Advertisement

Computational Statistics

, Volume 18, Issue 1, pp 19–37 | Cite as

A Comparison of Estimation Methods for Multilevel Logistic Models

  • Mirjam Moerbeek
  • Gerard J. P. Van Breukelen
  • Martijn P. F. Berger
Article

Summary

In this paper a comparison between Penalized Quasi Likelihood (PQL) and estimation by numerical integration is made for the analysis of two-level experimental binary data with two treatment conditions. The comparison between the estimation methods is made for three situations: randomization to treatment conditions at the cluster level, randomization at the person level with treatment by cluster interaction, and without such interaction. Criteria for comparison are convergence of the estimation process and improper estimates (i.e. unrealistic high point estimates), criteria concerning the point estimation (bias, variance, and mean squared error) and testing (bias of the point estimates and of the variances as reported by the software) of the treatment effect. The results show that non-convergence occurs more often when estimation is done by numerical integration. This method may also lead to improper estimates. First order PQL performs best in terms of point estimation of the treatment effect, but should not be used for testing. For the latter purpose second order PQL is more applicable.

Keywords

Penalized Quasi Likelihood numerical integration experimental data multilevel logistic model test statistic 

References

  1. Bryk, A. S., Raudenbush, S. W., & Congdon, R. T, Jr. (1996). HLM: Hierarchical Linear and Nonlinear Modeling with the HLM/2L and HLM/3L Programs. Scientific Software International-Chicago.Google Scholar
  2. Gibbons, R. D., & Hedeker, D. (1997). Random effects probit and logistic regression models for three-level data. Biometrics, 53, 1527–1537.CrossRefGoogle Scholar
  3. Goldstein, H. (1986). Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika, 73, 43–56.MathSciNetCrossRefGoogle Scholar
  4. Goldstein, H. (1989). Restricted unbiased iterative generalized least squares estimation. Biometrika, 76, 622–623.MathSciNetCrossRefGoogle Scholar
  5. Goldstein, H. (1991). Nonlinear multilevel models, with an application to discrete response data. Biometrika, 78, 45–51.MathSciNetCrossRefGoogle Scholar
  6. Goldstein, H. (1995). Multilevel statistical models, 2nd edn., Edward Arnold-London.Google Scholar
  7. Goldstein, H., & Rasbash, J. (1996). Improved approximations for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 159, 505–513.MathSciNetCrossRefGoogle Scholar
  8. Goldstein, H, Rasbash, J, Plewis, I, Draper, D., Browne, W., Yang, M., Woodhouse, G., & Healy, M. (1998). A user’s guide to MLwiN. Institute of Education-London.Google Scholar
  9. Hedeker, D., & Gibbons, R. D. (1994). A random-effects ordinal regression model for multilevel analysis. Biometrics, 50, 933–944.CrossRefGoogle Scholar
  10. Hedeker, D., & Gibbons, R. D. (1996). MIXOR: a computer program for mixed-effects ordinal regression analysis. Computer Methods and Programs in Biomedicine, 49, 157–176.CrossRefGoogle Scholar
  11. Lindstrom, M. J., & Bates, D. M. (1990). Nonlinear mixed effects models for repeated measures data. Biometrics, 46, 673–687.MathSciNetCrossRefGoogle Scholar
  12. Moerbeek, M., Van Breukelen, G. J. P, & Berger, M. P. F. (2001). Optimal experimental designs for multilevel logistic models. The Statistician, 50, 17–30.MathSciNetzbMATHGoogle Scholar
  13. Pinheiro, J. C., & Bates, D.M. (2000). Mixed-effects models in S and S-PLUS, 1st. edn., Springer-Verlag, New York.CrossRefGoogle Scholar
  14. Rodríguez, G., & Goldman, N. (1995). An assessment of estimation procedures for multilevel models with binary data. Journal of the Royal Statistical Society, Series A, 158, 73–89.CrossRefGoogle Scholar
  15. Rodríguez, G., & Goldman, N. (2001). Improved estimation procedures for multilevel models with binary response: a case-study. Journal of the Royal Statistical Society, Series A, 164, 339–355.MathSciNetCrossRefGoogle Scholar
  16. Zhou, X.-H., Perkins, A. K., & Hui, S. L. (1999). Comparisons of software packages for generalized linear multilevel models. The American Statistician, 53, 3.Google Scholar

Copyright information

© Physica-Verlag 2003

Authors and Affiliations

  • Mirjam Moerbeek
    • 1
  • Gerard J. P. Van Breukelen
    • 2
  • Martijn P. F. Berger
    • 2
  1. 1.Department of Methodology and StatisticsUtrecht UniversityMaastrichtThe Netherlands
  2. 2.Department of Methodology and StatisticsMaastricht UniversityMaastrichtThe Netherlands

Personalised recommendations