4th edition. Econometric Software, Plainview, NY, 2002.

    Google Scholar 

  41. P. Haan and A. Uhlendorff. Estimation of multinomial logit models with unobserved heterogeneity using maximum simulated likelihood. The Stata Journal, 6:229–245, 2006.

    Google Scholar 

  42. A. Han and J. A. Hausman. Flexible parametric estimation of duration and competing risk models. Journal of Applied Econometrics, 5:1–28, 1990.

    Google Scholar 

  43. J. Hartzel, A. Agresti, and B. Caffo. Multinomial logit random effects models. Statistical Modelling, 1:81–102, 2001.

    MATH  Google Scholar 

  44. D. A. Harville and R. W. Mee. A mixed-model procedure for analyzing ordered categorical data. Biometrics, 40:393–408, 1984.

    MATH  MathSciNet  Google Scholar 

  45. P. J. Heagerty and S. L. Zeger. Marginalized multilevel models and likelihood inference. Statisical Science, 15:1–26, 2000. (with discussion)swallowdot.

    Google Scholar 

  46. D. Hedeker. MIXNO: A computer program for mixed-effects nominal logistic regression. Journal of Statistical Software, 4(5):1–92, 1999.

    MathSciNet  Google Scholar 

  47. D. Hedeker. A mixed-effects multinomial logistic regression model. Statistics in Medicine, 21:1433–1446, 2003.

    Google Scholar 

  48. D. Hedeker and R. D. Gibbons. A random-effects ordinal regression model for multilevel analysis. Biometrics, 50:933–944, 1994.

    MATH  Google Scholar 

  49. D. Hedeker and R. D. Gibbons. MIXOR: A computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine, 49:157–176, 1996.

    Google Scholar 

  50. D. Hedeker and R. D. Gibbons. Application of random-effects pattern-mixture models for missing data in longitudinal studies. Psychological Methods, 2:64–78, 1997.

    Google Scholar 

  51. D. Hedeker and R. D. Gibbons. Longitudinal Data Analysis. Wiley, New York, 2006.

    MATH  Google Scholar 

  52. D. Hedeker and R. J. Mermelstein. A multilevel thresholds of change model for analysis of stages of change data. Multivariate Behavioral Research, 33:427–455, 1998.

    Google Scholar 

  53. D. Hedeker and R. J. Mermelstein. Analysis of longitudinal substance use outcomes using random-effects regression models. Addiction, 95(Supplement 3):S381–S394, 2000.

    Google Scholar 

  54. D. Hedeker, O. Siddiqui, and F. B. Hu. Random-effects regression analysis of correlated grouped-time survival data. Statistical Methods in Medical Research, 9:161–179, 2000.

    MATH  Google Scholar 

  55. R. L. Hough, S. Harmon, H. Tarke, S. Yamashiro, R. Quinlivan, P. Landau-Cox, M. S. Hurlburt, P. A. Wood, R. Milone, V. Renker, A. Crowell, and E. Morris. Supported independent housing: Implementation issues and solutions in the San Diego McKinney homeless demonstration research project. In W. R. Breakey and J. W. Thompson, editors, Mentally Ill and Homeless: Special Programs for Special Needs, pages 95–117. Harwood, New York, 1997.

    Google Scholar 

  56. M. S. Hurlburt, P. A. Wood, and R. L. Hough. Providing independent housing for the homeless mentally ill: A novel approach to evaluating long-term longitudinal housing patterns. Journal of Community Psychology, 24:291–310, 1996.

    Google Scholar 

  57. H. Ishwaran. Univariate and multirater ordinal cumulative link regression with covariate specific cutpoints. Canadian Journal of Statistics, 28:715–730, 2000.

    MATH  MathSciNet  Google Scholar 

  58. J. Jansen. On the statistical analysis of ordinal data when extravariation is present. Applied Statistics, 39:75–84, 1990.

    MATH  MathSciNet  Google Scholar 

  59. E. Läärä and J. N. S. Matthews. The equivalence of two models for ordinal data. Biometrika, 72:206–207, 1985.

    Google Scholar 

  60. N. M. Laird. Missing data in longitudinal studies. Statistics in Medicine, 7:305–315, 1988.

    Google Scholar 

  61. E. Lesaffre and B. Spiessens. On the effect of the number of quadrature points in a logistic random-effects model: An example. Applied Statistics, 50:325–335, 2001.

    MATH  MathSciNet  Google Scholar 

  62. J. K. Lindsey and P. Lambert. On the appropriateness of marginal models for repeated measurements in clinical trials. Statistics in Medicine, 17:447–469, 1998.

    Google Scholar 

  63. L. C. Liu and D. Hedeker. A mixed-effects regression model for longitudinal multivariate ordinal data. Biometrics, 62:261–268, 2006.

    MATH  MathSciNet  Google Scholar 

  64. Q. Liu and D. A. Pierce. A note on Gauss-Hermite quadrature. Biometrika, 81:624–629, 1994.

    MATH  MathSciNet  Google Scholar 

  65. J. S. Long. Regression Models for Categorical and Limited Dependent Variables. Sage, Thousand Oaks, CA, 1997.

    MATH  Google Scholar 

  66. F. M. Lord. Applications of Item Response Theory to Practical Testing Problems. Erlbaum, Hillside, NJ, 1980.

    Google Scholar 

  67. R. D. Luce. Individual Choice Behavior. Wiley, New York, 1959.

    MATH  Google Scholar 

  68. G. S. Maddala. Limited-Dependent and Qualitative Variables in Econometrics. Cambridge University Press, Cambridge, UK, 1983.

    MATH  Google Scholar 

  69. J. R. Magnus. Linear Structures. Charles Griffin, London, 1988.

    MATH  Google Scholar 

  70. E. C. Marshall and D. Spiegelhalter. Institutional performance. In A. H. Leyland and H. Goldstein, editors, Multilevel Modelling of Health Statistics, pages 127–142. Wiley, New York, 2001.

    Google Scholar 

  71. P. McCullagh. Regression models for ordinal data. Journal of the Royal Statistical Society, Series B, 42:109–142, 1980. (with discussion)swallowdot.

    MATH  MathSciNet  Google Scholar 

  72. P. McCullagh and J. A. Nelder. Generalized Linear Models, 2nd edition. Chapman & Hall, London, 1989.

    Google Scholar 

  73. D. McFadden. Conditional logit analysis of qualitative choice behavior. In P. Zarembka, editor, Frontiers in Econometrics. Academic Press, New York, 1973.

    Google Scholar 

  74. D. McFadden. Qualitative response models. In W. Hildenbrand, editor, Advances in Econometrics, pages 1–37. Cambridge University Press, Cambridge, UK, 1980.

    Google Scholar 

  75. J. M. Neuhaus, J. D. Kalbfleisch, and W. W. Hauck. A comparison of cluster-specific and population-averaged approaches for analyzing correlated binary data. International Statistical Review, 59:25–35, 1991.

    Google Scholar 

  76. J. F. Pendergast, S. J. Gange, M. A. Newton, M. J. Lindstrom, M. Palta, and M. R. Fisher. A survey of methods for analyzing clustered binary response data. International Statistical Review, 64:89–118, 1996.

    MATH  Google Scholar 

  77. B. Peterson and F. E. Harrell. Partial proportional odds models for ordinal response variables. Applied Statistics, 39:205–217, 1990.

    MATH  Google Scholar 

  78. J. C. Pinheiro and D. M. Bates. Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4:12–35, 1995.

    Google Scholar 

  79. R. L. Prentice and L. A. Gloeckler. Regression analysis of grouped survival data with application to breast cancer data. Biometrics, 34:57–67, 1978.

    MATH  Google Scholar 

  80. S. Rabe-Hesketh, A. Skrondal, and A. Pickles. Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal, 2:1–21, 2002.

    Google Scholar 

  81. S. Rabe-Hesketh, A. Skrondal, and A. Pickles. GLLAMM manual. Working Paper 160, U.C. Berkeley Division of Biostatistics, Berkeley, CA, 2004. (Downloadable from urlhttp://www.bepress.com/ucbbiostat/paper160/)swallowdot.

    Google Scholar 

  82. S. Rabe-Hesketh, A. Skrondal, and A. Pickles. Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics, 128:301–323, 2005.

    MathSciNet  Google Scholar 

  83. R. Raman and D. Hedeker. A mixed-effects regression model for three-level ordinal response data. Statistics in Medicine, 24:3331–3345, 2005.

    MathSciNet  Google Scholar 

  84. J. Rasbash, F. Steele, W. J. Browne, and B. Prosser. A User's Guide to MLwiN. Version 2.0. Centre for Multilevel Modelling, University of Bristol, Bristol, UK, 2005.

    Google Scholar 

  85. S. W. Raudenbush and A. S. Bryk. Hierarchical Linear Models: Applications and Data Analysis Methods, 2nd edition. Sage, Thousand Oaks, CA, 2002.

    Google Scholar 

  86. S. W. Raudenbush, A. S. Bryk, Y. F. Cheong, and R. Congdon. HLM 6: Hierarchical Linear and Nonlinear Modeling. Scientific Software International, Chicago, 2004.

    Google Scholar 

  87. S. W. Raudenbush, M.-L. Yang, and M. Yosef. Maximum likelihood for generalized linear models with nested random effects via high-order, multivariate Laplace approximation. Journal of Computational and Graphical Statistics, 9:141–157, 2000.

    MathSciNet  Google Scholar 

  88. D. Revelt and K. Train. Mixed logit with repeated choices: Household’s choices of appliance efficiency level. Review of Economics and Statistics, 80:647–657, 1998.

    Google Scholar 

  89. F. Rijmen, F. Tuerlinckx, P. De Boeck, and P. Kuppens. A nonlinear mixed model framework for item response theory. Psychological Methods, 8:185–205, 2003.

    Google Scholar 

  90. G. Rodríguez and N. Goldman. An assessment of estimation procedures for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 158:73–89, 1995.

    Google Scholar 

  91. D. B. Rubin. Inference and missing data. Biometrika, 63:581–592, 1976. (with discussion)swallowdot.

    Google Scholar 

  92. F. Samejima. Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph No. 17, 1969.

    Google Scholar 

  93. SAS/Stat. SAS/Stat User's Guide, version 9.1. SAS Institute, Cary, NC, 2004.

    Google Scholar 

  94. T. H. Scheike and T. K. Jensen. A discrete survival model with random effects: An application to time to pregnancy. Biometrics, 53:318–329, 1997.

    MATH  Google Scholar 

  95. J. D. Singer and J. B. Willett. It's about time: Using discrete-time survival analysis to study duration and the timing of events. Journal of Educational and Behavioral Statistics, 18:155–195, 1993.

    Google Scholar 

  96. J. D. Singer and J. B. Willett. Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. Oxford University Press, Oxford, UK, 2003.

    Google Scholar 

  97. A. Skrondal and S. Rabe-Hesketh. Multilevel logistic regression for polytomous data and rankings. Psychometrika, 68:267–287, 2003.

    MathSciNet  Google Scholar 

  98. A. Skrondal and S. Rabe-Hesketh. Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models. Chapman & Hall/CRC, Boca Raton, FL, 2004.

    MATH  Google Scholar 

  99. T. A. B. Snijders and R. J. Bosker. Multilevel Analysis: An Introduction to Basic and Advanced Multilevel Modeling. Sage, Thousand Oaks, CA, 1999.

    MATH  Google Scholar 

  100. D. J. Spiegelhalter, A. Thomas, N. G. Best, and W. R. Gilks. BUGS: Bayesian Inference Using Gibbs Sampling, Version 0.50. Technical Report, MRC Biostatistics Unit, Cambridge, UK, 1995.

    Google Scholar 

  101. StataCorp. Stata Statistical Software: Release 9. Stata Corporation, College Station, TX, 2005.

    Google Scholar 

  102. S. Stern. Simulation-based estimation. Journal of Economic Literature, 35:2006–2039, 1997.

    Google Scholar 

  103. R. Stiratelli, N. M. Laird, and J. H. Ware. Random-effects models for serial observations with binary response. Biometrics, 40:961–971, 1984.

    Google Scholar 

  104. A. H. Stroud and D. Secrest. Gaussian Quadrature Formulas. Prentice Hall, Englewood Cliffs, NJ, 1966.

    MATH  Google Scholar 

  105. M. A. Tanner. Tools for Statistical Inference: Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd edition. Springer, New York, 1996.

    MATH  Google Scholar 

  106. T. R. Ten Have. A mixed effects model for multivariate ordinal response data including correlated discrete failure times with ordinal responses. Biometrics, 52:473–491, 1996.

    MATH  MathSciNet  Google Scholar 

  107. T. R. Ten Have, A. R. Kunselman, and L. A. Tran. A comparison of mixed effects logistic regression models for binary response data with two nested levels of clustering. Statistics in Medicine, 18:947–960, 1999.

    Google Scholar 

  108. T. R. Ten Have and D. H. Uttal. Subject-specific and population-averaged continuation ratio logit models for multiple discrete time survival profiles. Applied Statistics, 43:371–384, 1994.

    MATH  Google Scholar 

  109. J. Terza. Ordinal probit: A generalization. Communications in Statistics: Theory and Methods, 14:1–11, 1985.

    Google Scholar 

  110. D. Thissen and L. Steinberg. A taxonomy of item response models. Psychometrika, 51:567–577, 1986.

    MATH  Google Scholar 

  111. L. L. Thurstone. Psychophysical analysis. American Journal of Psychology, 38:368–389, 1927.

    Google Scholar 

  112. K. E. Train. Discrete Choice Methods with Simulation. Cambridge University Press, Cambridge, UK, 2003.

    MATH  Google Scholar 

  113. G. Tutz and W. Hennevogl. Random effects in ordinal regression models. Computational Statistics & Data Analysis, 22:537–557, 1996.

    MATH  Google Scholar 

  114. M. A. Wakefield, F. J. Chaloupka, N. J. Kaufman, C. T. Orleans, D. C. Barker, and E. E. Ruel. Effect of restrictions on smoking at home, at school, and in public places on teenage smoking: Cross sectional study. British Medical Journal, 321:333–337, 2001.

    Google Scholar 

  115. G. Y. Wong and W. M. Mason. The hierarchical logistic regression model for multilevel analysis. Journal of the American Statistical Association, 80:513–524, 1985.

    Google Scholar 

  116. B. D. Wright. Solving measurement problems with the Rasch model. Journal of Educational Measurement, 14:97–116, 1977.

    Google Scholar 

  117. H. Xie, G. McHugo, A. Sengupta, D. Hedeker, and R. Drake. An application of the thresholds of change model to the analysis of mental health data. Mental Health Services Research, 3:107–114, 2001.

    Google Scholar 

  118. S. L. Zeger, K.-Y. Liang, and P. S. Albert. Models for longitudinal data: A generalized estimating equation approach. Biometrics, 44:1049–1060, 1988.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Illinois at Chicago, USA

    Donald Hedeker

Authors
  1. Donald Hedeker
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Statistics, University of California at Los Angeles, Los Angeles, CA 90095-1554, USA

    Jan de Leeuw

  2. RAND Corporation, 1776 Main St, PO Box 2138, Santa Monica, CA 90407-2138, USA

    Erik Meijer

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer Science+Business Media, LLC

About this chapter

Cite this chapter

Hedeker, D. (2008). Multilevel Models for Ordinal and Nominal Variables. In: Leeuw, J.d., Meijer, E. (eds) Handbook of Multilevel Analysis. Springer, New York, NY. https://doi.org/10.1007/978-0-387-73186-5_6

Download citation

Publish with us

Policies and ethics

Search

Navigation