Bayesian Hierarchical Modeling for Longitudinal Frequency DataJoseph JordanAdvisor: John C. Kern IIDepartment of Mathematics and Computer ScienceDuquesne UniversityMay 6, 2005

Outline

Motivation The Model Model Simulation Model Implementation Metropolis-Hastings Sampling Algorithm Results Conclusion References

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

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)

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

Motivation:Actual Subject Profile

Motivation:Actual Subject Profile

Mean Hot Flush Frequencies

The Model:

The Model:Prior Distributions

The Model:Prior Distributions (Non-Informative)

Model Simulation:j=.5, j=.9, 2

j=.5

Model Simulation:j=.5, j=.5, 2

j=.5

Model Implementation:Markov Chain Monte Carlo

Metropolis-Hastings Sampling:

Gibbs Sampling:

Metropolis-Hastings Sampling:Requirements

MUST know posterior distribution for parameter (product of likelihood and prior distributions)

Computational precision issues – utilize natural logs

For example:

Metropolis-Hastings Sampling: Algorithm

Gibbs Sampling:Requirements

Requirement: MUST know full conditional distribution for parameter

Sample from full conditional distribution; ALWAYS accept *

I

For Example:

Gibbs Sampling:Full Conditional Distributions

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

Metropolis-HastingsPrior for ij

Metropolis-HastingsDifference in log posterior densities evaluated at *

ij and cij

Metropolis-HastingsLikelihood for j

Metropolis-HastingsPrior for j

Metropolis-HastingsDifference in log posterior densities evaluated at *

j and cj

Metropolis-HastingsUpdating j

Same likelihood as j

Metropolis-HastingsUpdating 2

j

Same likelihood as j

Metropolis-HastingsUpdating 0j

Same posterior as ij’s

Metropolis-HastingsLikelihood Distribution for

Metropolis-HastingsPrior Distribution for

Metropolis-HastingsUpdating

Same likelihood as Uniform prior

Metropolis-HastingsUpdating a and b

Uniform Prior Same likelihood and prior for b

Hastings Ratios

ResultsTreatment Group

ResultsTreatment Group

ResultsPlacebo Group

ResultsPlacebo Group

ResultsEducation Group

ResultsEducation Group

ResultsBoxplot for 0’s

ResultsBoxplot for Exponentiated 0

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.