Multivariate Statistical Modelling Based on Generalized Linear Models

by Fahrmeir, Ludwig; Tutz, Gerhard

• ISBN13: 9780387951874
• ISBN10: 0387951873
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-04-01
• Publisher: Springer Verlag

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.

Table of Contents

Preface to the Second Edition	v
Preface to the First Edition	vii
List of Examples	xvii
List of Figures	xxi
List of Tables	xxv
Introduction	1	(14)
Outline and Examples	2	(11)
Remarks on Notation	13	(1)
Notes and Further Reading	14	(1)
Modelling and Analysis of Cross-Sectional Data: A Review of Univariate Generalized Linear Models	15	(54)
Univariate Generalized Linear Models	16	(22)
Data	16	(1)
Coding of Covariates	16	(1)
Grouped and Ungrouped Data	17	(1)
Definition of Univariate Generalized Linear Models	18	(4)
Models for Continuous Responses	22	(1)
Normal Distribution	22	(1)
Gamma Distribution	23	(1)
Inverse Gaussian Distribution	24	(1)
Models for Binary and Binomial Responses	24	(1)
Linear Probability Model	25	(1)
Probit Model	26	(1)
Logit Model	26	(1)
Complementary Log-Log Model	26	(1)
Complementary Log-Model	26	(3)
Binary Models as Threshold Models of Latent Linear Models	29	(1)
Parameter Interpretation	29	(6)
Overdispersion	35	(1)
Models for Count Data	36	(1)
Log-linear Poisson Model	36	(1)
Linear Poisson Model	36	(2)
Likelihood Inference	38	(17)
Maximum Likelihood Estimation	38	(1)
Log-likelihood, Score Function and Information Matrix	39	(2)
Numerical Computation of the MLE by Iterative Methods	41	(2)
Uniqueness and Existence of MLEs*	43	(1)
Asymptotic Properties	44	(2)
Discussion of Regularity Assumptions*	46	(1)
Additional Scale or Overdispersion Parameter	47	(1)
Hypothesis Testing and Goodness-of-Fit Statistics	47	(3)
Goodness-of-Fit Statistics	50	(5)
Some Extensions	55	(12)
Quasi-likelihood Models	55	(1)
Basic Models	55	(3)
Variance Functions with Unknown Parameters	58	(1)
Nonconstant Dispersion Parameter	59	(1)
Bayesian Models	60	(5)
Nonlinear and Nonexponential Family Regression Models*	65	(2)
Notes and Further Reading	67	(2)
Models for Multicategorical Responses: Multivariate Extensions of Generalized Linear Models	69	(70)
Multicategorical Response Models	70	(7)
Multinomial Distribution	70	(1)
Data	71	(1)
The Multivariate Model	72	(3)
Multivariate Generalized Linear Models	75	(2)
Models for Nominal Responses	77	(4)
The Principle of Maximum Random Utility	77	(2)
Modelling of Explanatory Variables: Choice of Design Matrix	79	(2)
Models for Ordinal Responses	81	(24)
Cumulative Models: The Threshold Approach	83	(1)
Cumulative Logistic Model or Proportional Odds Model	83	(3)
Grouped Cox Model or Proportional Hazards Model	86	(1)
Extreme Maximal-value Distribution Model	86	(1)
Extended Versions of Cumulative Models	87	(1)
Link Functions and Design Matrices for Cumulative Models	88	(4)
Sequential Models	92	(3)
Generalized Sequential Models	95	(3)
Link Functions of Sequential Models	98	(1)
Strict Stochastic Ordering*	99	(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)
A	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