Generalized Linear Models: A Bayesian Perspective

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ISBN: 9780824790349 | 0824790340
Cover: Hardcover
Copyright: 5/25/2000

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This volume describes how to conceptualize, perform, and critique traditional generalized linear models (GLMs) from a Bayesian perspective and how to use modern computational methods to summarize inferences using simulation. Introducing dynamic modeling for GLMs and containing over 1000 references and equations, Generalized Linear Models considers parametric and semiparametric approaches to overdispersed GLMs, presents methods of analyzing correlated binary data using latent variables. It also proposes a semiparametric method to model link functions for binary response data, and identifies areas of important future research and new applications of GLMs.

I General Overview

(54)

Generalized Linear Models: A Bayesian View

(20)

A. Gelfand

M. Ghosh

Introduction

(1)

GLMs and Bayesian Models

(4)

GLMs

(1)

Bayesian Models

(3)

Propriety of Posteriors

(2)

Semiparametric GLMs

(2)

Overdispersed Generalized Linear Models

(2)

Model Determination Approaches

(9)

Random Effects in Generalized Linear Mixed Models (GLMMs)

(18)

D. Sun

P. L. Speckman

R. K. Tsutakawa

Introduction

(1)

The Model

(2)

Random Effects

(5)

Independent Random Effects

(1)

Correlated Random Effects

(3)

Strongly Correlated Random Effects

(2)

Some Examples of the AR(d) Model

(1)

Hierarchical GLMMs

(5)

Bayesian Computation

(5)

Prior Elicitation and Variables Selection for Generalized Linear Mixed Models

(14)

J. Ibrahim

M. H. Chen

Introduction

(2)

Generalized Linear Mixed Models

(5)

Models

(1)

The Prior Distributions

(2)

Propriety of the Prior Distribution

(1)

Specifying the Hyperparameters

(1)

The Posterior Distribution and its Computation

(1)

Bayesian Variables Selection

(3)

Pediatric Pain Data

(1)

Discussion

(3)

II Extending the GLMs

(56)

Dynamic Generalized Linear Models

(16)

M. A. R. Ferreira

D. Gamerman

Introduction

(1)

Dynamic linear models

(1)

Definition and first approaches to inference

(3)

Linear Bayes Approach

(1)

Piecewise Linear Approximation

(1)

Posterior Mode Estimation

(1)

Other Approaches and Models

(1)

MCMC-based Approaches

(3)

Gibbs Sampling

(1)

Metropolis-Hasting Algorithm

(1)

Applications

(5)

Application 1: Meningococcic Meningitis

(2)

Application 2: Respiratory Diseases and Level of Pollutants

(2)

Discussions and Extensions

(3)

Bayesian Approaches for Overdispersion in Generalized Linear Models

(16)

D. K. Dey

N. Ravishanker

Introduction

(2)

Classes of Overdispersed General Linear Models

(3)

Fitting OGLM in the Parametric Bayesian Framework

(3)

Model Fitting

(1)

Example: Overdispersed Poisson Regression Model

(1)

Model Determination for Parametric OGLM's

(1)

Modeling Overdispersion in the Nonparametric Bayesian Framework

(3)

Fitting DP Mixed GLM and OGLM

(2)

Example: Overdispersed Binomial Regression Model

(1)

Model Determination for Dirichlet Process Mixed Models

(1)

Overdispersion in Multistage GLM

(5)

Bayesian Generalized Linear Models for Inference About Small Areas

(22)

B. Nandram

Introduction

(2)

Logistic Regression Models

(3)

Poisson Regression Models

(2)

Computational Issues

(4)

Models of the U.S. Mortality Data

100

(2)

Challenges in Small Area Estimation

102

(2)

Concluding Remarks

104

(7)

III Categorical and Longitudinal Data

111

(104)

Bayesian Methods for Correlated Binary Data

113

(20)

S. Chib

Introduction

113

(1)

The Multivariate Probit Model

114

(5)

Dependence Structures

116

(1)

Student-t Specification

116

(1)

Estimation of the MVP Model

117

(1)

Fitting of the Multivariate t-link Model

118

(1)

Longitudinal Binary Data

119

(5)

Probit (or logit) Normal Model

119

(1)

Inference

120

(1)

Computations for the Probit-Normal Model

120

(2)

Binary Response Hierarchical Model

122

(1)

Other Models

123

(1)

Comparison of Alternative Models

124

(3)

Likelihood Ordinate

124

(1)

Posterior Ordinate

125

(2)

Concluding Remarks

127

(1)

Appendix

127

(6)

Algorithm 1

127

(1)

Algorithm 2

128

(1)

Algorithm 3

129

(4)

Bayesian Analysis for Correlated Ordinal Data Models

133

(26)

M. H. Chen

D. K. Dey

Introduction

133

(2)

Models

135

(3)

Prior Distributions and Posterior Computations

138

(4)

Prior Distributions

138

(1)

Posterior Computations

138

(4)

Model Determination

142

(6)

Model Comparisons

143

(3)

Model Diagnostics

146

(2)

Item Response Data Example

148

(7)

Concluding Remarks

155

(4)

Bayesian Methods for Time Series Count Data

159

(14)

J. Ibrahim

M. H. Chen

Introduction

159

(1)

The Method

160

(5)

The Likelihood Function

160

(2)

The Prior Distributions

162

(2)

Specifying the Hyperparameters

164

(1)

Prior Distributions on the Model Space

164

(1)

Computation of Model Probabilities

165

(2)

Example: Pollen Data

167

(1)

Discussion

168

(5)

Item Response Modeling

173

(22)

J. Albert

M. Ghosh

Introduction

173

(2)

An Item Response Curve

175

(1)

Administering an Exam to a Group of Students

176

(2)

Prior Distributions

178

(3)

Noninformative Priors and Propriety of the Posterior Distribution

178

(2)

Choosing an Informative Prior

180

(1)

Bayesian Fitting of Item Response Models

181

(4)

Fitting of the Two-parameter Model Using Gibbs Sampling

181

(2)

Implementation of Gibbs Sampling for General F

183

(1)

Gibbs Sampling for a Probit Link Using Data Augmentation

184

(1)

Bayesian Fitting of the One-parameter Model

185

(1)

Inferences from the Model

185

(1)

Model Checking

186

(2)

Bayesian Residuals

186

(1)

Posterior Predictive Checks

187

(1)

The Mathematical Placement Test Example

188

(3)

Further Reading

191

(4)

Developing and Applying Medical Practice Guidelines Following Acute Myocardial Infarction: A Case Study Using Bayesian Probit and Logit Models

195

(20)

M. B. Landrum

S. Normand

Background and Significance

195

(2)

Developing of Appropriateness Ratings

197

(3)

Elicitation of Appropriateness Ratings

197

(1)

Combining the Angiography Panel Data

198

(1)

Estimation

199

(1)

Defining the Standard of Care

199

(1)

Results

200

(1)

Applying the Practice Guidelines

200

(9)

Study Population

200

(2)

Modeling Adherence to Practice Guidelines

202

(1)

Estimation

203

(1)

Profiling Hospitals

203

(1)

Explaining Variability in Quality of Care

204

(1)

Results

205

(4)

Discussion

209

(6)

IV Semiparametric Approaches

215

(56)

Semiparametric Generalized Linear Models: Bayesian Approaches

217

(14)

N. L. Mallick

D. G. T. Denison

A.F. M. Smith

Introduction

217

(1)

Modeling the Link Function g

218

(1)

Binary Response Regression

218

(1)

General Regression

219

(1)

Modeling the Systematic Part η

219

(1)

Model with Random Effects

220

(1)

Model with Deterministic Error

220

(1)

Models Using Curves and Surfaces

220

(1)

GLMs using Bayesian MARS

221

(3)

Classical MARS

221

(1)

Bayesian MARS

222

(2)

Bayesian MARS for GLMs

224

(1)

Examples of Bayesian MARS for GLMs

224

(7)

Motivating Examples

224

(1)

Pima Indian Example

225

(6)

Binary Response Regression with Normal Scale Mixture Links

231

(12)

S. Basu

S. Mukhopadhyay

Introduction

231

(2)

The Finite Mixture Model

233

(1)

General Mixtures and a Dirichlet Process Prior

234

(2)

Model Diagnostic

236

(1)

Basic Goal

236

(1)

Diagnostic Tools

237

(1)

Computational Methods

237

(1)

Application: Student Retention at the University of Arkansas

237

(2)

Discussion

239

(4)

Binary Regression Using Data Adaptive Robust Link Functions

243

(12)

R. Haro-Lopez

B. K. Mallick

A. F. M. Smith

Introduction

243

(1)

The Binary Regression Model

244

(4)

Detection of Outliers and Model Comparison

248

(1)

Numerical Illustration

248

(2)

Discussion

250

(5)

A Mixture-Model Approaches to the Analysis of Survival Data

255

(16)

L. Kuo

F. Peng

Introduction

255

(2)

Likelihood

257

(1)

EM and Monte Carlo EM

257

(2)

Gibbs Sampler

259

(1)

Model Selection

260

(1)

Example

261

(10)

EM Algorithm for the Specific Example

262

(2)

Gibbs Samplers for the Specific Example

264

(1)

Numerical Results

265

(6)

V Model Diagnostics and Variable Selection in GLMs

271

(58)

Bayesian Variable Selection Using the Gibbs Sampler

273

(14)

P. Dellaportas

J. J. Forster

I. Ntzoufras

Introduction

273

(1)

Gibbs Sampler Based Variable Selection Strategies

274

(4)

Carlin and Chib's Method

275

(1)

Stochastic Search Variable Selection

276

(1)

Unconditional Priors for Variable Selection

277

(1)

Gibbs Variable Selection

277

(1)

Summary of Variable Selection Strategies

278

(1)

Illustrative Example: 2 x 2 x 2 Contingency Table

278

(3)

Log-Linear models

280

(1)

Logistic Regression Models

280

(1)

Discussion

281

(1)

Appendix: Bugs Codes

282

(5)

Code for Log-linear Models for 23 Contingency Table

282

(1)

Code for Logistic Models with 2 Binary Explanatory Factors

283

(4)

Bayesian Methods for Variables Selection in the Cox Model

287

(26)

J. Ibrahim

M. H. Chen

Introduction

287

(2)

The Method

289

(10)

Model and Notation

289

(1)

Prior Distribution for hb(.)

289

(3)

The Likelihood Function

292

(1)

Prior Distribution for the Regression Coefficients

293

(4)

Prior Distribution on the Model Space

297

(2)

Computational Implementation

299

(6)

Computing the Marginal Distribution of the Data

299

(3)

Sampling from the Posterior Distribution of (β(m), Δ)

302

(3)

Example: Simulation Study

305

(4)

Discussion

309

(4)

Bayesian Model Diagnostics for Correlated Binary Data

313

(16)

D. K. Dey

M. H. Chen

Introduction

313

(1)

The Models

314

(2)

Stratified and Mixtures Models

314

(1)

Conditional Models

314

(1)

Multivariate Probit Models

315

(1)

Multivariate t-Link Models

315

(1)

The Prior Distributions and Posterior Computations

316

(4)

Prior Distributions

316

(1)

Posterior Computations

317

(3)

Model Adequacy for Correlated Binary Data

320

(4)

Voter Behavior Data example

324

(1)

Concluding Remarks

325

(4)

VI Challenging Approaches in GLMs

329

(78)

Bayesian Errors-in-Variables Modeling

331

(18)

J. Wakefield

D. Stephens

Introduction

331

(2)

Illustrative Example: Case-control Study with Deprivation

333

(4)

Classical approaches

337

(2)

Basic Formulation

337

(2)

Modeling and Analysis: Classical Extensional and Procedures

339

(1)

Bayesian Approaches

339

(1)

General Framework

339

(1)

Implementation

340

(1)

Previous Work

340

(1)

Example revisited

340

(2)

Conclusions and Discussion

342

(7)

Bayesian Analysis of Compositional Data

349

(16)

M. Iyengar

D. K. Dey

Introduction

349

(2)

A Parametric Approach

351

(1)

Simulation Based Model Determination

352

(2)

A Semiparametric Approach

354

(1)

Posterior Distributions and Estimation

355

(4)

Results

359

(2)

Conclusions

361

(4)

Classification Trees

365

(8)

D. G. T. Denison

B. K. Mallick

Introduction

365

(1)

The Classification Tree Model

366

(3)

The Basis Functions

366

(1)

The Classical Approach

367

(1)

The Bayesian Approach

368

(1)

Real data example

369

(2)

Discussion

371

(2)

Modeling and Inference for Point-Referenced Binary Spatial Data

373

(14)

A. E. Gelfand

N. Ravishanker

M. Ecker

Introduction

373

(2)

Modeling Details

375

(3)

Computational Issues

378

(2)

An Illustration

380

(4)

Related Remarks

384

(3)

Bayesian Graphical Models and Software for GLMs

387

(20)

N. Best

A. Thomas

Bayesian Graphical Models and Conditional Independence Structures

387

(2)

Computation of Bayesian Graphical Models

388

(1)

Constructing Software from Graphical Models

389

(1)

Implementing GLM's Using WinBUGS

389

(2)

GLMs with Non-canonical Links

391

(1)

Generalized Linear Mixed Models (GLMMs)

391

(3)

Exchangeable Random Effects

392

(1)

Correlated Random Effects

392

(2)

Polytomous Responses

394

(2)

Ordered Categories

395

(1)

Adding Complexity in GLMs/GLMMs

396

(2)

Missing Data

396

(1)

Informative Missing Data

396

(1)

Prediction

397

(1)

Covariate Measurement Error

397

(1)

General Advice on Modeling Using WinBUGSs

398

(4)

Parameterization

398

(2)

Prior Specification

400

(1)

Convergence and posterior Sample Size

401

(1)

Model Checking

402

(1)

Extending the WinBUGS Software

402

(5)

Index

407

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Generalized Linear Models: A Bayesian Perspective

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