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Generalized, Linear, and Mixed Models, 2nd Edition

ISBN: 978-0-470-07371-1
424 pages
June 2008
Generalized, Linear, and Mixed Models, 2nd Edition (0470073713) cover image

Description

An accessible and self-contained introduction to statistical models-now in a modernized new edition

Generalized, Linear, and Mixed Models, Second Edition provides an up-to-date treatment of the essential techniques for developing and applying a wide variety of statistical models. The book presents thorough and unified coverage of the theory behind generalized, linear, and mixed models and highlights their similarities and differences in various construction, application, and computational aspects.

A clear introduction to the basic ideas of fixed effects models, random effects models, and mixed models is maintained throughout, and each chapter illustrates how these models are applicable in a wide array of contexts. In addition, a discussion of general methods for the analysis of such models is presented with an emphasis on the method of maximum likelihood for the estimation of parameters. The authors also provide comprehensive coverage of the latest statistical models for correlated, non-normally distributed data. Thoroughly updated to reflect the latest developments in the field, the Second Edition features:
  • A new chapter that covers omitted covariates, incorrect random effects distribution, correlation of covariates and random effects, and robust variance estimation
  • A new chapter that treats shared random effects models, latent class models, and properties of models
  • A revised chapter on longitudinal data, which now includes a discussion of generalized linear models, modern advances in longitudinal data analysis, and the use between and within covariate decompositions
  • Expanded coverage of marginal versus conditional models
  • Numerous new and updated examples

With its accessible style and wealth of illustrative exercises, Generalized, Linear, and Mixed Models, Second Edition is an ideal book for courses on generalized linear and mixed models at the upper-undergraduate and beginning-graduate levels. It also serves as a valuable reference for applied statisticians, industrial practitioners, and researchers.
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Table of Contents

Preface.

Preface to the First Edition.

1. Introduction.

1.1 Models.

1.2 Factors, Levels, Cells, Effects And Data.

1.3 Fixed Effects Models.

1.4 Random Effects Models.

1.5 Linear Mixed Models (Lmms).

1.6 Fixed Or Random?

1.7 Inference.

1.8 Computer Software.

1.9 Exercises.

2. One-Way Classifications.

2.1 Normality And Fixed Effects.

2.2 Normality, Random Effects And MLE.

2.3 Normality, Random Effects And REM1.

2.4 More On Random Effects And Normality.

2.5 Binary Data: Fixed Effects.

2.6 Binary Data: Random Effects.

2.7 Computing.

2.8 Exercises.

3. Single-Predictor Regression.

3.1 Introduction.

3.2 Normality: Simple Linear Regression.

3.3 Normality: A Nonlinear Model.

3.4 Transforming Versus Linking.

3.5 Random Intercepts: Balanced Data.

3.6 Random Intercepts: Unbalanced Data.

3.7 Bernoulli - Logistic Regression.

3.8 Bernoulli - Logistic With Random Intercepts.

3.9 Exercises.

4. Linear Models (LMs).

4.1 A General Model.

4.2 A Linear Model For Fixed Effects.

4.3 Mle Under Normality.

4.4 Sufficient Statistics.

4.5 Many Apparent Estimators.

4.6 Estimable Functions.

4.7 A Numerical Example.

4.8 Estimating Residual Variance.

4.9 Comments On The 1- And 2-Way Classifications.

4.10 Testing Linear Hypotheses.

4.11 T-Tests And Confidence Intervals.

4.12 Unique Estimation Using Restrictions.

4.13 Exercises.

5. Generalized Linear Models (GLMs).

5.1 Introduction.

5.2 Structure Of The Model.

5.3 Transforming Versus Linking.

5.4 Estimation By Maximum Likelihood.

5.5 Tests Of Hypotheses.

5.6 Maximum Quasi-Likelihood.

5.7 Exercises.

6. Linear Mixed Models (LMMs).

6.1 A General Model.

6.2 Attributing Structure To VAR(y).

6.3 Estimating Fixed Effects For V Known.

6.4 Estimating Fixed Effects For V Unknown.

6.5 Predicting Random Effects For V Known.

6.6 Predicting Random Effects For V Unknown.

6.7 Anova Estimation Of Variance Components.

6.8 Maximum Likelihood (Ml) Estimation.

6.9 Restricted Maximum Likelihood (REMl).

6.10 Notes And Extensions.

6.11 Appendix For Chapter 6.

6.12 Exercises.

7. Generalized Linear Mixed Models.

7.1 Introduction.

7.2 Structure Of The Model.

7.3 Consequences Of Having Random Effects.

7.4 Estimation By Maximum Likelihood.

7.5 Other Methods Of Estimation.

7.6 Tests Of Hypotheses.

7.7 Illustration: Chestnut Leaf Blight.

7.8 Exercises.

8. Models for Longitudinal data.

8.1 Introduction.

8.2 A Model For Balanced Data.

8.3 A Mixed Model Approach.

8.4 Random Intercept And Slope Models.

8.5 Predicting Random Effects.

8.6 Estimating Parameters.

8.7 Unbalanced Data.

8.8 Models For Non-Normal Responses.

8.9 A Summary Of Results.

8.10 Appendix.

8.11 Exercises.

9. Marginal Models.

9.1 Introduction.

9.2 Examples Of Marginal Regression Models.

9.3 Generalized Estimating Equations.

9.4 Contrasting Marginal And Conditional Models.

9.5 Exercises.

10. Multivariate Models.

10.1 Introduction.

10.2 Multivariate Normal Outcomes.

10.3 Non-Normally Distributed Outcomes.

10.4 Correlated Random Effects.

10.5 Likelihood Based Analysis.

10.6 Example: Osteoarthritis Initiative.

10.7 Notes And Extensions.

10.8 Exercises.

11. Nonlinear Models.

11.1 Introduction.

11.2 Example: Corn Photosynthesis.

11.3 Pharmacokinetic Models.

11.4 Computations For Nonlinear Mixed Models.

11.5 Exercises.

12. Departures From Assumptions.

12.1 Introduction.

12.2 Misspecifications Of Conditional Model For Response.

12.3 Misspecifications Of Random Effects Distribution.

12.4 Methods To Diagnose And Correct For Misspecifications.

12.5 Exercises.

13. Prediction.

13.1 Introduction.

13.2 Best Prediction (BP).

13.3 Best Linear Prediction (BLP).

13.4 Linear Mixed Model Prediction (BLUP).

13.5 Required Assumptions.

13.6 Estimated Best Prediction.

13.7 Henderson’s Mixed Model Equations.

13.8 Appendix.

13.9 Exercises.

14. Computing.

14.1 Introduction.

14.2 Computing Ml Estimates For LMMs.

14.3 Computing Ml Estimates For GLMMs.

14.4 Penalized Quasi-Likelihood And Laplace.

14.5 Exercises.

Appendix M: Some Matrix Results.

M.1 Vectors And Matrices Of Ones.

M.2 Kronecker (Or Direct) Products.

M.3 A Matrix Notation.

M.4 Generalized Inverses.

M.5 Differential Calculus.

Appendix S: Some Statistical Results.

S.1 Moments.

S.2 Normal Distributions.

S.3 Exponential Families.

S.4 Maximum Likelihood.

S.5 Likelihood Ratio Tests.

S.6 MLE Under Normality.

References.

Index.

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Author Information

Charles E. McCulloch, PhD, is Professor and Head of the Division of Biostatistics in the School of Medicine at the University of California, San Francisco. A Fellow of the American Statistical Association, Dr. McCulloch is the author of numerous published articles in the areas of longitudinal data analysis, generalized linear mixed models, and latent class models and their applications.

Shayle R. Searle, PhD, is Professor Emeritus in the Department of Biological Statistics and Computational Biology at Cornell University. Dr. Searle is the author of Linear Models, Linear Models for Unbalanced Data, Matrix Algebra Useful for Statistics, and Variance Components, all published by Wiley.

John M. Neuhaus, PhD, is Professor of Biostatistics in the School of Medicine at the University of California, San Francisco. A Fellow of the American Statistical Association and the Royal Statistical Society, Dr. Neuhaus has authored or coauthored numerous journal articles on statistical methods for analyzing correlated response data and assessments on the effects of statistical model misspecification.

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New to This Edition

  • A new chapter, "Departures from Assumptions", has been added.  Sections within the chapter include the following topics: omitted covariates, incorrect random effects distribution, correlation of covariates and random effects, and robust variance estimator, among others.
  • A second new chapter, "Models for Multivariate Responses" has also been added, and provides the following topical sections: shared random effects models, latent class models, and properties of models.
  • The section of marginal versus conditional models has been greatly expanded. 
  • Numerous new examples have been added and existing examples have been updated.
  • The second edition has been completely updated and revised to reflect new developments in the field.
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The Wiley Advantage

  • Models for non-normal data, i.e. binary or count data, and generalized linear and nonlinear models are described and illustrated.
  • The chapter on longitudinal data has been completely revised.  While this chapter in the first edition focused primarily on Gaussian linear models, the second edition has been updated to include generalized linear models, modern advances in longitudinal data analysis, as well as the use between and within covariate decompositions.
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