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Bayesian Hierarchical Modeling for Longitudinal Frequency DataJoseph JordanAdvisor: John C. Kern IIDepartment of Mathematics and Computer ScienceDuquesne UniversityMay 6, 2005
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Outline
Motivation The Model Model Simulation Model Implementation Metropolis-Hastings Sampling Algorithm Results Conclusion References
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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
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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)
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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
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Motivation:Actual Subject Profile
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Motivation:Actual Subject Profile
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Mean Hot Flush Frequencies
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The Model:
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The Model:Prior Distributions
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The Model:Prior Distributions (Non-Informative)
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Model Simulation:j=.5, j=.9, 2
j=.5
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Model Simulation:j=.5, j=.5, 2
j=.5
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Model Implementation:Markov Chain Monte Carlo
Metropolis-Hastings Sampling:
Gibbs Sampling:
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Metropolis-Hastings Sampling:Requirements
MUST know posterior distribution for parameter (product of likelihood and prior distributions)
Computational precision issues – utilize natural logs
For example:
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Metropolis-Hastings Sampling: Algorithm
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Gibbs Sampling:Requirements
Requirement: MUST know full conditional distribution for parameter
Sample from full conditional distribution; ALWAYS accept *
I
For Example:
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Gibbs Sampling:Full Conditional Distributions
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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
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Metropolis-HastingsPrior for ij
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Metropolis-HastingsDifference in log posterior densities evaluated at *
ij and cij
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Metropolis-HastingsLikelihood for j
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Metropolis-HastingsPrior for j
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Metropolis-HastingsDifference in log posterior densities evaluated at *
j and cj
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Metropolis-HastingsUpdating j
Same likelihood as j
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Metropolis-HastingsUpdating 2
j
Same likelihood as j
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Metropolis-HastingsUpdating 0j
Same posterior as ij’s
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Metropolis-HastingsLikelihood Distribution for
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Metropolis-HastingsPrior Distribution for
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Metropolis-HastingsUpdating
Same likelihood as Uniform prior
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Metropolis-HastingsUpdating a and b
Uniform Prior Same likelihood and prior for b
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Hastings Ratios
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ResultsTreatment Group
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ResultsTreatment Group
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ResultsPlacebo Group
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ResultsPlacebo Group
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ResultsEducation Group
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ResultsEducation Group
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ResultsBoxplot for 0’s
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ResultsBoxplot for Exponentiated 0
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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.