Skip to main content
SpringerLink
Log in

Ordinal Logistic Regression

  • Chapter
Regression Modeling Strategies

Part of the book series: Springer Series in Statistics ((SSS))

Abstract

Many medical and epidemiologic studies incorporate an ordinal response variable. In some cases an ordinal response Y represents levels of a standard measurement scale such as severity of pain (none, mild, moderate, severe). In other cases, ordinal responses are constructed by specifying a hierarchy of separate endpoints. For example, clinicians may specify an ordering of the severity of several component events and assign patients to the worst event present from among none, heart attack, disabling stroke, and death. Still another use of ordinal response methods is the application of rank-based methods to continuous responses so as to obtain robust inferences. For example, the proportional odds model described later allows for a continuous Y and is really a generalization of the Wilcoxon–Mann–Whitney rank test. Thus the semiparametric proportional odds model is a direct competitor of ordinary linear models.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 64.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 84.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 129.99
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

Notes

  1. 1.

    If \(\hat{\beta }\) were derived from separate binary fits, all \(\bar{U}_{\cdot m} \equiv 0\).

  2. 2.

    If Y does not have very many levels, the median will be a discontinuous function of X and may not be satisfactory.

References

  1. A. Agresti. A survey of models for repeated ordered categorical response data. Stat Med, 8:1209–1224, 1989.

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. J. A. Anderson. Regression and ordered categorical variables. J Roy Stat Soc B, 46:1–30, 1984.

    MATH  Google Scholar 

  4. J. A. Anderson and P. R. Philips. Regression, discrimination and measurement models for ordered categorical variables. Appl Stat, 30:22–31, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. G. Armstrong and M. Sloan. Ordinal regression models for epidemiologic data. Am J Epi, 129:191–204, 1989. See letter to editor by Peterson.

    Google Scholar 

  6. D. Ashby, C. R. West, and D. Ames. The ordered logistic regression model in psychiatry: Rising prevalence of dementia in old people’s homes. Stat Med, 8:1317–1326, 1989.

    Article  Google Scholar 

  7. R. Bender and A. Benner. Calculating ordinal regression models in SAS and S-Plus. Biometrical J, 42:677–699, 2000.

    Article  Google Scholar 

  8. D. M. Berridge and J. Whitehead. Analysis of failure time data with ordinal categories of response. Stat Med, 10:1703–1710, 1991.

    Article  Google Scholar 

  9. R. Brant. Assessing proportionality in the proportional odds model for ordinal logistic regression. Biometrics, 46:1171–1178, 1990.

    Article  Google Scholar 

  10. S. R. Brazer, F. S. Pancotto, T. T. Long III, F. E. Harrell, K. L. Lee, M. P. Tyor, and D. B. Pryor. Using ordinal logistic regression to estimate the likelihood of colorectal neoplasia. J Clin Epi, 44:1263–1270, 1991.

    Article  Google Scholar 

  11. W. S. Cleveland. Robust locally weighted regression and smoothing scatterplots. J Am Stat Assoc, 74:829–836, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  12. T. J. Cole, C. J. Morley, A. J. Thornton, M. A. Fowler, and P. H. Hewson. A scoring system to quantify illness in babies under 6 months of age. J Roy Stat Soc A, 154:287–304, 1991.

    Article  Google Scholar 

  13. D. Collett. Modelling Binary Data. Chapman and Hall, London, second edition, 2002.

    Google Scholar 

  14. C. Cox. Location-scale cumulative odds models for ordinal data: A generalized non-linear model approach. Stat Med, 14:1191–1203, 1995.

    Article  Google Scholar 

  15. D. R. Cox. Regression models and life-tables (with discussion). J Roy Stat Soc B, 34:187–220, 1972.

    Google Scholar 

  16. M. W. Fagerland and D. W. Hosmer. A goodness-of-fit test for the proportional odds regression model. Stat Med, 32(13):2235–2249, 2013.

    Article  MathSciNet  Google Scholar 

  17. J. J. Faraway. The cost of data analysis. J Comp Graph Stat, 1:213–229, 1992.

    Google Scholar 

  18. S. E. Fienberg. The Analysis of Cross-Classified Categorical Data. Springer, New York, second edition, 2007.

    Google Scholar 

  19. P. Grambsch and T. Therneau. Proportional hazards tests and diagnostics based on weighted residuals. Biometrika, 81:515–526, 1994. Amendment and corrections in 82: 668 (1995).

    Google Scholar 

  20. S. Greenland. Alternative models for ordinal logistic regression. Stat Med, 13:1665–1677, 1994.

    Article  Google Scholar 

  21. A. Guisan and F. E. Harrell. Ordinal response regression models in ecology. J Veg Sci, 11:617–626, 2000.

    Article  Google Scholar 

  22. T. J. Hastie, J. L. Botha, and C. M. Schnitzler. Regression with an ordered categorical response. Stat Med, 8:785–794, 1989.

    Article  Google Scholar 

  23. G. G. Koch, I. A. Amara, and J. M. Singer. A two-stage procedure for the analysis of ordinal categorical data. In P. K. Sen, editor, BIOSTATISTICS: Statistics in Biomedical, Public Health and Environmental Sciences. Elsevier Science Publishers B. V. (North-Holland), Amsterdam, 1985.

    Google Scholar 

  24. J. M. Landwehr, D. Pregibon, and A. C. Shoemaker. Graphical methods for assessing logistic regression models (with discussion). J Am Stat Assoc, 79:61–83, 1984.

    Article  MATH  Google Scholar 

  25. C. Li and B. E. Shepherd. A new residual for ordinal outcomes. Biometrika, 99(2):473–480, 2012.

    Article  MathSciNet  MATH  Google Scholar 

  26. D. J. Lunn, J. Wakefield, and A. Racine-Poon. Cumulative logit models for ordinal data: a case study involving allergic rhinitis severity scores. Stat Med, 20:2261–2285, 2001.

    Article  Google Scholar 

  27. P. McCullagh. Regression models for ordinal data. J Roy Stat Soc B, 42:109–142, 1980.

    MathSciNet  Google Scholar 

  28. H. Murad, A. Fleischman, S. Sadetzki, O. Geyer, and L. S. Freedman. Small samples and ordered logistic regression: Does it help to collapse categories of outcome? Am Statistician, 57:155–160, 2003.

    Article  MathSciNet  Google Scholar 

  29. B. Peterson and F. E. Harrell. Partial proportional odds models for ordinal response variables. Appl Stat, 39:205–217, 1990.

    Article  MATH  Google Scholar 

  30. SAS Institute, Inc. SAS/STAT User’s Guide, volume 2. SAS Institute, Inc., Cary NC, fourth edition, 1990.

    Google Scholar 

  31. D. Schoenfeld. Partial residuals for the proportional hazards regression model. Biometrika, 69:239–241, 1982.

    Article  Google Scholar 

  32. S. C. Scott, M. S. Goldberg, and N. E. Mayo. Statistical assessment of ordinal outcomes in comparative studies. J Clin Epi, 50:45–55, 1997.

    Article  Google Scholar 

  33. S. Senn and S. Julious. Measurement in clinical trials: A neglected issue for statisticians? (with discussion). Stat Med, 28:3189–3225, 2009.

    Article  MathSciNet  Google Scholar 

  34. S. H. Walker and D. B. Duncan. Estimation of the probability of an event as a function of several independent variables. Biometrika, 54:167–178, 1967.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. Whitehead, R. Z. Omar, J. P. T. Higgins, E. Savaluny, R. M. Turner, and S. G. Thompson. Meta-analysis of ordinal outcomes using individual patient data. Stat Med, 20:2243–2260, 2001.

    Article  Google Scholar 

  36. J. Whitehead. Sample size calculations for ordered categorical data. Stat Med, 12:2257–2271, 1993. See letter to editor SM 15:1065-6 for binary case;see errata in SM 13:871 1994;see kol95com, jul96sam.

    Google Scholar 

  37. T. W. Yee and C. J. Wild. Vector generalized additive models. J Roy Stat Soc B, 58:481–493, 1996.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Biostatistics, School of Medicine Vanderbilt University, Nashville, TN, USA

    Frank E. Harrell Jr.

Authors
  1. Frank E. Harrell Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Harrell, F.E. (2015). Ordinal Logistic Regression. In: Regression Modeling Strategies. Springer Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-19425-7_13

Download citation

Publish with us

Policies and ethics

Search

Navigation