9780387950273

Linear Mixed Models for Longitudinal Data

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  • ISBN13:

    9780387950273

  • ISBN10:

    0387950273

  • Format: Hardcover
  • Copyright: 2000-06-01
  • Publisher: Springer Verlag
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Summary

This book provides a comprehensive treatment of linear mixed models for continuous longitudinal data. Next to model formulation, this edition puts major emphasis on exploratory data analysis for all aspects of the model, such as the marginal model, subject-specific profiles, and residual covariance structure. Further, model diagnostics and missing data receive extensive treatment. Sensitivity analysis for incomplete data is given a prominent place. Several variations to the conventional linear mixed model are discussed (a heterogeity model, condional linear mid models).This book will be of interest to applied statisticians and biomedical researchers in industry, public health organizations, contract research organizations, and academia. The book is explanatory rather than mathematically rigorous. Most analyses were done with the MIXED procedure of the SAS software package, and many of its features are clearly elucidated. How3ever, some other commercially available packages are discussed as well. Great care has been taken in presenting the data analyses in a software-independent fashion.Geert Verbeke is Assistant Professor at the Biostistical Centre of the Katholieke Universiteit Leuven in Belgium. He received the B.S. degree in mathematics (1989) from the Katholieke Universiteit Leuven, the M.S. in biostatistics (1992) from the Limburgs Universitair Centrum, and earned a Ph.D. in biostatistics (1995) from the Katholieke Universiteit Leuven. Dr. Verbeke wrote his dissertation, as well as a number of methodological articles, on various aspects of linear mixed models for longitudinal data analysis. He has held visiting positions at the Gerontology Research Center and the Johns Hopkins University.Geert Molenberghs is Assistant Professor of Biostatistics at the Limburgs Universitair Centrum in Belgium. He received the B.S. degree in mathematics (1988) and a Ph.D. in biostatistics (1993) from the Universiteit Antwerpen. Dr. Molenberghs published methodological work on the analysis of non-response in clinical and epidemiological studies. He serves as an associate editor for Biometrics, Applied Statistics, and Biostatistics, and is an officer of the Belgian Statistical Society. He has held visiting positions at the Harvard School of Public Health.

Table of Contents

Preface vii
Acknowledgments ix
Introduction
1(6)
Examples
7(12)
The Rat Data
7(2)
The Toenail Data (TDO)
9(1)
The Baltimore Longitudinal Study of Aging (BLSA)
10(5)
The Prostate Data
11(3)
The Hearing Data
14(1)
The Vorozole Study
15(1)
Heights of Schoolgirls
16(1)
Growth Data
16(2)
Mastitis in Dairy Cattle
18(1)
A Model for Longitudinal Data
19(12)
Introduction
19(1)
A Two-Stage Analysis
20(3)
Stage 1
20(1)
Stage 2
20(1)
Example: The Rat Data
21(1)
Example: The Prostate Data
21(1)
Two-Stage Analysis
22(1)
The General Linear Mixed-Effects Model
23(8)
The Model
23(2)
Example: The Rat Data
25(1)
Example: The Prostate Data
26(1)
A Model for the Residual Covariance Structure
26(5)
Exploratory Data Analysis
31(10)
Introduction
31(1)
Exploring the Marginal Distribution
31(4)
The Average Evolution
31(2)
The Variance Structure
33(1)
The Correlation Structure
34(1)
Exploring Subject-Specific Profiles
35(6)
Measuring the Overall Goodness-of-Fit
35(2)
Testing for the Need of a Model Extension
37(1)
Example: The Rat Data
38(1)
Example: The Prostate Data
39(2)
Estimation of the Marginal Model
41(14)
Introduction
41(1)
Maximum Likelihood Estimation
42(1)
Restricted Maximum Likelihood Estimation
43(4)
Variance Estimation of Normal Populations
43(1)
Estimation of Residual Variance in Linear Regression
43(1)
REML Estimation for the Linear Mixed Model
44(2)
Justification of REML Estimation
46(1)
Comparison Between ML and REML Estimation
46(1)
Model-Fitting Procedures
47(1)
Example: The Prostate Data
48(2)
Estimation Problems
50(5)
Small Variance Components
50(2)
Model Misspecifications
52(3)
Inference for the Marginal Model
55(22)
Introduction
55(1)
Inference for the Fixed Effects
55(9)
Approximate Wald Tests
56(1)
Approximate t-Tests and F-Tests
56(1)
Example: The Prostate Data
57(4)
Robust Inference
61(1)
Likelihood Ratio Tests
62(2)
Inference for the Variance Components
64(10)
Approximate Wald Tests
64(1)
Likelihood Ratio Tests
65(1)
Example: The Rat Data
66(3)
Marginal Testing for the Need of Random Effects
69(3)
Example: The Prostate Data
72(2)
Information Criteria
74(3)
Inference for the Random Effects
77(16)
Introduction
77(1)
Empirical Bayes Inference
78(1)
Henderson's Mixed-Model Equations
79(1)
Best Linear Unbiased Prediction (BLUP)
80(1)
Shrinkage
80(1)
Example: The Random-Intercepts Model
81(1)
Example: The Prostate Data
82(1)
The Normality Assumption for Random Effects
83(10)
Introduction
83(2)
Impact on EB Estimates
85(2)
Impact on the Estimation of the Marginal Model
87(2)
Checking the Normality Assumption
89(4)
Fitting Linear Mixed Models with SAS
93(28)
Introduction
93(1)
The SAS Program
94(10)
The PROC MIXED Statement
95(1)
The Class Statement
96(1)
The Model Statement
96(1)
The ID Statement
97(1)
The Random Statement
97(1)
The Repeated Statement
98(3)
The Contrast Statement
101(1)
The Estimate Statement
101(1)
The Make Statement
102(1)
Some Additional Statements and Options
102(2)
The SAS Output
104(10)
Information on the Iteration Procedure
104(1)
Information on the Model Fit
105(2)
Information Criteria
107(1)
Inference for the Variance Components
107(4)
Inference for the Fixed Effects
111(2)
Inference for the Random Effects
113(1)
Note on the Mean Parameterization
114(3)
The Random and Repeated Statements
117(2)
Proc Mixed versus Proc GLM
119(2)
General Guidelines for Model Building
121(14)
Introduction
121(2)
Selection of a Preliminary Mean Structure
123(2)
Selection of a Preliminary Random-Effects Structure
125(3)
Selection of a Residual Covariance Structure
128(4)
Model Reduction
132(3)
Exploring Serial Correlation
135(16)
Introduction
135(1)
An Informal Check for Serial Correlation
136(1)
Flexible Models for Serial Correlation
137(4)
Introduction
137(1)
Fractional Polynomials
137(1)
Example: The Prostate Data
138(3)
The Semi-Variogram
141(7)
Introduction
141(1)
The Semi-Variogram for Random-Intercepts Models
142(2)
Example: The Vorozole Study
144(1)
The Semi-Variogram for Random-Effects Models
144(3)
Example: The Prostate Data
147(1)
Some Remarks
148(3)
Local Influence for the Linear Mixed Model
151(18)
Introduction
151(2)
Local Influence
153(5)
The Detection of Influential Subjects
158(4)
Example: The Prostate Data
162(5)
Local Influence Under REML Estimation
167(2)
The Heterogeneity Model
169(20)
Introduction
169(2)
The Heterogeneity Model
171(2)
Estimation of the Heterogeneity Model
173(4)
Classification of Longitudinal Profiles
177(1)
Goodness-of-Fit Checks
178(2)
Example: The Prostate Data
180(3)
Example: The Heights of Schoolgirls
183(6)
Conditional Linear Mixed Models
189(12)
Introduction
189(1)
A Linear Mixed Model for the Hearing Data
190(4)
Conditional Linear Mixed Models
194(3)
Applied to the Hearing Data
197(1)
Relation with Fixed-Effects Models
198(3)
Exploring Incomplete Data
201(8)
Joint Modeling of Measurements and Missingness
209(12)
Introduction
209(1)
The Impact of Incompleteness
210(1)
Simple ad hoc Methods
211(1)
Modeling Incompleteness
212(2)
Terminology
214(1)
Missing Data Patterns
215(1)
Missing Data Mechanisms
215(2)
Ignorability
217(1)
A Special Case: Dropout
218(3)
Simple Missing Data Methods
221(10)
Introduction
221(2)
Complete Case Analysis
223(1)
Simple Forms of Imputation
223(4)
Last Observation Carried Forward
224(1)
Imputing Unconditional Means
225(1)
Buck's Method: Conditional Mean Imputation
225(1)
Discussion of Imputation Techniques
226(1)
Available Case Methods
227(1)
MCAR Analysis of Toenail Data
227(4)
Selection Models
231(44)
Introduction
231(2)
A Selection Model for the Toenail Data
233(6)
MAR Analysis
233(1)
MNAR analysis
234(5)
Scope of Ignorability
239(1)
Growth Data
240(29)
Analysis of Complete Growth Data
240(16)
Frequentist Analysis of Incomplete Growth Data
256(1)
Likelihood Analysis of Incomplete Growth Data
257(10)
Missingness Process for the Growth Data
267(2)
A Selection Model for Nonrandom Dropout
269(1)
A Selection Model for the Vorozole Study
270(5)
Pattern-Mixture Models
275(20)
Introduction
275(5)
A Simple Illustration
275(3)
A Paradox
278(2)
Pattern-Mixture Models
280(1)
Pattern-Mixture Model for the Toenail Data
281(6)
A Pattern-Mixture Model for the Vorozole Study
287(4)
Some Reflections
291(4)
Sensitivity Analysis for Selection Models
295(36)
Introduction
295(2)
A Modified Selection Model for Nonrandom Dropout
297(1)
Local Influence
298(9)
Review of the Theory
299(1)
Applied to the Model of Diggle and Kenward
300(2)
Special Case: Compound Symmetry
302(4)
Serial Correlation
306(1)
Analysis of the Rat Data
307(5)
Mastitis in Dairy Cattle
312(14)
Informal Sensitivity Analysis
312(7)
Local Influence Approach
319(7)
Alternative Local Influence Approaches
326(2)
Random-Coefficient-based Models
328(2)
Concluding Remarks
330(1)
Sensitivity Analysis for Pattern-Mixture Models
331(44)
Introduction
331(1)
Pattern-Mixture Models and MAR
332(4)
MAR and ACMV
333(2)
Nonmonotone Patterns: A Counterexample
335(1)
Multiple Imputation
336(3)
Parameter and Precision Estimation
338(1)
Hypothesis Testing
338(1)
Pattern-Mixture Models and Sensitivity Analysis
339(4)
Identifying Restrictions Strategies
343(9)
Strategy Outline
343(1)
Identifying Restrictions
344(3)
ACMV Restrictions
347(3)
Drawing from the Conditional Densities
350(2)
Analysis of the Vorozole Study
352(21)
Fitting a Model
352(14)
Hypothesis Testing
366(5)
Model Reduction
371(2)
Thoughts
373(2)
How Ignorable Is Missing At Random?
375(12)
Introduction
375(2)
Information and Sampling Distributions
377(2)
Illustration
379(4)
Example
383(2)
Implications for PROC Mixed
385(2)
The Expectation-Maximization Algorithm
387(4)
Design Considerations
391(14)
Introduction
391(1)
Power Calculations Under Linear Mixed Models
392(1)
Example: The Rat Data
393(1)
Power Calculations When Dropout Is to Be Expected
394(3)
Example: The Rat Data
397(8)
Constant pj,k\≥k, Varying nj
399(2)
Constant pj,k\≥k, Constant nj
401(1)
Increasing pj,k\≥k over Time, Constant nj
402(3)
Case Studies
405(118)
Blood Pressures
405(6)
The Heat Shock Study
411(9)
Introduction
411(4)
Analysis of Heat Shock Data
415(5)
The Validation of Surrogate Endpoints from Multiple Trials
420(26)
Introduction
420(1)
Validation Criteria
421(3)
Notation and Motivating Examples
424(5)
A Meta-Analytic Approach
429(5)
Data Analysis
434(5)
Computational Issues
439(3)
Extensions
442(1)
Reflections on Surrogacy
443(1)
Precdiction Intervals
444(1)
SAS Code for Random-Effects Model
445(1)
The Milk Protein Content Trial
446(24)
Introduction
446(2)
Informal Sensitivity Analaysis
448(9)
Formal Sensitivity Analysis
457(13)
Hepatitis B Vaccination
470(15)
Time Evolution of Antibodies
472(9)
Prediction at Year 12
481(1)
SAS Code for Vaccination Models
482(3)
Appendix
A Software
485(30)
A.1 The SAS System
485(1)
A.1.1 Standard Applications
485(1)
A.1.2 New Features in SAS Version 7.0
485(4)
A.2 Fitting Mixed Models Using MLwiN
489(4)
A.3 Fitting Mixed Models Using SPlus
493(1)
A.3.1 Standard SPlus Functions
494(3)
A.3.2 OSWALD for Nonrandom Nonresponse
497(18)
B Technical Details for Sensitivity Analysis
515(8)
B.1 Local Influence: Derivation of Components of Δi
515(3)
B.2 Proof of Theorem 20.1
518(5)
References 523(31)
Index 554

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