Bayesian Hierarchical Modeling for Longitudinal Frequency Data Joseph
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 n n n n Motivation The Model Simulation Model Implementation Metropolis-Hastings Sampling Algorithm Results Conclusion References
Motivation n n 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
Model Simulations: Study Information Individuals randomly assigned to 1 of 3 groups n Length of Study: 13 weeks (1 week baseline followed by 12 weeks of “treatment” n Measurement: Hot flush frequency (91 observations) n
Motivation: Study Samples Education Group: 6 individuals given weekly educational sessions n Treatment Group: 16 individuals given weekly acupuncture on effective bodily areas n Placebo Group: 17 individuals given weekly acupuncture on non-effective bodily areas n
Motivation: Actual Subject Profile
Motivation: Actual Subject Profile
Mean Hot Flush Frequencies
The Model: n n
The Model: Prior Distributions n n n
The Model: Prior Distributions (Non-Informative) n n
Model Simulation: j=. 5, j=. 9, 2 j=. 5
Model Simulation: j=. 5, 2 j=. 5
Model Implementation: Markov Chain Monte Carlo n Metropolis-Hastings Sampling: n Gibbs Sampling:
Metropolis-Hastings Sampling: Requirements n n n MUST know posterior distribution for parameter (product of likelihood and prior distributions) Computational precision issues – utilize natural logs For example:
Metropolis-Hastings Sampling: Algorithm n n n
Gibbs Sampling: Requirements Requirement: MUST know full conditional distribution for parameter n Sample from full conditional distribution; ALWAYS accept *I n For Example: n n
Gibbs Sampling: Full Conditional Distributions
Metropolis-Hastings Likelihood for ij n ij: mean hot flush freq on days i and 2 i-1 for i=1, …, 44, with 45 j representing the mean hot flush freq for days 89, 90, 91
Metropolis-Hastings Prior for ij
Metropolis-Hastings Difference in log posterior densities evaluated at *ij and cij
Metropolis-Hastings Likelihood for j
Metropolis-Hastings Prior for j
Metropolis-Hastings Difference in log posterior densities evaluated at *j and cj
Metropolis-Hastings Updating j n Same likelihood as j
Metropolis-Hastings Updating 2 j n Same likelihood as j
Metropolis-Hastings Updating 0 j n Same posterior as ij’s
Metropolis-Hastings Likelihood Distribution for
Metropolis-Hastings Prior Distribution for
Metropolis-Hastings Updating Same likelihood as n Uniform prior n
Metropolis-Hastings Updating a and b n n Uniform Prior Same likelihood and prior for b
Hastings Ratios
Results Treatment Group
Results Treatment Group
Results Placebo Group
Results Placebo Group
Results Education Group
Results Education Group
Results Boxplot for 0’s
Results Boxplot for Exponentiated 0
References n Borgesi, J. 2004. A Piecewise Linear Generalized Poisson Regression Approach to Modeling Longitudinal Frequency Data. Unpublished masters thesis, Duquesne University, Pittsburgh, PA, USA. n Gelman, A. , Carlin, J. B. , Stern, H. S. , and Rubin, D. B. 1995. Bayesian Data Analysis. London: Chapman and Hall. n Gilks, W. R. , Richardson, S. , and Spiegelhalter, D. J. 1996. Markov Chain Monte Carlo in Practice. London: Chapman and Hall. n 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.
- Slides: 41