Bayesian Hierarchical Modeling for Longitudinal Frequency Data

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.