Log-Link Regression Models for Ordinal Responses

Affiliation(s)

Menzies Research Institute Tasmania, University of Tasmania, Hobart, Australia.

Flinders Clinical Effectiveness, Flinders University, Adelaide, Australia.

Department of Public Health, University of Massachusetts, Amherst, USA.

Menzies Research Institute Tasmania, University of Tasmania, Hobart, Australia.

Flinders Clinical Effectiveness, Flinders University, Adelaide, Australia.

Department of Public Health, University of Massachusetts, Amherst, USA.

ABSTRACT

The adjacent-categories, continuation-ratio and proportional odds logit-link regression models provide useful extensions of the multinomial logistic model to ordinal response data. We propose fitting these models with a logarithmic link to allow estimation of different forms of the risk ratio. Each of the resulting ordinal response log-link models is a constrained version of the log multinomial model, the log-link counterpart of the multinomial logistic model. These models can be estimated using software that allows the user to specify the log likelihood as the objective function to be maximized and to impose constraints on the parameter estimates. In example data with a dichotomous covariate, the unconstrained models produced valid coefficient estimates and standard errors, and the constrained models produced plausible results. Models with a single continuous covariate performed well in data simulations, with low bias and mean squared error on average and appropriate confidence interval coverage in admissible solutions. In an application to real data, practical aspects of the fitting of the models are investigated. We conclude that it is feasible to obtain adjusted estimates of the risk ratio for ordinal outcome data.

Cite this paper

C. Blizzard, S. Quinn, J. Canary and D. Hosmer, "Log-Link Regression Models for Ordinal Responses,"*Open Journal of Statistics*, Vol. 3 No. 4, 2013, pp. 16-25. doi: 10.4236/ojs.2013.34A003.

C. Blizzard, S. Quinn, J. Canary and D. Hosmer, "Log-Link Regression Models for Ordinal Responses,"

References

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[12] C. C. Clogg and E. S. Shihadeh, “Statistical Models for Ordinal Variables,” Sage, Thousand Oaks, 1994.

[13] R. Bender and A. Benner, “Calculating Ordinal Regression Models in SAS and S-Plus,” Biometrical Journal, Vol. 42, No. 6, 2000, pp. 677-699. doi:10.1002/1521-4036(200010)42:6<677::AID-BIMJ677>3.0.CO;2-O

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[15] B. G. Armstrong and M. Sloan, “Ordinal Regression Models for Epidemiologic Data,” American Journal of Epidemiology, Vol. 129, No. 1, 1989, pp. 191-204.

[16] D. Katz, J. Baptista, S. P. Azen and M. C. Pike, “Obtaining Confidence Intervals for the Risk Ratio in Cohort Studies,” Biometrics, Vol. 34, 1978, pp. 469-474. doi:10.2307/2530610.

[17] C. R. Rao, “Linear Statistical Inference and Its Applications,” 2nd Edition, Wiley, New York, 2002.

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[19] C. R. Rao, “Large Sample Tests of Statistical Hypotheses Concerning Several Parameters with Applications to Problems of Estimation,” Proceedings of the Cambridge Philosophical Society, Vol. 44, No. 1, 1948, pp. 50-57. doi:10.1017/S0305004100023987.

[20] A. Wald, “Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations Is Large,” Transactions of the American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482. doi:10.1090/S0002-9947-1943-0012401-3.

[21] B. Peterson and F. E. Harrrell, “Partial Proportional Odds Models for Ordinal Response Variables,” Applied Statistics, Vol. 39, No. 2, 1990, pp. 2005-2017. doi:10.2307/2347760.

[22] P. McCullough and J. Nelder, “Generalized Linear Models,” 2nd Edition, Chapman Hall, New York, 1989.

[1] L. A. Goodman. “The Analysis of Dependence in CrossClassifications Having Ordered Categories, Using LogLinear Models for Frequencies and Log-Linear Models for Odds,” Biometrics, Vol. 39, No. 1, 1983, pp. 149-160. doi:10.2307/2530815

[2] S. E. Fienberg, “The Analysis of Cross-Classified Data,” 2nd Edition, MIT Press, Cambridge, 1980.

[3] S. H. Walker and D. B. Duncan, “Estimation of the Probability of an Event as a Function of Several Independent Variables,” Biometrika, Vol. 54, No. 1, 1967, pp. 167-79.

[4] P. McCullough, “Regression Models for Ordinal Data,” Journal of the Royal Statistical Society B, Vol. 42, No. 2, 1980, pp. 109-142.

[5] D. McFadden, “Econometric Models for Probabilistic Choice among Products,” Journal of Business, Vol. 53, No. 3, 1980, pp. S13-S29. doi:10.1086/296093

[6] D. W. Hosmer and S. Lemeshow, “Applied Logistic Regression,” 2nd Edition, Wiley, New York, 2000. doi:10.1002/0471722146

[7] L. Blizzard and D. W. Hosmer, “Parameter Estimation and Goodness-of-Fit in Log Binomial Regression,” Biometrical Journal, Vol. 48, No. 6, 2006, pp. 5-22.

[8] O. S. Miettenen, “Theoretical Epidemiology,” Wiley, New York, 1985.

[9] S. Greenland, “Interpretation and Choice of Effect Measures in Epidemiologic Analysis,” American Journal of Epidemiology, Vol. 125, No. 5, 1987, pp. 761-768.

[10] O. S. Miettenen and E. F. Cook, “Confounding: essence and detection,” American Journal of Epidemiology, Vol. 114, No. 4, 1981, pp. 593-603.

[11] J. Lee, “Odds Ratio or Relative Risk for Cross-Sectional Data?” International Journal of Epidemiology, Vol. 23, No. 1, 1994, pp. 201-203. doi:10.1093/ije/23.1.201

[12] C. C. Clogg and E. S. Shihadeh, “Statistical Models for Ordinal Variables,” Sage, Thousand Oaks, 1994.

[13] R. Bender and A. Benner, “Calculating Ordinal Regression Models in SAS and S-Plus,” Biometrical Journal, Vol. 42, No. 6, 2000, pp. 677-699. doi:10.1002/1521-4036(200010)42:6<677::AID-BIMJ677>3.0.CO;2-O

[14] A. A. O’Connell, “Logistic Regression Models for Ordinal Response Variables,” Sage, Thousand Oaks, 2006.

[15] B. G. Armstrong and M. Sloan, “Ordinal Regression Models for Epidemiologic Data,” American Journal of Epidemiology, Vol. 129, No. 1, 1989, pp. 191-204.

[16] D. Katz, J. Baptista, S. P. Azen and M. C. Pike, “Obtaining Confidence Intervals for the Risk Ratio in Cohort Studies,” Biometrics, Vol. 34, 1978, pp. 469-474. doi:10.2307/2530610.

[17] C. R. Rao, “Linear Statistical Inference and Its Applications,” 2nd Edition, Wiley, New York, 2002.

[18] J. Neyman and E. S. Pearson “On the Use and Interpretation of Certain Test Criteria,” Biometrika, Vol. 20, No. 1/2, 3/4, 1928, pp. 175-240, 263-294.

[19] C. R. Rao, “Large Sample Tests of Statistical Hypotheses Concerning Several Parameters with Applications to Problems of Estimation,” Proceedings of the Cambridge Philosophical Society, Vol. 44, No. 1, 1948, pp. 50-57. doi:10.1017/S0305004100023987.

[20] A. Wald, “Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations Is Large,” Transactions of the American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482. doi:10.1090/S0002-9947-1943-0012401-3.

[21] B. Peterson and F. E. Harrrell, “Partial Proportional Odds Models for Ordinal Response Variables,” Applied Statistics, Vol. 39, No. 2, 1990, pp. 2005-2017. doi:10.2307/2347760.

[22] P. McCullough and J. Nelder, “Generalized Linear Models,” 2nd Edition, Chapman Hall, New York, 1989.