Multivariate Statistical Modelling Based on Generalized Linear Models

by Fahrmeir, Ludwig; Tutz, Gerhard

Edition: 2nd

ISBN13: 9780387951874

ISBN10: 0387951873

Format: Hardcover

Pub. Date: 2001-04-01

Publisher(s): Springer Verlag

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Summary

The first edition of Multivariate Statistical Modelling provided an extension of classical models for regression, time series, and longitudinal data to a much broader class including categorical data and smoothing concepts. Generalized linear modesl for univariate and multivariate analysis build the central concept, which for the modelling of complex data is widened to much more general modelling approaches. The primary aim of the new edition is to bring the book up-to-date and to reflect the major new developments over the past years. The authors give a detailed introductory survey of the subject based on the alaysis of real data drawn from a variety of subjects, including the biological sciences, economics, and the social sciences. Technical details and proofs are deferred to an appendix in order to provide an accessible account for non-experts. The appendix serves as a reference or brief tutorial for the concepts of EM algorithm, numberical integration, MCMC and others. The topics covered inlude: Models for multi-categorial responses, model checking, semi- and nonparametric modelling, time series and longitudinal data, random effects models, state-space models, and survival analysis. In the new edition Bayesian concepts which are of growing importance in statistics are treated more extensively. The chapter on nonparametric and semiparametric generalized regression has been rewritten totally, random effects models now cover nonparametric maximum likelihood and fully Bayesian approaches, and state-space and hidden Markov models have been supplemented with an extension to models that can accommodate for spatial and spatiotemporal data. The authors have taken great pains to discuss the underlying theoretical ideas in ways that relate well to the data at hand. As a result, this book is ideally suited for applied statisticians, graduate students of statistics, and students and researchers with a strong interest in statistics and data analysis from econometrics, biometrics and the social sciences.

Author Biography

Gerhard Tutz is Full Professor of Statistics at the Department of Statistics, Ludwig Maximalians University, Munich, Germany.

Preface to the Second Edition

Preface to the First Edition

vii

List of Examples

xvii

List of Figures

xxi

List of Tables

xxv

Introduction

(14)

Outline and Examples

(11)

Remarks on Notation

(1)

Notes and Further Reading

(1)

Modelling and Analysis of Cross-Sectional Data: A Review of Univariate Generalized Linear Models

(54)

Univariate Generalized Linear Models

(22)

Data

(1)

Coding of Covariates

(1)

Grouped and Ungrouped Data

(1)

Definition of Univariate Generalized Linear Models

(4)

Models for Continuous Responses

(1)

Normal Distribution

(1)

Gamma Distribution

(1)

Inverse Gaussian Distribution

(1)

Models for Binary and Binomial Responses

(1)

Linear Probability Model

(1)

Probit Model

(1)

Logit Model

(1)

Complementary Log-Log Model

(1)

Complementary Log-Model

(3)

Binary Models as Threshold Models of Latent Linear Models

(1)

Parameter Interpretation

(6)

Overdispersion

(1)

Models for Count Data

(1)

Log-linear Poisson Model

(1)

Linear Poisson Model

(2)

Likelihood Inference

(17)

Maximum Likelihood Estimation

(1)

Log-likelihood, Score Function and Information Matrix

(2)

Numerical Computation of the MLE by Iterative Methods

(2)

Uniqueness and Existence of MLEs*

(1)

Asymptotic Properties

(2)

Discussion of Regularity Assumptions*

(1)

Additional Scale or Overdispersion Parameter

(1)

Hypothesis Testing and Goodness-of-Fit Statistics

(3)

Goodness-of-Fit Statistics

(5)

Some Extensions

(12)

Quasi-likelihood Models

(1)

Basic Models

(3)

Variance Functions with Unknown Parameters

(1)

Nonconstant Dispersion Parameter

(1)

Bayesian Models

(5)

Nonlinear and Nonexponential Family Regression Models*

(2)

Notes and Further Reading

(2)

Models for Multicategorical Responses: Multivariate Extensions of Generalized Linear Models

(70)

Multicategorical Response Models

(7)

Multinomial Distribution

(1)

Data

(1)

The Multivariate Model

(3)

Multivariate Generalized Linear Models

(2)

Models for Nominal Responses

(4)

The Principle of Maximum Random Utility

(2)

Modelling of Explanatory Variables: Choice of Design Matrix

(2)

Models for Ordinal Responses

(24)

Cumulative Models: The Threshold Approach

(1)

Cumulative Logistic Model or Proportional Odds Model

(3)

Grouped Cox Model or Proportional Hazards Model

(1)

Extreme Maximal-value Distribution Model

(1)

Extended Versions of Cumulative Models

(1)

Link Functions and Design Matrices for Cumulative Models

(4)

Sequential Models

(3)

Generalized Sequential Models

(3)

Link Functions of Sequential Models

(1)

Strict Stochastic Ordering*

(1)

Two-Step Models

100

(2)

Link Function and Design Matrix for Two-Step Models

102

(1)

Alternative Approaches

103

(2)

Statistical Inference

105

(7)

Maximum Likelihood Estimation

105

(2)

Numerical Computation

107

(1)

Testing and Goodness-of-Fit

107

(1)

Testing of Linear Hypotheses

107

(1)

Goodness-of-Fit Statistics

107

(2)

Power-Divergence Family*

109

(2)

Asymptotic Properties under Classical ``Fixed Cells'' Assumptions

111

(1)

Sparseness and ``Increasing-Cells'' Asymptotics

112

(1)

Multivariate Models for Correlated Responses

112

(24)

Conditional Models

114

(1)

Asymmetric Models

114

(2)

Symmetric Models

116

(3)

Marginal Models

119

(1)

Marginal Models for Correlated Univariate Responses

120

(3)

The Generalized Estimating Approach for Statistical Inference

123

(6)

Marginal Models for Correlated Categorical Responses

129

(6)

Likelihood-based Inference for Marginal Models

135

(1)

Notes and Further Reading

136

(3)

Bayesian Inference

136

(3)

Selecting and Checking Models

139

(34)

Variable Selection

139

(6)

Selection Criteria

140

(2)

Selection Procedures

142

(1)

All-Subsets Selection

142

(1)

Stepwise Backward and Forward Selection

143

(2)

Diagnostics

145

(16)

Diagnostic Tools for the Classical Linear Model

146

(1)

Generalized Hat Matrix

147

(4)

Residuals and Goodness-of-Fit Statistics

151

(5)

Case Deletion

156

(5)

General Tests for Misspecification*

161

(9)

Estimation under Model Misspecification

162

(3)

Hausman-type Tests

165

(1)

Hausman Tests

165

(1)

Information Matrix Test

166

(1)

Tests for Nonnested Hypotheses

167

(1)

Tests Based on Artificial Nesting

168

(1)

Generalized Wald and Score Tests

168

(2)

Notes and Further Reading

170

(3)

Bayesian Model Determination

170

(2)

Robust Estimates

172

(1)

Model Tests Against Smooth Alternatives

172

(1)

Semi- and Nonparametric Approaches to Regression Analysis

173

(68)

Smoothing Techniques for Continuous Responses

174

(19)

Regression Splines and Other Basis Functions

174

(2)

Regression Splines

176

(2)

Other Basis Functions

178

(1)

Regularization

179

(2)

Smoothing Splines

181

(2)

Local Estimators

183

(1)

Simple Neighborhood Smoothers

183

(1)

Local Regression

184

(3)

Bias-Variance Trade-off

187

(2)

Relation to Other Smoothers

189

(1)

Selection of Smoothing Parameters

190

(3)

Smoothing for Non-Gaussian Data

193

(9)

Basis Function Approach

193

(1)

Fisher Scoring for Penalized Likelihood*

194

(1)

Penalization and Spline Smoothing

195

(1)

Fisher Scoring for Generalized Spline Smoothing*

196

(1)

Choice of Smoothing Parameter

197

(1)

Localizing Generalized Linear Models

198

(3)

Local Fitting by Weighted Scoring

201

(1)

Modelling with Multiple Covariates

202

(19)

Modelling Approaches

207

(1)

Generalized Additive Models

207

(1)

Partially Linear Models

208

(1)

Varying-Coefficient Models

208

(1)

Projection Pursuit Regression

209

(1)

Basis Function Approach

210

(3)

Estimation Concepts

213

(1)

Backfitting Algorithm for Generalized Additive Models

213

(4)

Backfitting with Spline Functions

217

(3)

Choice of Smoothing Parameter

220

(1)

Partial Linear Models

220

(1)

Semiparametric Bayesian Inference for Generalized Regression

221

(18)

Gaussian Responses

221

(1)

Smoothness Priors Approaches

221

(6)

Basis Function Approaches

227

(1)

Models with Multiple Covariates

228

(3)

Non-Gaussian Responses

231

(3)

Latent Variable Models for Categorical Responses

234

(5)

Notes and Further Reading

239

(2)

Fixed Parameter Models for Time Series and Longitudinal Data

241

(42)

Time Series

242

(18)

Conditional Models

242

(1)

Generalized Autoregressive Models

242

(4)

Quasi-Likelihood Models and Generalized Autoregression Moving Average Models

246

(3)

Statistical Inference for Conditional Models

249

(6)

Marginal Models

255

(3)

Estimation of Marginal Models

258

(2)

Longitudinal Data

260

(18)

Conditional Models

261

(1)

Generalized Autoregressive Models, Quasi-Likelihood Models

261

(1)

Statistical Inference

262

(2)

Transition Models

264

(1)

Subject-specific Approaches and Conditional Likelihood

264

(3)

Marginal Models

267

(1)

Statistical Inference

268

(6)

Generalized Additive Models for Longitudinal Data

274

(4)

Notes and Further Reading

278

(5)

Random Effects Models

283

(48)

Linear Random Effects Models for Normal Data

285

(7)

Two-stage Random Effects Models

285

(1)

Random Intercepts

286

(1)

Random Slopes

287

(1)

Multilevel Models

288

(1)

Statistical Inference

289

(1)

Known Variance-Covariance Components

289

(1)

Unknown Variance-Covariance Components

289

(2)

Derivation of the EM algorithm*

291

(1)

Random Effects in Generalized Linear Models

292

(6)

Generalized Linear Models with Random Effects

293

(1)

Examples

294

(4)

Estimation Based on Posterior Modes

298

(5)

Known Variance-Covariance Components

298

(1)

Unknown Variance-Covariance Components

299

(1)

Algorithmic Details*

300

(1)

Fisher Scoring for Given Variance-Covariance Components

300

(2)

EM Type Algorithm

302

(1)

Estimation by Integration Techniques

303

(15)

Maximum Likelihood Estimation of Fixed Parameters

303

(2)

Direct Maximization Using Fitting Techniques for GLMs

305

(3)

Nonparametric Maximum Likelihood for Finite Mixtures

308

(2)

Posterior Mean Estimation of Random Effects

310

(1)

Indirect Maximization Based on the EM Algorithm*

311

(4)

Algorithmic Details for Posterior Mean Estimation*

315

(3)

Examples

318

(3)

Bayesian Mixed Models

321

(4)

Bayesian Generalized Mixed Models

321

(1)

Generalized Additive Mixed Models

322

(3)

Marginal Estimation Approach to Random Effects Models

325

(3)

Notes and Further Reading

328

(3)

State Space and Hidden Markov Models

331

(54)

Linear State Space Models and the Kalman Filter

332

(13)

Linear State Space Models

332

(5)

Statistical Inference

337

(1)

Linear Kalman Filtering and Smoothing

338

(2)

Kalman Filtering and Smoothing as Posterior Mode Estimation*

340

(2)

Unknown Hyperparameters

342

(1)

EM Algorithm for Estimating Hyperparameters*

343

(2)

Non-Normal and Nonlinear State Space Models

345

(5)

Dynamic Generalized Linear Models

345

(2)

Categorical Time Series

347

(2)

Nonlinear and Nonexponential Family Models*

349

(1)

Non-Normal Filtering and Smoothing

350

(19)

Posterior Mode Estimation

351

(1)

Generalized Extended Kalman Filter and Smoother*

352

(2)

Gauss-Newton and Fisher-Scoring Filtering and Smoothing*

354

(2)

Estimation of Hyperparameters*

356

(1)

Some Applications

356

(5)

Markov Chain Monte Carlo and Integration-based Approaches

361

(1)

MCMC Inference

362

(3)

Integration-based Approaches

365

(4)

Longitudinal Data

369

(7)

State Space Modelling of Longitudinal Data

369

(3)

Inference For Dynamic Generalized Linear Mixed Models

372

(4)

Spatial and Spatio-temporal Data

376

(7)

Notes and Further Reading

383

(2)

Survival Models

385

(48)

Models for Continuous Time

385

(11)

Basic Models

385

(1)

Exponential Distribution

386

(1)

Weibull Distribution

387

(1)

Piecewise Exponential Model

388

(1)

Parametric Regression Models

388

(1)

Location-Scale Models for log T

388

(1)

Proportional Hazards Models

389

(1)

Linear Transformation Models and Binary Regression Models

390

(1)

Censoring

391

(1)

Random Censoring

391

(1)

Type I Censoring

392

(1)

Estimation

393

(1)

Exponential Model

394

(1)

Weibull Model

394

(1)

Piecewise Exponential Model

395

(1)

Models for Discrete Time

396

(18)

Life Table Estimates

397

(3)

Parametric Regression Models

400

(1)

The Grouped Proportional Hazards Model

400

(2)

A Generalized Version: The Model of Aranda-Ordaz

402

(1)

The Logistic Model

403

(1)

Sequential Model and Parameterization of the Baseline Hazard

403

(1)

Maximum Likelihood Estimation

404

(4)

Time-varying Covariates

408

(3)

Internal Covariates*

411

(1)

Maximum Likelihood Estimation*

412

(2)

Discrete Models for Multiple Modes of Failure

414

(6)

Basic Models

414

(3)

Maximum Likelihood Estimation

417

(3)

Smoothing in Discrete Survival Analysis

420

(9)

Smoothing Life Table Estimates

420

(2)

Smoothing with Covariates

422

(1)

Dynamic Discrete-Time Survival Models

423

(1)

Posterior Mode Smoothing

423

(2)

Fully Bayesian Inference via MCMC

425

(4)

Remarks and Further Reading

429

(4)

433

(22)

Exponential Families and Generalized Linear Models

433

(4)

Basic Ideas for Asymptotics

437

(5)

EM Algorithm

442

(1)

Numerical Integration

443

(6)

Monte Carlo Methods

449

(6)

B. Software for Fitting Generalized Linear Models and Extensions

455

(12)

Bibliography

467

(38)

Author Index

505

(7)

Subject Index

512

Multivariate Statistical Modelling Based on Generalized Linear Models

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