Bayesian Multivariate Logistic Regression by Sean OBrien and David Dunson Biometrics, 2004

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Bayesian Multivariate Logistic Regressionby
Sean OBrien and David Dunson(Biometrics, 2004 )

• Presented by Lihan He
• ECE, Duke University
• May 16, 2008

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Outlines

• Univariate logistic regression
• Multivariate logistic regression
• Prior specification and convergence
• Posterior computation
• Experimental result
• Conclusions

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Univariate Logistic Regression Model
Equivalent
zi latent variable
L( ) logistic density
logistic density
CDF
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Univariate Logistic Regression Model
Approximation using t distribution
set
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Multivariate Logistic Regression Model
Binary variable for each output
-- marginal pdf has univariate logistic density
with
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Multivariate Logistic Regression Model
Property

• The marginal univariate densities of zj, for
  j1,,p, have univariate logistic form
• p1, reduce to the univariate logistic density
• R is a correlation matrix (with 1s on the
  diagonal), reflecting the correlations between
  zj, and hence the correlations between yj
• Rdiag(1,,1), reduce to a product of univariate
  logistic densities, and the elements of z are
  uncorrelated
• Good convergence property for MCMC sampling

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Multivariate Logistic Regression Model
Likelihood
M-ary variable for each output (ordered)
Assume
Define
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Prior specification and convergence
or
R uniform density -1,1 for each element in
non-diagonal position
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Posterior Computation
Posterior
Prior and likelihood are not conjugate
Proposal distribution

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Posterior Computation
Update R using a Metropolis step (accept/reject)
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Posterior Computation
Importance weights for inference
weights
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Application
Subject 584 twin pregnancies
Output small for gestational age (SGA), defined
as a birthweight below the 10th percentile for a
given gestational age in a reference population.
Binary output, yij0,1, i1,,584, j1, 2
Covariates xij for the ith pregnancy and the jth
infant
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Application

• Obtain nearly identical estimates to the study of
  AP for the regression coefficients.
• Female gender (ß1), prior preterm delivery (ß4,
  ß5) and smoking (ß8) are associated with an
  increased risk of SGA.
• Outcomes for twins are highly correlated,
  represented by R.

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Conclusions

• Propose a multivariate logistic density for
  multivariate logistic regression model.
• The proposed multivariate logistic density is
  closely approximated by a multivariate t
  distribution.
• Has properties that facilitate efficient sampling
  and guaranteed convergence.
• The marginals are univariate logistic densities.
• Embed the correlation structure within the model.