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Bayesian Hierarchical Modeling for Longitudinal Frequency Data

Bayesian Hierarchical Modeling for Longitudinal Frequency Data. Joseph Jordan Advisor: John C. Kern II Department of Mathematics and Computer Science Duquesne University May 6, 2005. Outline. Motivation The Model Model Simulation Model Implementation

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Bayesian Hierarchical Modeling for Longitudinal Frequency Data

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  1. Bayesian Hierarchical Modeling for Longitudinal Frequency Data Joseph Jordan Advisor: John C. Kern II Department of Mathematics and Computer Science Duquesne University May 6, 2005

  2. Outline • Motivation • The Model • Model Simulation • Model Implementation • Metropolis-Hastings Sampling Algorithm • Results • Conclusion • References

  3. Motivation • Yale University Study: The Patrick & Catherine Weldon Donaghue Medical Foundation • Menopausal women in breast cancer remission • Acupuncture relief of menopausal symptoms • Unlike previous models, this model explicitly recognizes time dependence through prior distributions

  4. Model Simulations:Study Information • Individuals randomly assigned to 1 of 3 groups • Length of Study: 13 weeks (1 week baseline followed by 12 weeks of “treatment” • Measurement: Hot flush frequency (91 observations)

  5. Motivation:Study Samples • Education Group: 6 individuals given weekly educational sessions • Treatment Group: 16 individuals given weekly acupuncture on effective bodily areas • Placebo Group: 17 individuals given weekly acupuncture on non-effective bodily areas

  6. Motivation:Actual Subject Profile

  7. Motivation:Actual Subject Profile

  8. Mean Hot Flush Frequencies

  9. The Model:

  10. The Model:Prior Distributions

  11. The Model:Prior Distributions (Non-Informative)

  12. Model Simulation:j=.5, j=.9, 2j=.5

  13. Model Simulation:j=.5, j=.5, 2j=.5

  14. Model Implementation:Markov Chain Monte Carlo • Metropolis-Hastings Sampling: • Gibbs Sampling:

  15. Metropolis-Hastings Sampling:Requirements • MUST know posterior distribution for parameter (product of likelihood and prior distributions) • Computational precision issues – utilize natural logs • For example:

  16. Metropolis-Hastings Sampling: Algorithm

  17. Gibbs Sampling:Requirements • Requirement: MUST know full conditional distribution for parameter • Sample from full conditional distribution; ALWAYS accept *I • For Example:

  18. Gibbs Sampling:Full Conditional Distributions

  19. Metropolis-HastingsLikelihood for ij • ij: mean hot flush freq on days i and 2i-1 for i=1,…,44, with 45j representing the mean hot flush freq for days 89, 90, 91

  20. Metropolis-HastingsPrior for ij

  21. Metropolis-HastingsDifference in log posterior densities evaluated at *ij and cij

  22. Metropolis-HastingsLikelihood for j

  23. Metropolis-HastingsPrior for j

  24. Metropolis-HastingsDifference in log posterior densities evaluated at *j and cj

  25. Metropolis-HastingsUpdating j • Same likelihood as j

  26. Metropolis-HastingsUpdating 2j • Same likelihood as j

  27. Metropolis-HastingsUpdating 0j • Same posterior as ij’s

  28. Metropolis-HastingsLikelihood Distribution for 

  29. Metropolis-HastingsPrior Distribution for 

  30. Metropolis-HastingsUpdating  • Same likelihood as  • Uniform prior

  31. Metropolis-HastingsUpdating a and b • Uniform Prior • Same likelihood and prior for b

  32. Hastings Ratios

  33. ResultsTreatment Group

  34. ResultsTreatment Group

  35. ResultsPlacebo Group

  36. ResultsPlacebo Group

  37. ResultsEducation Group

  38. ResultsEducation Group

  39. ResultsBoxplot for 0’s

  40. ResultsBoxplot for Exponentiated 0

  41. References • Borgesi, J. 2004. A Piecewise Linear Generalized Poisson Regression Approach to Modeling Longitudinal Frequency Data. Unpublished masters thesis, Duquesne University, Pittsburgh, PA, USA. • Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. 1995. Bayesian Data Analysis. London: Chapman and Hall. • Gilks, W.R., Richardson, S., and Spiegelhalter, D.J. 1996. Markov Chain Monte Carlo in Practice. London: Chapman and Hall. • Kern, J. and S.M. Cohen. 2005. Menopausal symptom relief with acupuncture: modeling longitudinal frequency data. Vol 34, 3: Communications in Statistics: Simulation and Computation.

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