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A (poor) Gibbs Sampling Approach to Logistic Regression
Kyle BogdanGrant Brown
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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.
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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
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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
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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
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Initial Problems
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Sampler Output/Diagostics
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Sampler Output/Diagnostics
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Sampler Output/Diagnostics
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Sampler Output/Diagnostics
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Sampler Output/Diagnostics
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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
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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[]);}
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WinBUGS Output: Beta0 (1,0)
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WinBUGS Output: Beta0 (1,1)
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WinBUGS Output: Beta0 (1,-2)
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Comparison
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WinBUGS Wins
Uses proportions instead of Individual Y[i,j]’s
Convergence is Better WinBUGS appears more precise
(more trials needed) Also, much faster.
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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