Archives

Archives
Review Articles

Objective Bayesian hypothesis testing and estimation for the intraclass model

Duo Zhang ,

Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, USA

Daojiang He ,

School of Mathematics and Computer Science, Anhui Normal University, Wuhu, People's Republic of China

Xiaoqian Sun ,

Department of Mathematical Sciences, Clemson University, Clemson, SC, USA

Min Wang

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA

min.wang@ttu.edu minwang@mtu.edu

Pages 37-47 | Received 06 Nov. 2017, Accepted 24 May. 2018, Published online: 07 Jun. 2018,
  • Abstract
  • Full Article
  • References
  • Citations

ABSTRACT

The intraclass correlation coefficient (ICC) plays an important role in various fields of study as a coefficient of reliability. In this paper, we consider objective Bayesian analysis for the ICC in the context of normal linear regression model. We first derive two objective priors for the unknown parameters and show that both result in proper posterior distributions. Within a Bayesian decision-theoretic framework, we then propose an objective Bayesian solution to the problems of hypothesis testing and point estimation of the ICC based on a combined use of the intrinsic discrepancy loss function and objective priors. The proposed solution has an appealing invariance property under one-to-one reparametrisation of the quantity of interest. Simulation studies are conducted to investigate the performance the proposed solution. Finally, a real data application is provided for illustrative purposes.

References

  1. Barkto, J. (1966). The intraclass correlation coefficient as a measure of reliability. Psychological Reports, 19, 2–11. [Google Scholar]
  2. Berger, J. O., & Bernardo, J. M. (1992a). On the development of reference priors. In J. M. Bernardo, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian statistics, 4 (Peñíscola, 1991) (pp. 35–60). New York: Oxford University Press. [Google Scholar]
  3. Berger, J. O., & Bernardo, J. M. (1992b). Reference priors in a variance components problem. In P. K. Goel & N. Sreenivas Iyengar (Eds.), Lecture notes in statistics: Vol. 75. Bayesian analysis in statistics and econometrics (Bangalore, 1988) (pp. 177–194). New York: Springer. [Google Scholar]
  4. Bernardo, J. (2010). Integrated objective Bayesian estimation and hypothesis testing. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, & M. West (Eds.), Bayesian statistics, 9. Proceedings of the ninth valencia international meeting (pp. 1–68). New York: Oxford University Press. [Google Scholar]
  5. Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference. Journal of the Royal Statistical Society. Series B (Methodological), 41, 113–147. [Google Scholar]
  6. Bernardo, J. M. (1999). Nested hypothesis testing: The Bayesian reference criterion. In J. M. Bernardo, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian statistics, 6 (Alcoceber, 1998) (pp. 101–130). New York: Oxford University Press. [Google Scholar]
  7. Bernardo, J. M. (2005). Reference analysis. In D. K. Dey & C. R. Rao (Eds.), Handbook of Statistics: Vol. 25. Bayesian thinking: Modeling and computation (pp. 17–90). Amsterdam: Elsevier/North-Holland. [Google Scholar]
  8. Bernardo, J. M., & Juárez, M. A. (2003). Intrinsic estimation. In J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith, & M. West (Eds.), Bayesian statistics, 7 (Tenerife, 2002) (pp. 465–476). New York: Oxford University Press. [Google Scholar]
  9. Bernardo, J. M., & Pérez, S. (2007). Comparing normal means: New methods for an old problem. Bayesian Analysis, 2, 45–58. doi: 10.1214/07-BA202 [Google Scholar]
  10. Bernardo, J. M., & Rueda, R. (2002). Bayesian hypothesis testing: A reference approach. International Statistical Review, 70, 351–372. doi: 10.1111/j.1751-5823.2002.tb00175.x [Google Scholar]
  11. Bernardo, J. M., & Smith, A. F. (1994). Bayesian theory. Chichester: Wiley. [Google Scholar]
  12. Box, G. E. P., & Tiao, G. C. (1973). Bayesian inference in statistical analysis. Reading, MA: Addison-Wesley. [Google Scholar]
  13. Chung, Y., & Dey, D. K. (1998). Bayesian approach to estimation of intraclass correlation using reference prior. Communications in Statistics: Theory and Methods, 27, 2241–2255. doi: 10.1080/03610929808832225 [Taylor & Francis Online], [Google Scholar]
  14. Datta, G. S., & Ghosh, J. K. (1995a). Noninformative priors for maximal invariant parameter in group models. TEST, 4, 95–114. doi: 10.1007/BF02563105 [Google Scholar]
  15. Datta, G. S., & Ghosh, J. K. (1995b). On priors providing frequentist validity for Bayesian inference. Biometrika, 82, 37–45. doi: 10.2307/2337625 [Google Scholar]
  16. Datta, G. S., & Mukerjee, R. (2004). Probability matching priors: Higher order asymptotics. Lecture notes in statistics: Vol. 178. New York: Springer-Verlag. [Google Scholar]
  17. Fleiss, J. (1986). The design and analysis of clinical experiments. New York: Wiley. [Google Scholar]
  18. Frees, E. W. (2004). Longitudinal and panel data: Analysis and applications in the social sciences. New York: Cambridge University Press. [Google Scholar]
  19. Ghosh, M., & Heo, J. (2003). Noninformative priors, credible sets and Bayesian hypothesis testing for the intraclass model. Journal of Statistical Planning and Inference, 112, 133–146. doi: 10.1016/S0378-3758(02)00328-2 [Google Scholar]
  20. Jelenkowska, T. H. (1998). Bayesian estimation of the intraclass correlation coefficients in the mixed linear model. Applications of Mathematics, 43, 103–110. doi: 10.1023/A:1023210900467 [Google Scholar]
  21. Lin, L., Hedayat, A. S., Sinha, B., & Yang, M. (2002). Statistical methods in assessing agreement: Models, issues, and tools. Journal of the American Statistical Association, 97, 257–270. doi: 10.1198/016214502753479392 [Taylor & Francis Online], [Google Scholar]
  22. Paul, S. R. (1990). Maximum likelihood estimation of intraclass correlation in the analysis of familial data: Estimating equation approach. Biometrika, 77, 549–555. doi: 10.1093/biomet/77.3.549 [Google Scholar]
  23. Paul, S. R. (1996). Score tests for interclass correlation in familial data. Biometrics, 52, 955–963. doi: 10.2307/2533056 [Google Scholar]
  24. Robert, C. P. (1996). Intrinsic losses. Theory and Decision, 40, 191–214. doi: 10.1007/BF00133173 [Google Scholar]
  25. Sun, D., & Ye, K. (1996). Frequentist validity of posterior quantiles for a two-parameter exponential family. Biometrika, 83, 55–65. doi: 10.1093/biomet/83.1.55 [Google Scholar]

Authors

About the Journal

Links

3663 N.Zhongshan Rd., Shanghai 200062  Tel.: 86-21-62233333