Robust Bayesian Hierarchical Model Using Monte-Carlo Simulation

  • Geng ChenEmail author
  • Sheng Luo
Part of the ICSA Book Series in Statistics book series (ICSABSS)


Many clinical trials collect information on multiple longitudinal outcomes. Depending on the nature of the disease and its symptoms, the longitudinal outcomes can be of mixed types, e.g., binary, ordinal and continuous. Clinical studies on Parkinson’s disease (PD) are good examples of this case. Due to the multidimensional nature of PD, it is difficult to identify a single outcome to represent the overall disease status and severity. Thus, clinical studies that search for treatments for PD usually collect multiple outcomes at different visits. In this chapter, we will introduce the multilevel item response theory (MLIRT) models that account for all the information from multiple longitudinal outcomes and provide valid inference for the overall treatment effects. We will also introduce the normal/independent (NI) distributions, which can be easily implemented into the MLIRT model hierarchically, to handle the outlier and heavy tails problems to produce robust inference. Other data features such as dependent censoring and skewness will also be discussed under the MLIRT framework.


Clinical trial Item-response theory Latent variable MCMC Outliers Joint model Robust distribution Multivariate longitudinal data 


  1. Baghfalaki, T., Ganjali, M., & Berridge, D. (2013). Robust joint modeling of longitudinal measurements and time to event data using normal/independent distributions: A Bayesian approach. Biometrical Journal, 55(6), 844–865.Google Scholar
  2. Bushnell, D. M., & Martin, M. L. (1999). Quality of life and Parkinson’s disease: Translation and validation of the US Parkinson’s disease questionnaire (PDQ-39). Quality of Life Research, 8(4), 345–350.Google Scholar
  3. Carlin, B. P., & Louis, T. A. (2011). Bayesian methods for data analysis. Boca Raton, FL: Chapman & Hall.Google Scholar
  4. Chen, G., & Luo, S. (2016). Robust Bayesian hierarchical model using normal/independent distributions. Biometrical Journal, 58(4), 831–851.Google Scholar
  5. Chen, M. H., Shao, Q. M., & Ibrahim, J. G. (2000). Monte Carlo methods in Bayesian computation. New York: Springer Series in Statistics.Google Scholar
  6. Cummings, J. L. (1992). Depression and Parkinson’s disease: A review. The American Journal of Psychiatry, 149(4), 443–454.Google Scholar
  7. David, D. (2007). Dunson. Bayesian methods for latent trait modelling of longitudinal data. Statistical Methods in Medical Research, 16(5), 399–415.MathSciNetCrossRefGoogle Scholar
  8. Elm, J. J., & The NINDS NET-PD Investigators. (2012). Design innovations and baseline findings in a long-term Parkinson’s trial: The National Institute of Neurological Disorders and Stroke exploratory trials in Parkinson’s Disease Long-Term study-1. Movement Disorders, 27(12), 1513–1521.Google Scholar
  9. Fahn, S., Oakes, D., Shoulson, I., Kieburtz, K., Rudolph, A., Lang, A., et al. (2004). Levodopa and the progression of Parkinson’s disease. The New England Journal of Medicine, 351(24), 2498–2508.Google Scholar
  10. Fox, J. P. (2010). Bayesian item response modeling: Theory and applications. New York: Springer.Google Scholar
  11. Geisser, S. (1993). Predictive inference: An introduction (Vol. 55). Boca Raton, FL: CRC Press.Google Scholar
  12. Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian data analysis. CRC Press.Google Scholar
  13. He, B., & Luo, S. (2013). Joint modeling of multivariate longitudinal measurements and survival data with applications to Parkinson’s disease. Statistical Methods in Medical Research.Google Scholar
  14. Henderson, R., Diggle, P., & Dobson, A. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics, 1(4), 465–480.Google Scholar
  15. Huang, P., Tilley, B. C., Woolson, R. F., & Lipsitz, S. (2005). Adjusting O’Brien’s test to control type I error for the generalized nonparametric Behrens-Fisher problem. Biometrics, 61(2), 532–539.Google Scholar
  16. Jasra, A., Holmes, C. C., & Stephens, D. A. (2005). Markov chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Statistical Science, 20(1), 50–67.MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kamata, A. (2001). Item analysis by the hierarchical generalized linear model. Journal of Educational Measurement, 38(1), 79–93.Google Scholar
  18. Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90(430), 773–795.Google Scholar
  19. Lachos, V. H., Bandyopadhyay, D., & Dey, D. K. (2011). Linear and nonlinear mixed-effects models for censored HIV viral loads using normal/independent distributions. Biometrics, 67(4), 1594–1604.Google Scholar
  20. Lachos, V. H., Castro, L. M., & Dey, D. K. (2013). Bayesian inference in nonlinear mixed-effects models using normal independent distributions. Computational Statistics & Data Analysis, 64, 237–252.Google Scholar
  21. Lachos, V. H., Dey, D. K., & Cancho, V. G. (2009). Robust linear mixed models with skew-normal independent distributions from a Bayesian perspective. Journal of Statistical Planning and Inference, 139(12), 4098–4110.Google Scholar
  22. Lange, K., & Sinsheimer, J. S. (1993). Normal/independent distributions and their applications in robust regression. Journal of Computational and Graphical Statistics, 2(2), 175–198.Google Scholar
  23. Lavine, M., & Schervish, M. J. (1999). Bayes factors: What they are and what they are not. The American Statistician, 53(2), 119–122.Google Scholar
  24. Lee, S.-Y., & Song, X.-Y. (2004). Evaluation of the bayesian and maximum likelihood approaches in analyzing structural equation models with small sample sizes. Multivariate Behavioral Research, 39(4), 653–686.Google Scholar
  25. Lewis, S. M., & Raftery, A. E. (1997). Estimating Bayes factors via posterior simulation with the Laplace-Metropolis estimator. Journal of the American Statistical Association, 92(438), 648–655.Google Scholar
  26. Liu, C. (1996). Bayesian robust multivariate linear regression with incomplete data. Journal of the American Statistical Association, 91(435), 1219–1227.Google Scholar
  27. Lord, F. M., Novick, M. R., & Birnbaum, A. (1968). Statistical theories of mental test scores. Boston, MA: Addison-Wesley.Google Scholar
  28. Luo, S. (2014). A Bayesian approach to joint analysis of multivariate longitudinal data and parametric accelerated failure time. Statistics in Medicine, 33(4), 580–594.Google Scholar
  29. Luo, S., & Wang, J. (2014). Bayesian hierarchical model for multiple repeated measures and survival data: An application to parkinson’s disease. Statistics in Medicine, 33(24), 4279–4291.Google Scholar
  30. Luo, S., Lawson, A. B., He, B., Elm, J. J., & Tilley, B. C. (2012). Bayesian multiple imputation for missing multivariate longitudinal data from a Parkinson’s disease clinical trial. Statistical Methods in Medical Research.Google Scholar
  31. Luo, S., Ma, J., & Kieburtz, K. D. (2013). Robust Bayesian inference for multivariate longitudinal data by using normal/independent distributions. Statistics in Medicine, 32(22), 3812–3828.Google Scholar
  32. Maier, K. S. (2001). A rasch hierarchical measurement model. Journal of Educational and Behavioral Statistics, 26(3), 307–330.Google Scholar
  33. McRae, C., Diem, G., Vo, A., O’Brien, C., & Seeberger, Lauren. (2000). Schwab & England: Standardization of administration. Movement Disorders, 15(2), 335–336.Google Scholar
  34. Miller, T. M., Balsis, S., Lowe, D. A., Benge, J. F., & Doody, R. S. (2012). Item response theory reveals variability of functional impairment within clinical dementia rating scale stages. Dementia and Geriatric Cognitive Disorders, 32(5), 362–366.Google Scholar
  35. Müller, J., Wenning, G. K., Jellinger, K., McKee, A., Poewe, W., & Litvan, I. (2000). Progression of Hoehn and Yahr stages in Parkinsonian disorders: A clinicopathologic study. Neurology, 55(6), 888–891.Google Scholar
  36. Parkinson Study Group. (1989). DATATOP: A multicenter controlled clinical trial in early Parkinson’s disease. Archives of Neurology, 46(10), 1052–1060.Google Scholar
  37. Parkinson Study Group. (1993). Effects of tocopherol and deprenyl on the progression of disability in early Parkinson’s disease. The New England Journal of Medicine, 328(3), 176–183.Google Scholar
  38. Parkinson Study Group. (2002). A controlled trial of rasagiline in early Parkinson disease: The TEMPO study. Archives of Neurology, 59(12), 1937.Google Scholar
  39. Rijmen, F., Tuerlinckx, F., De Boeck, P., & Kuppens, P. (2003). A nonlinear mixed model framework for item response theory. Psychological methods, 8(2), 185.Google Scholar
  40. Rosa, G. J. M., Padovani, C. R., & Gianola, D. (2003). Robust linear mixed models with normal/independent distributions and Bayesian MCMC implementation. Biometrical Journal, 45(5), 573–590.Google Scholar
  41. Sahu, S. K., Dey, D. K., & Branco, M. D. (2003). A new class of multivariate skew distributions with applications to Bayesian regression models. Canadian Journal of Statistics, 31(2), 129–150.Google Scholar
  42. Samejima, F. (1997). Graded response model. New York: Springer.Google Scholar
  43. Snitz, B. E., Yu, L., Crane, P. K., Chang, C.-C. H., Hughes, T. F., & Ganguli, M.(2012). Subjective cognitive complaints of older adults at the population level: An item response theory analysis. Alzheimer Disease & Associated Disorders, 26(4), 344–351.Google Scholar
  44. Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4), 583–639.Google Scholar
  45. Vaccarino, A. L., Anderson, K., Borowsky, B., Duff, K., Joseph, G., Mark, G., et al. (2011). An item response analysis of the motor and behavioral subscales of the unified Huntington’s disease rating scale in Huntington disease gene expansion carriers. Movement Disorders, 26(5), 877–884.Google Scholar
  46. Wang, C., Douglas, J., & Anderson, S. (2002). Item response models for joint analysis of quality of life and survival. Statistics in Medicine, 21(1), 129–142.Google Scholar
  47. Weisscher, N., Glas, C. A., Vermeulen, M., & De Haan, R. J. (2010). The use of an item response theory-based disability item bank across diseases: accounting for differential item functioning. Journal of Clinical Epidemiology, 63(5), 543–549.Google Scholar

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© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Clinical Statistics, GlaxoSmithKlineCollegevilleUSA
  2. 2.Department of BiostatisticsThe University of Texas Health Science Center at HoustonHoustonUSA

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