Mixed Effects Models in S and S-Plus

This book provides an overview of the theory and application of linear and nonlinear mixed-effects models in the analysis of grouped data, such as longitudinal data, repeated measures, and multilevel data. A unified model-building strategy for both linear and nonlinear models is presented and applied to the analysis of over 20 real datasets from a wide variety of areas, including pharmacokinetics, agriculture, and manufacturing. A strong emphasis is placed on the use of graphical displays at the various phases of the model-building process, starting with exploratory plots of the data and concluding with diagnostic plots to assess the adequacy of a fitted model. Over 170 figures are included in the book. The NLME library for analyzing mixed-effects models in S and S-PLUS, developed by the authors, provides the underlying software for implementing the methods presented in the text, being described and illustrated in detail throughout the book. The balanced mix of real data examples, modeling software, and theory makes this book a useful reference for practitioners who use, or intend to use, mixed-effects models in their data analyses. It can also be used as a text for a one-semester graduate-level applied course in mixed-effects models. Researchers in statistical computing will also find this book appealing for its presentation of novel and efficient computational methods for fitting linear and nonlinear mixed-effects models. Jos C. Pinheiro has been a member of the technical staff in statistics research at Bell Laboratories since 1996. He received his Ph. D. in Statistics in 1994 from, and worked for two years in the Department of Biostatistics at the University of Wisconsin, Madison. The author of several articles in mixed-effects models, he is a memeber of the American Statistical Association and the Biometric Society. Douglas M. Bates is Professor of Statistics at the University of Wisconsin, Madison. He, with Donald G. Watts, is the author of "Nonlinear Regression Analysis and Its Applications", a Fellow of the American Statistical Association, and former chair of its Statistical Computing Section. Copyright......Page 5 Preface......Page 7 Contents......Page 11 Part I - Linear Mixed-Effects Models......Page 17 1 Linear Mixed-Effects Models: Basic Concepts and Examples......Page 18 1.1 A Simple Example of Random Effects......Page 19 1.1.1 Fitting the Random-Effects Model With lme......Page 23 1.1.2 Assessing the Fitted Model......Page 26 1.2 A Randomized Block Design......Page 27 1.2.1ChoosingContrastsforFixed-EffectsTerms......Page 29 1.2.2ExaminingtheModel......Page 34 1.3 Mixed-Effects Models for Replicated, Blocked Designs......Page 36 1.3.1 Fitting Random Interaction Terms......Page 38 1.3.2 Unbalanced Data......Page 40 1.3.3 More General Models for the Random Interaction Effects......Page 42 1.4.1 Modeling Simple Linear Growth Curves......Page 45 1.4.2 Predictions of the Response and the Random Effects......Page 52 1.5 Models for Nested Classification Factors......Page 55 1.5.1 Model Building for Multilevel Models......Page 59 1.6 A Split-Plot Experiment......Page 60 Exercises......Page 67 2 Theory and Computational Methods for Linear Mixed-Effect Models......Page 72 2.1.1 Single Level of Grouping......Page 73 2.1.2 A Multilevel LME Model......Page 75 2.2.1 The Single-Level LME Likelihood Function......Page 77 2.2.2 Orthogonal-Triangular Decompositions......Page 81 2.2.3 Evaluating the Likelihood Through Decompositions......Page 83 2.2.4 Components of the Profiled Log-Likelihood......Page 86 2.2.5 Restricted Likelihood Estimation......Page 90 2.2.6 Multiple Levels of Random Effects......Page 92 2.2.7 Parameterizing Relative Precision Factors......Page 93 2.2.8 Optimization Algorithms......Page 94 2.3 Approximate Distributions......Page 96 2.4 Hypothesis Tests and Confidence Intervals......Page 97 2.4.1 Likelihood Ratio Tests......Page 98 2.4.2 Hypothesis Tests for Fixed-Effects Terms......Page 102 2.4.3 Confidence Intervals......Page 107 2.6 Chapter Summary......Page 109 Exercises......Page 111 3.1 The Display Formula and its Components......Page 112 3.2 Constructing groupedData Objects......Page 116 3.2.1 Roles of Other Experimental or Blocking Factors......Page 119 3.2.2 Constructors for Balanced Data......Page 123 3.3.1 Layout of the Trellis Plot......Page 125 3.3.2 Modifying the Vertical and Horizontal Scales......Page 128 3.3.3 Modifying the Panel Function......Page 129 3.3.4 Plots of Multiply-Nested Data......Page 131 3.4 Summaries......Page 135 Exercises......Page 145 4 Fitting Linear Mixed-Effects Models......Page 148 4.1 Fitting Linear Models in S with lm and lmList......Page 149 4.1.1 Separate lm Fits per Group: the lmList Function......Page 154 4.2.1 Fitting Single-Level Models......Page 161 4.2.2 Patterned Variance–Covariance Matrices for the Random Effects: The pdMat Classes......Page 172 4.2.3 Fitting Multilevel Models......Page 182 4.3.1 Assessing Assumptions on the Within-Group Error......Page 189 4.3.2 Assessing Assumptions on the Random Effects......Page 202 4.4 Chapter Summary......Page 211 Exercises......Page 212 5 Extending the Basic Linear Mixed-Effects Model......Page 215 5.1.1 Estimation and Computational Methods......Page 216 5.1.2 An Extended Linear Model with no Random Effects......Page 217 5.1.3 Decomposing the Within-Group Variance–Covariance Structure......Page 219 5.2 Variance Functions for Modeling Heteroscedasticity......Page 220 5.2.1 Variance Functions in nlme: The varFunc Classes......Page 222 5.2.2 Using Variance Functions with lme......Page 228 5.3.1 Serial Correlation Structures......Page 240 5.3.2 Spatial Correlation Structures......Page 244 5.3.3 Correlation Structures in nlme: The corStruct Classes......Page 246 5.3.4 Using Correlation Structures with lme......Page 253 5.4 Fitting Extended Linear Models with gls......Page 263 5.5 Chapter Summary......Page 280 Exercises......Page 281 Part II - Nonlinear Mixed-Effects Models......Page 285 6.1 LME Models vs. NLME Models......Page 286 6.2 Indomethicin Kinetics......Page 290 6.3 Growth of Soybean Plants......Page 300 6.4 Clinical Study of Phenobarbital Kinetics......Page 307 6.5 Chapter Summary......Page 313 Exercises......Page 314 7 Theory and Computational Methods for Nonlinear Mixed_effects Models......Page 318 7.1.1 Single-Level of Grouping......Page 319 7.1.2 Multilevel NLME Models......Page 322 7.1.3 Other NLME Models......Page 323 7.2.1 Likelihood Estimation......Page 325 7.2.2 Inference and Predictions......Page 335 7.3 Computational Methods......Page 337 7.4.1 General Formulation of the Extended NLME Model......Page 341 7.4.2 Estimation and Computational Methods......Page 342 7.5 An Extended Nonlinear Regression Model......Page 345 7.5.1 General Model Formulation......Page 346 7.5.2 Estimation and Computational Methods......Page 347 7.6 Chapter Summary......Page 349 8 Fitting Nonlinear Mixed-Effects Models......Page 350 8.1.1 Using the nls Function......Page 351 8.1.2 Self-Starting Nonlinear Model Functions......Page 355 8.1.3 Separate Nonlinear Fits by Group: The nlsList Function......Page 360 8.2.1 Fitting Single-Level nlme Models......Page 367 8.2.2 Using Covariates with nlme......Page 378 8.2.3 Fitting Multilevel nlme Models......Page 398 8.3.1 Variance Functions in nlme......Page 404 8.3.2 Correlation Structures in nlme......Page 408 8.3.3 Fitting Extended Nonlinear Regression Models with gnls......Page 414 8.4 Chapter Summary......Page 422 Exercises......Page 423 References......Page 428 Appendix A: Data used in Examples and Exercises......Page 435 A.2 Assay — Bioassay on Cell Culture Plate......Page 437 A.4 Cefamandole — Pharmacokinetics of Cefamandole......Page 439 A.5 CO2 — Carbon Dioxide Uptake......Page 440 A.7 DNase — Assay Data for the Protein DNase......Page 441 A.8 Earthquake — Earthquake Intensity......Page 442 A.9 ergoStool — Ergometrics Experiment with Stool Types......Page 443 A.10 Glucose2 — Glucose Levels Following Alcohol Ingestion......Page 444 A.12 Indometh — Indomethicin Kinetics......Page 445 A.13 Loblolly — Growth of Loblolly Pine Trees......Page 446 A.15 Oats — Split-plot Experiment on Varieties of Oats......Page 447 A.17 Orthodont — Orthodontic Growth Data......Page 448 A.20 Oxide — Variability in Semiconductor Manufacturing......Page 449 A.21 PBG — Effct of Phenylbiguanide on Blood Pressure......Page 450 A.22 PBIB — A Partially Balanced Incomplete Block Design......Page 451 A.24 Pixel — Pixel Intensity in Lymphnodes......Page 452 A.25 Quinidine — Quinidine Kinetics......Page 453 A.27 Soybean — Soybean Leaf Weight over Time......Page 455 A.29 Theoph — Theophylline Kinetics......Page 456 A.31 Wheat2 — Wheat Yield Trials......Page 460 Appendix B: S Functions and Classes......Page 462 C.1.1 Starting Estimates for SSasymp......Page 522 C.2.1 Starting Estimates for SSasympOff......Page 523 C.3.1 Starting Estimates for SSasympOrig......Page 524 C.4 SSbiexp—Biexponential Model......Page 525 C.4.1 Starting Estimates for SSbiexp......Page 526 C.5.1 Starting Estimates for SSfol......Page 527 C.6 SSfpl—Four-Parameter Logistic Model......Page 528 C.6.1 Starting Estimates for SSfpl......Page 529 C.7.1 Starting Estimates for SSlogis......Page 530 C.8 SSmicmen—Michaelis–Menten Model......Page 531 C.8.1StartingEstimatesforSSmicmen......Page 532 An overview of the theory and application of linear and nonlinear mixed-effects models in the analysis of grouped data, such as longitudinal data, repeated measures, and multilevel data. The authors present a unified model-building strategy for both models and apply this to the analysis of over 20 real datasets from a wide variety of areas, including pharmacokinetics, agriculture, and manufacturing. Much emphasis is placed on the use of graphical displays at the various phases of the model-building process, starting with exploratory plots of the data and concluding with diagnostic plots to assess the adequacy of a fitted model. The NLME library for analyzing mixed-effects models in S and S-PLUS, developed by the authors, provides the underlying software for implementing the methods presented. This balanced mix of real data examples, modeling software, and theory makes the book a useful reference for practitioners who use, or intend to use, mixed-effects models in their data analyses. It can also be used as a text for a one-semester graduate-level applied course. To Follow I. Linear Mixed-effects Models. 1. Linear Mixed-effects Models. 2. Theory And Computational Methods For Lme Models. 3. Describing The Structure Of Grouped Data. 4. Fitting Linear Mixed-effects Models. 5. Extending The Basic Linear Mixed-effects Model -- Ii. Nonlinear Mixed-effects Models. 6. Nlme Models: Basic Concepts And Motivating Examples. 7. Theory And Computational Methods For Nlme Models. 8. Fitting Nonlinear Mixed-effects Models. A. Data Used In Examples And Exercises -- B. S Functions And Classes -- C. A Collection Of Self-starting Nonlinear Regression Models. José C. Pinheiro, Douglas M. Bates. Includes Bibliographical References (p. [415]-421) And Index. “Over 170 figures are included in the book. ... the material covered in the book is self-contained .... The balanced mix of real data examples, modeling software, and theory makes this book a useful reference for practitioners who use, or intend to use, mixed-effects models in their data analyses. It can also be used as a text for a one-semester graduate-level applied course in mixed-effects models.” (E.M.Psyadlo, zbMATH 0953.62065, 2022) Provides an overview of the theory and application of linear and nonlinear mixed-effects models in the analysis of grouped data. This book also presents an unified model-building strategy for both linear and nonlinear models, applying it to the analysis of over 20 real datasets from areas including pharmacokinetics, agriculture and manufacturing Many common statistical models can be expressed as linear models that incorporate both fixed effects, which are parameters associated with an entire population or with certain repeatable levels of experimental factors, and random effects, which are associated with individual experimental units drawn at random from a population. The balanced mix of real data examples, modeling software, and theory makes this book a useful reference for practitioners who use mixed-effects models. Researchers in statistical computing will also learn novel and efficient computational methods for fitting linear and non-linear mixed effects models. 172 illus.