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