Generalized Linear Models: A Bayesian Perspective

by Dey; Dipak K.

• ISBN13: 9780824790349
• ISBN10: 0824790340
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-05-25
• Publisher: CRC Press

Summary

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.

Table of Contents

I General Overview	1	(54)
Generalized Linear Models: A Bayesian View	3	(20)
A. Gelfand
M. Ghosh
Introduction	3	(1)
GLMs and Bayesian Models	4	(4)
GLMs	4	(1)
Bayesian Models	5	(3)
Propriety of Posteriors	8	(2)
Semiparametric GLMs	10	(2)
Overdispersed Generalized Linear Models	12	(2)
Model Determination Approaches	14	(9)
Random Effects in Generalized Linear Mixed Models (GLMMs)	23	(18)
D. Sun
P. L. Speckman
R. K. Tsutakawa
Introduction	23	(1)
The Model	24	(2)
Random Effects	26	(5)
Independent Random Effects	26	(1)
Correlated Random Effects	26	(3)
Strongly Correlated Random Effects	29	(2)
Some Examples of the AR(d) Model	31	(1)
Hierarchical GLMMs	31	(5)
Bayesian Computation	36	(5)
Prior Elicitation and Variables Selection for Generalized Linear Mixed Models	41	(14)
J. Ibrahim
M. H. Chen
Introduction	41	(2)
Generalized Linear Mixed Models	43	(5)
Models	43	(1)
The Prior Distributions	44	(2)
Propriety of the Prior Distribution	46	(1)
Specifying the Hyperparameters	47	(1)
The Posterior Distribution and its Computation	48	(1)
Bayesian Variables Selection	48	(3)
Pediatric Pain Data	51	(1)
Discussion	52	(3)
II Extending the GLMs	55	(56)
Dynamic Generalized Linear Models	57	(16)
M. A. R. Ferreira
D. Gamerman
Introduction	57	(1)
Dynamic linear models	58	(1)
Definition and first approaches to inference	59	(3)
Linear Bayes Approach	60	(1)
Piecewise Linear Approximation	61	(1)
Posterior Mode Estimation	61	(1)
Other Approaches and Models	62	(1)
MCMC-based Approaches	62	(3)
Gibbs Sampling	63	(1)
Metropolis-Hasting Algorithm	64	(1)
Applications	65	(5)
Application 1: Meningococcic Meningitis	66	(2)
Application 2: Respiratory Diseases and Level of Pollutants	68	(2)
Discussions and Extensions	70	(3)
Bayesian Approaches for Overdispersion in Generalized Linear Models	73	(16)
D. K. Dey
N. Ravishanker
Introduction	73	(2)
Classes of Overdispersed General Linear Models	75	(3)
Fitting OGLM in the Parametric Bayesian Framework	78	(3)
Model Fitting	78	(1)
Example: Overdispersed Poisson Regression Model	79	(1)
Model Determination for Parametric OGLM's	80	(1)
Modeling Overdispersion in the Nonparametric Bayesian Framework	81	(3)
Fitting DP Mixed GLM and OGLM	81	(2)
Example: Overdispersed Binomial Regression Model	83	(1)
Model Determination for Dirichlet Process Mixed Models	83	(1)
Overdispersion in Multistage GLM	84	(5)
Bayesian Generalized Linear Models for Inference About Small Areas	89	(22)
B. Nandram
Introduction	89	(2)
Logistic Regression Models	91	(3)
Poisson Regression Models	94	(2)
Computational Issues	96	(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