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A (poor) Gibbs Sampling Approach to Logistic Regression

Kyle Bogdan Grant Brown. A (poor) Gibbs Sampling Approach to Logistic Regression. Data. Simulated based on known values of parameters (one covariate, ‘dose’). ‘rats’ given different dosages of imaginary chemical, 4 dose groups with 25 rats in each group.

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A (poor) Gibbs Sampling Approach to Logistic Regression

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  1. Kyle Bogdan Grant Brown A (poor) Gibbs Sampling Approach to Logistic Regression

  2. Data • Simulated based on known values of parameters (one covariate, ‘dose’). • ‘rats’ given different dosages of imaginary chemical, 4 dose groups with 25 rats in each group. • Data generated three times under different parameters, three chains used for each data set.

  3. Gibbs Sampling For Logistic Data? • Traditionally, binomial likelihood, prior on logit. • Full Conditionals have no coherent form. • Attractive, however, because it eliminates the need to reject iterations

  4. Algorithm • Groenewald and Mokgatlhe, 2005 • Create Uniform Latent Variables Based on Y[i,j] = 0, 1 • Draws from joint posterior of Betas and U[i,j] • pi[i] = p(uniform(01) <= logit-1(Beta*x[i])) • Written in R, refined in Python • Very inefficient • Draw new parameter for each Y[i,j] at each iteration

  5. Implementation • Three datasets • Three chains per set • 1 Million iterations per chain • Last 500k iterations sent to CODA • 9m total iterations, 4.5 m analyzed

  6. Initial Problems

  7. Sampler Output/Diagostics

  8. Sampler Output/Diagnostics

  9. Sampler Output/Diagnostics

  10. Sampler Output/Diagnostics

  11. Sampler Output/Diagnostics

  12. WinBUGS Model • Y[i,j]’s given binomial (instead of Bernoulli) likelihood • Betas regressed on logit of proportion • Locally uniform priors on beta1 and beta2

  13. WinBUGS Model model{ for (i in 1:N){ r[i] ~ dbin(p[i], n[i]);logit(p[i]) <- (beta1 + beta2*(x[i] - mean(x[]))); r.hat[i] <- (p[i] * n[i]);} beta1 ~ dflat(); beta2 ~ dflat(); beta1nocenter <- beta1 - beta2*mean(x[]);}

  14. WinBUGS Output: Beta0 (1,0)

  15. WinBUGS Output: Beta0 (1,1)

  16. WinBUGS Output: Beta0 (1,-2)

  17. Comparison

  18. WinBUGS Wins • Uses proportions instead of Individual Y[i,j]’s • Convergence is Better • WinBUGS appears more precise (more trials needed) • Also, much faster.

  19. Resources • Groenewald, Pieter C.N., and Lucky Mokgatlhe. "Bayesian computation for logistic regression.“ Computational Statistics & Data Analysis 48 (2005): 857-68. Science Direct. Elsevier. Web. <http://www.sciencedirect.com/>. • Professor Cowles

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