ISBN-10:
047119364X
ISBN-13:
9780471193647
Pub. Date:
01/28/2001
Publisher:
Wiley
Generalized, Linear, and Mixed Models / Edition 1

Generalized, Linear, and Mixed Models / Edition 1

by Charles E. McCulloch, Shayle R. Searle

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Product Details

ISBN-13: 9780471193647
Publisher: Wiley
Publication date: 01/28/2001
Series: Wiley Series in Probability and Statistics Series , #364
Edition description: Older Edition
Pages: 358
Product dimensions: 6.46(w) x 9.43(h) x 0.94(d)

About the Author

Charles E. McCulloch, PhD, is Professor and Head of theDivision of Biostatistics in the School of Medicine at theUniversity of California, San Francisco. A Fellow of the AmericanStatistical Association, Dr. McCulloch is the author of numerouspublished articles in the areas of longitudinal data analysis,generalized linear mixed models, and latent class models and theirapplications.

Shayle R. Searle, PhD, is Professor Emeritus in theDepartment of Biological Statistics and Computational Biology atCornell University. Dr. Searle is the author of LinearModels, Linear Models for Unbalanced Data, MatrixAlgebra Useful for Statistics, and Variance Components,all published by Wiley.

John M. Neuhaus, PhD, is Professor of Biostatistics inthe School of Medicine at the University of California, SanFrancisco. A Fellow of the American Statistical Association and theRoyal Statistical Society, Dr. Neuhaus has authored or coauthorednumerous journal articles on statistical methods for analyzingcorrelated response data and assessments on the effects ofstatistical model misspecification.

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Table of Contents


Preface     xxi
Preface to the First Edition     xxiii
Introduction     1
Models     1
Linear models (LM) and linear mixed models (LMM)     1
Generalized models (GLMs and GLMMs)     2
Factors, Levels, Cells, Effects and Data     2
Fixed Effects Models     5
Example 1: Placebo and a drug     6
Example 2: Comprehension of humor     7
Example 3: Four dose levels of a drug     8
Random Effects Models     8
Example 4: Clinics     8
Notation     9
Example 5: Ball bearings and calipers     12
Linear Mixed Models (LMMs)     13
Example 6: Medications and clinics     13
Example 7: Drying methods and fabrics     13
Example 8: Potomac River Fever     14
Regression models     14
Longitudinal data     14
Example 9: Osteoarthritis Initiative     16
Model equations     16
Fixed or Random?     16
Example 10: Clinical trials     17
Making a decision     17
Inference     19
Estimation     20
Testing     24
Prediction     25
Computer Software     25
Exercises     26
One-Way Classifications     28
Normality and Fixed Effects     29
Model     29
Estimation by ML     29
Generalized likelihood ratio test     31
Confidence intervals     32
Hypothesis tests     34
Normality, Random Effects and MLE     34
Model     34
Balanced data     37
Unbalanced data     42
Bias     44
Sampling variances     44
Normality, Random Effects and Reml     45
Balanced data     45
Unbalanced data     48
More on Random Effects and Normality     48
Tests and confidence intervals     48
Predicting random effects     49
Binary Data: Fixed Effects     51
Model equation     51
Likelihood     51
ML equations and their solutions     52
Likelihood ratio test     52
The usual chi-square test     52
Large-sample tests and confidence intervals     54
Exact tests and confidence intervals     55
Example: Snake strike data     56
Binary Data: Random Effects     57
Model equation     57
Beta-binomial model     57
Logit-normal model     64
Probit-normal model     68
Computing     68
Exercises     68
Single-Predictor Regression     72
Introduction     72
Normality: Simple Linear Regression     73
Model     73
Likelihood     74
Maximum likelihood estimators     74
Distributions of MLEs     75
Tests and confidence intervals     76
Illustration     76
Normality: A Nonlinear Model     77
Model     77
Likelihood     77
Maximum likelihood estimators     78
Distributions of MLEs     79
Transforming Versus Linking     80
Transforming     80
Linking     80
Comparisons     80
Random Intercepts: Balanced Data     81
The model     81
Estimating [mu] and [beta]     83
Estimating variances     86
Tests of hypotheses - using LRT     89
Illustration      92
Predicting the random intercepts     93
Random Intercepts: Unbalanced Data     95
The model     97
Estimating [mu] and [beta] when variances are known     98
Bernoulli - Logistic Regression     101
Logistic regression model     102
Likelihood     104
ML equations     104
Large-sample tests and confidence intervals     107
Bernoulli - Logistic with Random Intercepts     108
Model     108
Likelihood     109
Large-sample tests and confidence intervals     110
Prediction     110
Conditional Inference     111
Exercises     112
Linear Models (LMs)     114
A General Model     115
A Linear Model for Fixed Effects     116
Mle Under Normality     117
Sufficient Statistics     118
Many Apparent Estimators     119
General result     119
Mean and variance     120
Invariance properties     120
Distributions     121
Estimable Functions     121
Introduction     121
Definition     122
Properties      122
Estimation     123
A Numerical Example     123
Estimating Residual Variance     125
Estimation     125
Distribution of estimators     126
The One- and Two-Way Classifications     127
The one-way classification     127
The two-way classification     128
Testing Linear Hypotheses     129
Likelihood ratio test     130
Wald test     131
t-Tests and Confidence Intervals     131
Unique Estimation Using Restrictions     132
Exercises     134
Generalized Linear Models (GLMs)     136
Introduction     136
Structure of the Model     138
Distribution of y     138
Link function     139
Predictors     139
Linear models     140
Transforming Versus Linking     140
Estimation by Maximum Likelihood     140
Likelihood     140
Some useful identities     141
Likelihood equations     142
Large-sample variances     144
Solving the ML equations     144
Example: Potato flour dilutions     145
Tests of Hypotheses     148
Likelihood ratio tests     148
Wald tests     149
Illustration of tests     150
Confidence intervals     151
Illustration of confidence intervals     151
Maximum Quasi-Likelihood     152
Introduction     152
Definition     152
Exercises     156
Linear Mixed Models (LMMs)     157
A General Model     157
Introduction     157
Basic properties     158
Attributing Structure to Var(y)     159
Example     159
Taking covariances between factors as zero     159
The traditional variance components model     161
An LMM for longitudinal data     163
Estimating Fixed Effects for V Known     163
Estimating Fixed Effects for V Unknown     165
Estimation     165
Sampling variance     165
Bias in the variance     167
Approximate F-statistics     168
Predicting Random Effects for V Known     169
Predicting Random Effects for V Unknown     171
Estimation     171
Sampling variance     171
Bias in the variance     172
Anova Estimation of Variance Components     172
Balanced data     173
Unbalanced data     174
Maximum Likelihood (ML) Estimation     174
Estimators     174
Information matrix     176
Asymptotic sampling variances     176
Restricted Maximum Likelihood (REML)     177
Estimation     177
Sampling variances     178
Notes and Extensions     178
ML or REML?     178
Other methods for estimating variances     179
Appendix for Chapter 6     179
Differentiating a log likelihood     179
Differentiating a generalized inverse     182
Differentiation for the variance components model     183
Exercises     185
Generalized Linear Mixed Models     188
Introduction     188
Structure of the Model     189
Conditional distribution of y     189
Consequences of Having Random Effects     190
Marginal versus conditional distribution     190
Mean of y     190
Variances     191
Covariances and correlations     192
Estimation by Maximum Likelihood     193
Likelihood     193
Likelihood equations     195
Other Methods of Estimation     196
Penalized quasi-likelihood     196
Conditional likelihood     198
Simpler models     203
Tests of Hypotheses     204
Likelihood ratio tests     204
Asymptotic variances     204
Wald tests     204
Score tests     205
Illustration: Chestnut Leaf Blight     205
A random effects probit model     206
Exercises     210
Models for Longitudinal Data     212
Introduction     212
A Model for Balanced Data     213
Prescription     213
Estimating the mean     213
Estimating V[subscript 0]     214
A Mixed Model Approach     215
Fixed and random effects     215
Variances     215
Random Intercept and Slope Models     216
Variances     217
Within-subject correlations     217
Predicting Random Effects     219
Uncorrelated subjects     219
Uncorrelated between, and within, subjects      220
Uncorrelated between, and autocorrelated within     220
Random intercepts and slopes     221
Estimating Parameters     221
The general case     221
Uncorrelated subjects     222
Uncorrelated between, and autocorrelated within, subjects     223
Unbalanced Data     225
Example and model     225
Uncorrelated subjects     227
Models for Non-Normal Responses     228
Covariances and correlations     229
Estimation     229
Prediction of random effects     229
Binary responses, random intercepts and slopes     231
A Summary of Results     231
Balanced data     232
Unbalanced data     233
Appendix     233
For Section 8.4a     233
For Section 8.4b     234
Exercises     234
Marginal Models     236
Introduction     236
Examples of Marginal Regression Models     238
Generalized Estimating Equations     239
Models with marginal and conditional interpretations     244
Contrasting Marginal and Conditional Models     246
Exercises      247
Multivariate Models     249
Introduction     249
Multivariate Normal Outcomes     250
Non-Normally Distributed Outcomes     252
A multivariate binary model     252
A binary/normal example     253
A Poisson/Normal Example     257
Correlated Random Effects     260
Likelihood-Based Analysis     261
Example: Osteoarthritis Initiative     263
Notes and Extensions     264
Missing data     264
Efficiency     265
Exercises     265
Nonlinear Models     266
Introduction     266
Example: Corn Photosynthesis     266
Pharmacokinetic Models     269
Computations for Nonlinear Mixed Models     270
Exercises     270
Departures from Assumptions     271
Introduction     271
Incorrect Model for Response     272
Omitted covariates     272
Misspecified link functions     275
Misclassified binary outcomes     276
Informative cluster sizes     278
Incorrect Random Effects Distribution     281
Incorrect distributional family      282
Correlation of covariates and random effects     290
Covariate-dependent random effects variance     293
Diagnosing Misspecification     295
Conditional likelihood methods     295
Between/within cluster covariate decompositions     297
Specification tests     298
Nonparametric maximum likelihood     299
A Summary of Results     300
Exercises     301
Prediction     303
Introduction     303
Best Prediction (BP)     304
The best predictor     304
Mean and variance properties     305
A correlation property     305
Maximizing a mean     305
Normality     306
Best Linear Prediction (BLP)     306
BLP(u)     306
Example     307
Derivation     308
Ranking     309
Linear Mixed Model Prediction (BLUP)     310
BLUE(X[beta])     310
BLUP(t'X[beta] + s'u)     311
Two variances     312
Other derivations     312
Required Assumptions     313
Estimated Best Prediction     313
Henderson's Mixed Model Equations     314
Origin     314
Solutions     315
Use in ML estimation of variance components     316
Appendix     317
Verification of (13.5)     317
Verification of (13.7) and (13.8)     318
Exercises     318
Computing     320
Introduction     320
Computing ML Estimates for LMMs     320
The EM algorithm     320
Using E[u|y]     323
Newton-Raphson method     324
Computing ML Estimates for GLMMs     326
Numerical quadrature     326
EM algorithm     331
Markov chain Monte Carlo algorithms     333
Stochastic approximation algorithms     336
Simulated maximum likelihood     337
Penalized Quasi-Likelihood and Laplace     338
Iterative Bootstrap Bias Correction     342
Exercises     342
Some Matrix Results     344
Vectors and Matrices of Ones     344
Kronecker (or Direct) Products     345
A Matrix Notation in Terms of Elements     346
Generalized Inverses     346
Definition     346
Generalized inverses of X'X     347
Two results involving X(X'V[superscript -1]X)[superscript -]X'V[superscript -1]     348
Solving linear equations     349
Rank results     349
Vectors orthogonal to columns of X     349
A theorem for K' with K'X being null     350
Differential Calculus     350
Definition     350
Scalars     350
Vectors     351
Inner products     351
Quadratic forms     351
Inverse matrices     351
Determinants     352
Some Statistical Results     353
Moments     353
Conditional moments     353
Mean of a quadratic form     354
Moment generating function     354
Normal Distributions     355
Univariate     355
Multivariate     355
Quadratic forms in normal variables     356
Exponential Families     357
Maximum Likelihood     357
The likelihood function     357
Maximum likelihood estimation     358
Asymptotic variance-covariance matrix     358
Asymptotic distribution of MLEs     359
Likelihood Ratio Tests      359
MLE Under Normality     360
Estimation of [beta]     360
Estimation of variance components     361
Asymptotic variance-covariance matrix     361
Restricted maximum likelihood (REML)     362
References     364
Index     378

What People are Saying About This

From the Publisher

"I strongly recommend…[it] for inclusion in math and statistics libraries and in the personal libraries of professional statisticians." (Journal of the American Statistical Association, December 2006)

"…well written and suitable to be a textbook…I enjoyed reading this book and recommend it highly to statisticians." (Journal of Statistical Computation and Simulation, January 2006)

"This text is to be highly recommended as one that provides a modern perspective on fitting models to data." (Short Book Reviews, Vol. 21, No. 2, August 2001)

"For graduate students and statisticians, McCulloch and Searle begin by reviewing the basics of linear models and linear mixed models..." (SciTech Book News, Vol. 25, No. 4, December 2001)

"...a very good reference book." (Zentralblatt MATH, Vol. 964, 2001/14)

"...another fine contribution to the statistics literature from these respected authors..." (Technometrics, Vol. 45, No. 1, February 2003)

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