Mixed Models E-Book

Eugene Demidenko

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Serie: Wiley Series in Probability and Statistics
• Sprache: Englisch

Seitenzahl: 1369

Veröffentlichungsjahr: 2013

Beschreibung

Praise for the First Edition "This book will serve to greatly complement the growing number of texts dealing with mixed models, and I highly recommend including it in one's personal library." --Journal of the American Statistical Association Mixed modeling is a crucial area of statistics, enabling the analysis of clustered and longitudinal data. Mixed Models: Theory and Applications with R, Second Edition fills a gap in existing literature between mathematical and applied statistical books by presenting a powerful examination of mixed model theory and application with special attention given to the implementation in R. The new edition provides in-depth mathematical coverage of mixed models' statistical properties and numerical algorithms, as well as nontraditional applications, such as regrowth curves, shapes, and images. The book features the latest topics in statistics including modeling of complex clustered or longitudinal data, modeling data with multiple sources of variation, modeling biological variety and heterogeneity, Healthy Akaike Information Criterion (HAIC), parameter multidimensionality, and statistics of image processing. Mixed Models: Theory and Applications with R, Second Edition features unique applications of mixed model methodology, as well as: * Comprehensive theoretical discussions illustrated by examples and figures * Over 300 exercises, end-of-section problems, updated data sets, and R subroutines * Problems and extended projects requiring simulations in R intended to reinforce material * Summaries of major results and general points of discussion at the end of each chapter * Open problems in mixed modeling methodology, which can be used as the basis for research or PhD dissertations Ideal for graduate-level courses in mixed statistical modeling, the book is also an excellent reference for professionals in a range of fields, including cancer research, computer science, and engineering.

Leseprobe

Contents

Cover

Half Title page

Title page

Copyright page

Dedication

Preface

Preface to the Second Edition

R Software and Functions

Data Sets

Open Problems in Mixed Models

Chapter 1: Introduction: Why Mixed Models?

1.1 Mixed effects for clustered data

1.2 ANOVA, variance components, and the mixed model

1.3 Other special cases of the mixed effects model

1.4 Compromise between Bayesian and frequentist approaches

1.5 Penalized likelihood and mixed effects

1.6 Healthy Akaike information criterion

1.7 Penalized smoothing

1.8 Penalized polynomial fitting

1.9 Restraining parameters, or what to eat

1.10 Ill-posed problems, Tikhonov regularization, and mixed effects

1.11 Computerized tomography and linear image reconstruction

1.12 GLMM for PET

1.13 Maple leaf shape analysis

1.14 DNA Western blot analysis

1.15 Where does the wind blow?

1.16 Software and books

1.17 Summary points

Chapter 2: MLE for the LME Model

2.1 Example: weight versus height

2.2 The model and log-likelihood functions

2.3 Balanced random-coefficient model

2.4 LME model with random intercepts

2.5 Criterion for MLE existence

2.6 Criterion for the positive definiteness of matrix D

2.7 Pre-estimation bounds for variance parameters

2.8 Maximization algorithms

2.9 Derivatives of the log-likelihood function

2.10 Newton–Raphson algorithm

2.11 Fisher scoring algorithm

2.12 EM algorithm

2.13 Starting point

2.14 Algorithms for restricted MLE

2.15 Optimization on nonnegative definite matrices

2.16 lmeFS and lme in R

2.17 Appendix: proof of the existence of MLE

2.18 Summary points

Chapter 3: Statistical Properties of the LME Model

3.1 Introduction

3.2 Identifiability of the LME model

3.3 Information matrix for variance parameters

3.4 Profile-likelihood confidence intervals

3.5 Statistical testing of the presence of random effects

3.6 Statistical properties of MLE

3.7 Estimation of random effects

3.8 Hypothesis and membership testing

3.9 Ignoring random effects

3.10 MINQUE for variance parameters

3.11 Method of moments

3.12 Variance least squares estimator

3.13 Projection on + space

3.14 Comparison of the variance parameter estimation

3.15 Asymptotically efficient estimation for β

3.16 Summary points

Chapter 4: Growth Curve Model and Generalizations

4.1 Linear growth curve model

4.2 General linear growth curve model

4.3 Linear model with linear covariance structure

4.4 Robust linear mixed effects model

4.5 Appendix: derivation of the MM estimator

4.6 Summary points

Chapter 5: Meta-analysis Model

5.1 Simple meta-analysis model

5.2 Meta-analysis model with covariates

5.3 Multivariate meta-analysis model

5.4 Summary points

Chapter 6: Nonlinear Marginal Model

6.1 Fixed matrix of random effects

6.2 Varied matrix of random effects

6.3 Three types of nonlinear marginal models

6.4 Total generalized estimating equations approach

6.5 Summary points

Chapter 7: Generalized Linear Mixed Models

7.1 Regression models for binary data

7.2 Binary model with subject-specific intercept

7.3 Logistic regression with random intercept

7.4 Probit model with random intercept

7.5 Poisson model with random intercept

7.6 Random intercept model: overview

7.7 Mixed models with multiple random effects

7.8 GLMM and simulation methods

7.9 GEE for clustered marginal GLM

7.10 Criteria for MLE existence for a binary model

7.11 Summary points

Chapter 8: Nonlinear Mixed Effects Model

8.1 Introduction

8.2 The model

8.3 Example: height of girls and boys

8.4 Maximum likelihood estimation

8.5 Two-stage estimator

8.6 First-order approximation

8.7 Lindstrom–Bates estimator

8.8 Likelihood approximations

8.9 One-parameter exponential model

8.10 Asymptotic equivalence of the TS and LB estimators

8.11 Bias-corrected two-stage estimator

8.12 Distribution misspecification

8.13 Partially nonlinear marginal mixed model

8.14 Fixed sample likelihood approach

8.15 Estimation of random effects and hypothesis testing

8.16 Example (continued)

8.17 Practical recommendations

8.18 Appendix: Proof of theorem on equivalence

8.19 Summary points

Chapter 9: Diagnostics and Influence Analysis

9.1 Introduction

9.2 Influence analysis for linear regression

9.3 The idea of infinitesimal influence

9.4 Linear regression model

9.5 Nonlinear regression model

9.6 Logistic regression for binary outcome

9.7 Influence of correlation structure

9.8 Influence of measurement error

9.9 Influence analysis for the LME model

9.10 Appendix: MLE derivative with respect to σ2

9.11 Summary points

Chapter 10: Tumor Regrowth Curves

10.1 Survival curves

10.2 Double-exponential regrowth curve

10.3 Exponential growth with fixed regrowth time

10.4 General regrowth curve

10.5 Double-exponential transient regrowth curve

10.6 Gompertz transient regrowth curve

10.7 Summary points

Chapter 11: Statistical Analysis of Shape

11.1 Introduction

11.2 Statistical analysis of random triangles

11.3 Face recognition

11.4 Scale-irrelevant shape model

11.5 Gorilla vertebrae analysis

11.6 Procrustes estimation of the mean shape

11.7 Fourier descriptor analysis

11.8 Summary points

Chapter 12: Statistical Image Analysis

12.1 Introduction

12.2 Testing for uniform lighting

12.3 Kolmogorov–Smirnov image comparison

12.4 Multinomial statistical model for images

12.5 Image entropy

12.6 Ensemble of unstructured images

12.7 Image alignment and registration

12.8 Ensemble of structured images

12.9 Modeling spatial correlation

12.10 Summary points

Chapter 13: Appendix: Useful Facts and Formulas

13.1 Basic facts of asymptotic theory

13.2 Some formulas of matrix algebra

13.3 Basic facts of optimization theory

References

Index

Mixed Models

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Demidenko, Eugene, 1948–  Mixed models : theory and applications with R / Eugene Demidenko. — Second [edition].   p. cm. — (Wiley series in probability and statistics ; 893)  Includes bibliographical references and index.  ISBN 978-1-118-09157-9 (hardback)  1. Analysis of variance. I. Title.  QA279.D457 2013  519.538—dc23  2013001306

To my family

Preface

Technological advances change the world, and statistics is no exception. The cornerstone of classical statistics is the notion of sample. Today, data are richer: We may have repeated measurements with thousands of clusters; data may come in the form of shapes or images. This book is about statistical analysis of data that constitute a sample of samples. In the first ten chapters we discuss statistical models when data come in traditional form as a sequence of numbers. Chapter 11 deals with a sample (ensemble) of shapes, and in Chapter 12 we discuss how to analyze an ensemble of images.

We take the statistical model based approach to analyzing data. Then the method of analysis is a derivative. Although the method sometimes comes first, the model-based approach has obvious advantages: Assumptions are clearly formulated, and properties of several methods can be studied and compared. For example, least squares is a method of fitting, but its pros and cons can be fully understood only when a statistical model is put forward to describe how observations are obtained. Then least squares is deduced, for example, from maximum likelihood.

Statistical treatment is carried out under a unifying mixed effects approach. This approach becomes fruitful not only to analyze complex clustered data (a sample of samples) but also as a statistical model for penalization and a common ground for the Bayesian and frequentist camps.

Use of the mixed modeling technique in shape and image analysis is exciting and promising. Much work remains to reveal the full power of this statistical approach to these nontraditional statistical data.

The book is divided into three parts. The first eight chapters cover the theory of mixed models: the linear mixed effects (LME) model, the generalized linear mixed model (GLMM), and the nonlinear mixed effects (NLME) model. In Chapter 9 we discuss methods of model diagnostics and influential analysis. The final three chapters are devoted to applications: tumor regrowth, shape, and image. Major results and points of discussion in each chapter are written in lay language and are collected in Summary Points sections so that the reader can get a quick chapter overview.

I look forward to hearing from readers and invite them to visit the book web site at

http://www.dartmouth.edu/~eugened

where some additional information with data and images is presented.

I would like to thank the many people I worked with on various projects that have led up to this book. First, I would like to mention my long-term collaboration with Therese Stukel and Tor Tosteson and thank them for their support. I am grateful to Harold Swartz and Jack Hoopes for the exposure to biological problems, and to the team led by Keith Paulsen, including Alex Hartov, Paul Meaney, and Brian Pogue, all from Dartmouth, who introduced me to the world of image reconstruction. Many thanks to John Baron, Margaret Karagas, and Mark Israel for creating a friendly scientific atmosphere. I am grateful to Ed Vonesh for discussion and his helpful comments.

Finally, thanks to the Scientific Workplace, a WYSIWYG version of the LATEX typesetting system (http://www.mackichan.com)—it is hard to imagine writing this book without this software.

Eugene Demidenko

Hanover, New Hampshire Dartmouth College January 2004

Preface to the Second Edition

Time proved that mixed model is an indispensable tool in studying multilevel and clustered data. Mixed model became one of the mainstreams of moderns statistics, on both the theoretical and practical fronts. Several books on the topic have been published since the first edition; see Section 1.16 for a comprehensive list. Most of these books target applications of mixed models and illustrate the examples with popular statistical software packages, such as SAS and R. This book has a distinct theoretical and research flavor. It is intended to explain what is “under the hood” of the mixed model methodology. In particular, it may be used for educational purposes by graduate and Ph.D. students in statistics.

Two major additions have been made in the second edition:

Each section ends with a set of problems that should be important for an active understanding of the material. There are two type of problems: unmarked problems are regular problems, and problems marked with an asterisk are more difficult and are broader in scope. Usually, they involve an analytical derivation with further empirical confirmation through simulations. In many cases, I deliberately left the solution plan open so that students, together with their instructors, could use their own interpretation, and address questions to different depths. Some problems could be used for graduate or even Ph.D. research.

Most parts of the theoretical material and methods of estimation are accompanied by respective

R

codes. While the first edition used S-Plus/S+, the second edition switches to the

R

language. The data sets and

R

codes can be downloaded at the author’s web site,

www.dartmouth.edu/~eugened

It is suggested that they be saved on the hard drive in the directory

C:\MixedModels\

with a subdirectory that corresponds to the chapter in the book. All the codes can be distributed and modified freely.

The theory of mixed models has several important unsolved problems. I hope that the list that follows will stimulate research in this direction.

I would like to hear comments and suggestions from readers, including interesting solutions to the problems, and of course typos, which can be e-mailed to me at [email protected].

Eugene Demidenko

Hanover, New Hampshire January 2013

R Software and Functions

Data Sets

Open Problems in Mixed Models

It should be noted that some literature exists that deals with some of the problems outlined above. We have deliberately not tried to mention all existing publications in these directions because it would require much more space. Therefore, an important part of advancing along the lines of these problems will be a careful review of work already done.

Chapter 1

Introduction: Why Mixed Models?

Big ideas have many names and applications. Sometimes the mixed model is called the model for repeated measurements, sometimes a hierarchical model. Sometimes the mixed model is used to analyze clustered or panel data, sometimes longitudinal data.

Mixed model methodology brings statistics to the next level. In classical statistics a typical assumption is that observations are drawn from the same general population and are independent and identically distributed. Mixed model data have a more complex, multilevel, hierarchical structure. Observations between levels or clusters are independent, but observations within each cluster are dependent because they belong to the same subpopulation. Consequently, we speak of two sources of variation: between clusters and within clusters.

Mixed model is also well suited for the analysis of longitudinal data, where each time series constitutes an individual curve, a cluster. Mixed model is well suited for biological and medical data, which display notorious heterogeneity of responses to stimuli and treatment. An advantage of the mixed model is the ability to genuinely combine the data by introducing multilevel random effects. Mixed model is a nonlinear statistical model, due mainly to the presence of variance parameters, and thus it requires special theoretical treatment. The goal of this book is to provide systematic coverage and development of all spectra of mixed models: linear, generalized linear, and nonlinear.

The aim of this chapter is to show the variety of applications for which the mixed model methodology can be useful, or even a breakthrough. For example, application of mixed modeling methodology to shape and image analysis seems especially exciting and challenging.

Mixed models can be used for the following purposes:

To model complex clustered or longitudinal data.

To model data with multiple sources of variation.

To model biological variety and heterogeneity.

As a compromise between the frequentist and Bayesian approaches.

As a statistical model for the penalized log-likelihood.

To provide a theoretical basis for the Healthy Akaike Information Criterion (HAIC).

To cope with parameter multidimensionality.

As a statistical model to solve ill-posed problems, including image reconstruction problems.

To model shapes and images.

An important feature of this book is that it provides numerical algorithms as a realization of statistical methods that it develops. We strongly believe that an approach is not valuable without an appropriate efficient algorithm. Each chapter ends with a summary points section that may help the reader to quickly grasp the chapter’s major points.

1.1 Mixed effects for clustered data

FIGURE 1.1. Classical and mixed effects approaches lead to reverse conclusions. Left: In the classical approach, it is assumed that observations are independent and identically distributed, resulting in a negative relationship. The straight line shows simple regression estimated by ordinary least squares. Right: In the mixed effects approach, it is assumed that each commodity represents a cluster and therefore that an increase in price for a specific commodity leads to an increase in sales. The straight line shows the linear mixed effects model with population-averaged slope and commodity-specific intercept.

Classical statistics assumes the model

(1.1)

where the {εk} are independent and identically distributed random variables with zero mean and constant variance σ2. In other words, it is assumed that the data are collected from similar, homogeneous commodities. As follows from the right panel, the commodities are not homogeneous and vary substantially in terms of price and sales. An adequate model for the sales problem would be to assume that each commodity has its own commodity-specific sales (in statistical language, intercept); namely,

(1.2)

(1.3)

where α is the population-averaged sale (intercept) and bi is the random effect, or deviation of the commodity-specific sale from the population-averaged sale. Thus, on the one hand, we allow commodity-specific sales, but on the other hand, we assume that commodities represent the country market economy, and therefore one can speak of how an increase in price affects sales across all commodities. Coupled models (1.2) and (1.3) define a linear mixed effects model, parameters α and β are fixed effects (population-averaged parameters), and bi is the random effect with zero mean and variance σ2b independent of {εij}. This is a hierarchical model or a model with random coefficients. The model defined by equations (1.2) and (1.3) can be combined into one as

(1.4)

(1.5)

Observations {yi1, yi2,…, yi,ni} can also be interpreted as repeated measurements. Therefore, model (1.4) is sometimes called the model for repeated measurements. An important example of clustered data is that of longitudinal data when subjects are observed over time. In fact, the pioneering work by Laird and Ware (1982) on the linear mixed effects model was concerned with this kind of data. Model (1.4) belongs to the family of linear mixed effects (LME) models and is studied extensively in Chapters 2 through 4. Specifically, model (1.4) is called the LME model with random intercepts, and it has many nice properties (see Section 2.4). There is more on ignoring random effects in the LME model in Section 3.9.

Summing up, ignoring clustered structure may lead to false analysis. The linear mixed effects model is an adequate model for clustered (repeated) data that involve two sources of variation, within and between clusters.

1.2 ANOVA, variance components, and the mixed model

The mixed model may be viewed as a combination of analysis of variance (ANOVA), variance component (VARCOMP), and regression models. For example, the simplest, one-way ANOVA model deals with tabular data:

(1.6)

The ANOVA model is a special case of the linear regression model,

(1.7)

We come to a different statistical model when the {βi