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

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,  2 j =.5

13. Model Simulation:  j =.5,  j =.5,  2 j =.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-Hastings Likelihood 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-Hastings Prior for  ij

21. Metropolis-Hastings Difference in log posterior densities evaluated at  * ij and  c ij

22. Metropolis-Hastings Likelihood for  j

23. Metropolis-Hastings Prior for  j

24. Metropolis-Hastings Difference in log posterior densities evaluated at  * j and  c j

25. Metropolis-Hastings Updating  j Same likelihood as  j

26. Metropolis-Hastings Updating  2 j Same likelihood as  j

27. Metropolis-Hastings Updating  0j Same posterior as  ij ’s

28. Metropolis-Hastings Likelihood Distribution for 

29. Metropolis-Hastings Prior Distribution for 

30. Metropolis-Hastings Updating  Same likelihood as  Uniform prior

31. Metropolis-Hastings Updating a and b Uniform Prior Same likelihood and prior for b

32. Hastings Ratios

33. Results Treatment Group

35. Results Placebo Group

37. Results Education Group

39. Results Boxplot for  0 ’s

40. Results Boxplot 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.