Mixed Models

Theory and Applications with R

Eugene Demidenko

• Publisher: Wiley
• Year: 2013
• Print ISBN: 9781118091579
• eBook ISBN: 9781118592991
• Edition: 2
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

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.

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

The mixed effects approach copes with clustered data that can be viewed as a sample of samples. To illustrate, let us consider the relationship between price (x) and sales (y). Let {(x_k, y_k), k = 1,…, K} be the sample of observations collected on price and sales for several commodities. Plotting y versus x reveals that the relationship is close to linear with a negative slope; see the left-hand panel in Figure 1.1. In classical statistics it is assumed that pairs (x_k, y_k) are independent and identically distributed (iid) with the regression line E(y|x) = α + βx. However, one may argue that we deal with clustered data, where each cluster is a commodity. In the right-hand panel, we connect observation points for each commodity and obtain a reverse picture—increase in price leads to increase in sales. A paradox?

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 {ε_k} 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)

Note that we use a double index now because we are dealing with clustered/panel/tabular data: i corresponds to the ith commodity, j corresponds to the jth observation of the ith commodity, n_i is the number of observations for the ith commodity, and α_i is the ...

Table of 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?
• Chapter 2: MLE for the LME Model
• Chapter 3: Statistical Properties of the LME Model
• Chapter 4: Growth Curve Model and Generalizations
• Chapter 5: Meta-analysis Model
• Chapter 6: Nonlinear Marginal Model
• Chapter 7: Generalized Linear Mixed Models
• Chapter 8: Nonlinear Mixed Effects Model
• Chapter 9: Diagnostics and Influence Analysis
• Chapter 10: Tumor Regrowth Curves
• Chapter 11: Statistical Analysis of Shape
• Chapter 12: Statistical Image Analysis
• Chapter 13: Appendix: Useful Facts and Formulas
• References
• Index