Bayesian Multivariate Logistic Regression by Sean O’Brien and David Dunson (Biometrics, 2004 )

1. Bayesian Multivariate Logistic Regressionby Sean O’Brien and David Dunson(Biometrics, 2004 ) Presented by Lihan He ECE, Duke University May 16, 2008

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

3. Univariate Logistic Regression Model Equivalent: zi: latent variable L( ): logistic density logistic density: CDF:

4. Univariate Logistic Regression Model Approximation using t distribution set

5. , F-1( ) is the inverse CDF of density Multivariate Logistic Regression Model Binary variable for each output -- marginal pdf has univariate logistic density with

6. Multivariate Logistic Regression Model Property • The marginal univariate densities of zj, for j=1,…,p, have univariate logistic form • p=1, reduce to the univariate logistic density • R is a correlation matrix (with 1’s on the diagonal), reflecting the correlations between zj, and hence the correlations between yj • R=diag(1,…,1), reduce to a product of univariate logistic densities, and the elements of z are uncorrelated • Good convergence property for MCMC sampling

7. Multivariate Logistic Regression Model Likelihood M-ary variable for each output (ordered) Assume Define

8. Prior specification and convergence or R: uniform density [-1,1] for each element in non-diagonal position

9. Importance sampling: sample from a proposal distribution to approximate samples from , and use importance weights for exact inference. Use multivariate t distribution to approximate the multivariate logistic density in the likelihood part. Posterior Computation Posterior: Prior and likelihood are not conjugate Proposal distribution: =

10. Introduce latent variables and z, the proposal is expressed as z) Sample and z from the full conditionals since the likelihood is conjugate to prior. Set with probability Set otherwise Posterior Computation Update R using a Metropolis step (accept/reject)

11. Posterior Computation Importance weights for inference weights

12. 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, yij={0,1}, i=1,…,584, j=1, 2 Covariates: xij for the ith pregnancy and the jth infant

13. Application • Obtain nearly identical estimates to the study of AP for the regression coefficients. • Female gender (β1), prior preterm delivery (β4, β5) and smoking (β8) are associated with an increased risk of SGA. • Outcomes for twins are highly correlated, represented by R.

14. Conclusions • Propose a multivariate logistic density for multivariate logistic regression model. • The proposed multivariatelogistic 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.