Mixed Effects Models for Complex Data

✍ Scribed by Lang Wu

• Publisher: CRC Press
• Year: 2010
• Tongue: English
• Leaves: 440
• Series: Monographs on Statistics and Applied Probability 113
• Edition: 1
• Category: Library

✦ Synopsis

Although standard mixed effects models are useful in a range of studies, other approaches must often be used in correlation with them when studying complex or incomplete data. Mixed Effects Models for Complex Data discusses commonly used mixed effects models and presents appropriate approaches to address dropouts, missing data, measurement errors, censoring, and outliers. For each class of mixed effects model, the author reviews the corresponding class of regression model for cross-sectional data. An overview of general models and methods, along with motivating examples After presenting real data examples and outlining general approaches to the analysis of longitudinal/clustered data and incomplete data, the book introduces linear mixed effects (LME) models, generalized linear mixed models (GLMMs), nonlinear mixed effects (NLME) models, and semiparametric and nonparametric mixed effects models. It also includes general approaches for the analysis of complex data with missing values, measurement errors, censoring, and outliers. Self-contained coverage of specific topicsSubsequent chapters delve more deeply into missing data problems, covariate measurement errors, and censored responses in mixed effects models. Focusing on incomplete data, the book also covers survival and frailty models, joint models of survival and longitudinal data, robust methods for mixed effects models, marginal generalized estimating equation (GEE) models for longitudinal or clustered data, and Bayesian methods for mixed effects models. Background materialIn the appendix, the author provides background information, such as likelihood theory, the Gibbs sampler, rejection and importance sampling methods, numerical integration methods, optimization methods, bootstrap, and matrix algebra. Failure to properly address missing data, measurement errors, and other issues in statistical analyses can lead to severely biased or misleading results. This book explores the biases that arise when naïve methods are used and shows which approaches should be used to achieve accurate results in longitudinal data analysis.

✦ Table of Contents

Cover......Page 1
Semi-Title......Page 2
Series Title List......Page 4
Title......Page 8
Copyright......Page 9
Dedication......Page 10
Contents......Page 12
Preface......Page 20
1.1 Introduction......Page 22
1.2 Longitudinal Data and Clustered Data......Page 24
1.3.1 A Study on Mental Distress......Page 27
1.3.2 An AIDS Study......Page 30
1.3.3 A Study on Students' Performance......Page 33
1.3.4 A Study on Children's Growth......Page 34
1.4.1 General Concepts and Approaches......Page 35
1.4.2 Regression Models for Cross-Sectional Data......Page 37
Mixed Effects Models......Page 41
Marginal GEE Models......Page 42
Comparison of the Models......Page 43
1.4.4 Regression Models for Survival Data......Page 44
1.5.1 Motivating Examples......Page 45
1.5.2 LME Models......Page 48
1.5.3 GLMM, NLME and Frailty Models......Page 50
1.6 Complex or Incomplete Data......Page 51
1.6.1 Missing Data......Page 52
1.6.2 Censoring, Measurement Error, and Outliers......Page 53
1.6.3 Simple Methods......Page 54
1.7 Software......Page 55
Outline of the Book......Page 56
Notation Used in the Book......Page 57
2.1 Introduction......Page 60
2.2.1 Linear Regression Models......Page 62
2.2.2 LME Models......Page 63
Example 2.2 Mental distress data......Page 69
2.3.1 Nonlinear Regression Models......Page 70
Example 2.3 Growth curve models......Page 72
2.3.2 NLME Models......Page 73
The Model......Page 74
Statistical Inference......Page 75
Example 2.5 Mixed effects growth and pharmacokinetics models......Page 76
Example 2.6 HIV viral dynamic models......Page 77
2.4.1 Generalized Linear Models (GLMs)......Page 79
Example 2.7 Logistic regression models......Page 82
2.4.2 GLMMs......Page 83
Example 2.9 Logistic regression model with random effects......Page 85
Example 2.11 Mental distress data......Page 86
2.5.1 Nonparametric and Semiparametric Regression Models......Page 87
Regression Splines......Page 88
Kernel Smoothing and Local Polynomial Kernel Smoothing......Page 89
Smoothing Spline......Page 90
Choosing Smoothing Parameters......Page 91
Semiparametric Models......Page 92
Nonparametric Regression Models with Non-Normal Responses......Page 93
Nonparametric Regression Models with Multiple Comparisons......Page 94
Nonparametric Mixed Effects Models......Page 95
Semiparametric Mixed Effects Models......Page 97
Example 2.13 A semiparametric NLME model......Page 98
2.6 Computational Strategies......Page 99
2.6.1 "Exact" Methods......Page 102
2.6.2 EM Algorithms......Page 104
Linearization Methods......Page 106
Laplacian Approximations......Page 107
Discussion......Page 109
Selection of Randome Effects......Page 110
Further Topics......Page 113
2.7.2 Choosing an Mixed Effects Model and Method......Page 114
2.8 Software......Page 115
3.1 Introduction......Page 118
3.2.1 Missing Data Mechanisms......Page 120
3.2.2 Ignorability......Page 121
The Complete-Case (CC) Method......Page 123
The Mean Imputation Method......Page 124
Likelihood Inference Using EM Algorithm......Page 125
Multiple Imputation Methods......Page 126
Weighted GEE......Page 127
3.3.3 Sensitivity Analysis......Page 128
3.3.4 Selection Models versus Pattern-Mixture Models......Page 129
3.3.5 Choosing a Method for Missing Data......Page 130
3.4.1 Introduction......Page 131
Example 3.1 An EM algorithm for normal data......Page 132
Example 3.2 Mental distress data......Page 134
3.4.2 An EM Algorithm for Missing Covariates......Page 135
3.4.3 Properties and Extensions......Page 136
3.5.1 Introduction......Page 138
3.5.2 Multiple Imputation Methods......Page 139
Other Multiple Imputation Methods......Page 142
Example 3.5 Multiple imputation for mental distress data (Example 3.2 continued)......Page 143
3.6.1 Covariate Measurement Errors......Page 144
3.6.2 General Methods for Measurement Errors......Page 146
3.7.1 Outliers......Page 147
3.7.2 General Robust Methods......Page 148
3.8 Software......Page 149
4.1 Introduction......Page 152
The Response Model......Page 154
The Covariate Models......Page 155
The Observed-Data Likelihood......Page 156
A Monte-Carlo EM Algorithm......Page 157
Summary......Page 159
4.2.2 Non-Ignorable Missing Covariates......Page 160
4.2.3 Missing Data in Time-Dependence Covariates......Page 162
4.2.4 Multivariate, Semiparametric, and Nonparametric Models......Page 166
4.3 Approximate Methods......Page 168
4.3.1 Linearization......Page 169
4.3.2 Laplace Approximation......Page 172
4.4.1 Exact Likelihood Inference......Page 175
4.4.2 Approximate Likelihood Inference......Page 178
4.5.1 Advantages and Issues of Multiple Imputation Methods......Page 180
4.5.2 Multiple Imputation for Mixed Effects Models with Missing Data......Page 181
4.5.3 Computational Issues and Other Methods......Page 183
4.6.1 Sampling Methods......Page 184
4.6.2 Speed Up EM Algorithms......Page 187
4.6.3 Convergence......Page 189
Example 4.1 A GLMM with missing data......Page 190
Example 4.2 A NLME model with missing data......Page 193
Example 4.3 A Multiple Imputation Method......Page 195
5.1 Introduction......Page 198
5.2.1 Measurement Error Models......Page 200
Example 5.1 A measurement error model for CD4 count......Page 202
5.2.2 Measurement Error Methods......Page 205
5.2.3 Bias Analysis......Page 207
5.3.1 Two-Step Methods......Page 208
5.3.2 A Two-Step Method for NLME Models with Measurement Errors......Page 209
5.4.1 Joint Likelihood......Page 211
5.4.2 Estimation Based on Monte Carlo EM Algorithms......Page 212
5.5 Approximate Methods......Page 213
5.5.1 Linearization......Page 214
5.5.2 Laplace Approximation......Page 215
5.6 Measurement Error and Missing Data......Page 217
5.6.1 Measurement Errors and Missing Data in Covariates......Page 218
5.6.2 Measurement Errors in Covariates and Missing Data in Responses......Page 220
Example 5.2 Covariate measurement errors and missing responses......Page 222
6.1 Introduction......Page 224
6.2 Mixed Effects Models with Censored Responses......Page 225
6.2.1 LME Models......Page 227
Example 6.1 LME models with censored responses......Page 229
6.2.2 GLMM and NLME Models......Page 231
Single Imputation Methods......Page 232
Multiple Imputation Methods......Page 233
Discussion......Page 234
6.3.1 LME Models......Page 235
6.3.2 GLMM and NLME Models......Page 238
Censoring and Missing Data......Page 242
Censoring and Time-to-Event......Page 244
6.5 Appendix......Page 246
7.1 Introduction......Page 250
7.2 Survival Models......Page 252
7.2.1 Nonparametric Methods......Page 253
7.2.2 Semiparametric Models......Page 255
Weibull Distribution and Model......Page 257
Accelerated failure time (AFT) models......Page 259
Example 7.1 Modeling time to dropout......Page 261
Interval Censored Data......Page 263
7.3.1 Clustered Survival Data......Page 264
7.3.2 Models and Inference......Page 266
7.4.1 Survival Models with Missing Covariates......Page 267
7.4.2 Frailty Models with Missing Covariates......Page 269
The Models......Page 270
Likelihood Inference......Page 271
8.1 Introduction......Page 274
A Simple Two-Step Method......Page 275
Missing Data and Measurement Errors......Page 276
8.2.1 Joint Models with Right Censored Survival Data......Page 277
Survival Model......Page 278
Longitudinal Model......Page 279
Clustered Survival Data......Page 280
8.2.2 Joint Models with Interval Censored Survival Data......Page 281
Comments......Page 283
8.3.1 Simple Two-Step Methods......Page 284
8.3.2 Modified Two-Step Methods......Page 286
8.4.1 Exact Likelihood Inference......Page 287
8.4.2 Approximate Inference......Page 289
8.5 Joint Models with Incomplete Data......Page 291
8.5.1 Joint Models with Missing Data......Page 292
The Models......Page 293
Joint Likelihood Inference......Page 295
Example 8.1 Joint inference for an NLME model and survival model with missing data......Page 297
8.5.2 Joint Models with Measurement Errors......Page 299
8.6.1 Multivariate Mixed Effects Models with Incomplete Data......Page 303
Missing Data Problems......Page 304
Factorization of Joint Distributions......Page 307
Random Effects Approach......Page 308
Example 8.2 Joint transitional models......Page 309
8.6.3 Joint Longitudinal Models with Incomplete Data: A Summary......Page 310
9.1 Introduction......Page 314
9.2.1 Robust Distributions......Page 317
9.2.2 M-Estimators......Page 319
9.3.1 LME Models with Multivariate t-Disributions......Page 322
LME models with t-distributions......Page 323
Likelihood Inference......Page 324
9.3.2 GLMM and NLME Models with Multivariate t-Distributions......Page 326
9.3.3 Robust Models with Incomplete Data......Page 328
9.4.1 M-Estimators for GLM......Page 329
9.4.2 M-Estimators for Mixed Effects Models......Page 331
Asymptotics......Page 333
9.4.3 A Monte Carlo Newton-Raphson Method......Page 334
9.4.4 A Robust Approximate Method......Page 337
A2: Proof of Theorem 9.2......Page 340
A3: Proof of Theorem 9.3......Page 342
9.5.1 Robust Inference with Covariate Measurement Errors......Page 343
9.5.2 A Robust Approximate Method......Page 347
9.5.3 Robust Inference with Non-Ignorable Missing Data......Page 349
9.5.4 Appendix......Page 351
10.1 Introduction......Page 354
10.2.1 Quasi-Likelihood and GEE......Page 357
Example 10.1 Over-dispersion problems......Page 359
10.2.2 Marginal Models for Longitudinal Data or Cluster Data......Page 362
10.2.3 GEE for Marginal Models......Page 365
Example 10.3 Analyses of mental distress data......Page 367
10.3.1 Weighted GEE for Missing Data......Page 368
10.3.2 Weighted GEE for Measurement Errors and Missing Data......Page 370
10.4 Discussion......Page 372
11.1 Introduction......Page 374
11.2.1 General Concepts......Page 375
11.2.2 Prior Distributions......Page 377
11.3.1 Bayesian LME Models......Page 379
11.3.2 Bayesian GLMMs......Page 382
11.3.3 Bayesian NLME Models......Page 384
11.4.1 Bayesian Models with Missing Data......Page 388
11.4.2 Bayesian Mixed Models with Missing Data......Page 389
11.5.1 Bayesian Regression Models with Covariate Measurement Errors......Page 390
11.5.2 Bayesian Mixed Models with Covariate Measurement Errors......Page 392
11.6 Bayesian Joint Models of Longitudinal and Survival Data......Page 393
12.1 Likelihood Methods......Page 396
12.2 The Gibbs sampler and MCMC Methods......Page 401
12.3 Rejection Sampling and Importance Sampling methods......Page 404
12.4 Numerical Integration and the Guass-Hermite Quadrature Method......Page 406
12.6 Bootstrap Methods......Page 408
12.7 Matrix Algebra for Vector Differential Calculus......Page 410
References......Page 414
Index......Page 435
Abstract......Page 440