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Generalized, Linear, and Mixed Models

ISBN-10: 0470073713
ISBN-13: 9780470073711
Edition: 2nd 2008
List price: $192.00
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Description: This book provides a unified treatment of the use of mixed models for analyzing correlated data. Models for non-normal data, i.e., binary or count data, and generalized linear and nonlinear models are described and illustrated, while an accessible  More...

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Book details

List price: $192.00
Edition: 2nd
Copyright year: 2008
Publisher: John Wiley & Sons, Incorporated
Publication date: 6/30/2008
Binding: Hardcover
Pages: 424
Size: 6.25" wide x 9.50" long x 1.00" tall
Weight: 1.804
Language: English

This book provides a unified treatment of the use of mixed models for analyzing correlated data. Models for non-normal data, i.e., binary or count data, and generalized linear and nonlinear models are described and illustrated, while an accessible treatment of many of the newer statistical models for correlated, non-normally distributed data is also provided. The books unified approach addresses the needs of applications-oriented users of statistical packages and graduate students in statistics alike.

Preface
Preface to the First Edition
Introduction
Models
Linear models (LM) and linear mixed models (LMM)
Generalized models (GLMs and GLMMs)
Factors, Levels, Cells, Effects and Data
Fixed Effects Models
Example 1: Placebo and a drug
Example 2: Comprehension of humor
Example 3: Four dose levels of a drug
Random Effects Models
Example 4: Clinics
Notation
Example 5: Ball bearings and calipers
Linear Mixed Models (LMMs)
Example 6: Medications and clinics
Example 7: Drying methods and fabrics
Example 8: Potomac River Fever
Regression models
Longitudinal data
Example 9: Osteoarthritis Initiative
Model equations
Fixed or Random?
Example 10: Clinical trials
Making a decision
Inference
Estimation
Testing
Prediction
Computer Software
Exercises
One-Way Classifications
Normality and Fixed Effects
Model
Estimation by ML
Generalized likelihood ratio test
Confidence intervals
Hypothesis tests
Normality, Random Effects and MLE
Model
Balanced data
Unbalanced data
Bias
Sampling variances
Normality, Random Effects and Reml
Balanced data
Unbalanced data
More on Random Effects and Normality
Tests and confidence intervals
Predicting random effects
Binary Data: Fixed Effects
Model equation
Likelihood
ML equations and their solutions
Likelihood ratio test
The usual chi-square test
Large-sample tests and confidence intervals
Exact tests and confidence intervals
Example: Snake strike data
Binary Data: Random Effects
Model equation
Beta-binomial model
Logit-normal model
Probit-normal model
Computing
Exercises
Single-Predictor Regression
Introduction
Normality: Simple Linear Regression
Model
Likelihood
Maximum likelihood estimators
Distributions of MLEs
Tests and confidence intervals
Illustration
Normality: A Nonlinear Model
Model
Likelihood
Maximum likelihood estimators
Distributions of MLEs
Transforming Versus Linking
Transforming
Linking
Comparisons
Random Intercepts: Balanced Data
The model
Estimating [mu] and [beta]
Estimating variances
Tests of hypotheses - using LRT
Illustration
Predicting the random intercepts
Random Intercepts: Unbalanced Data
The model
Estimating [mu] and [beta] when variances are known
Bernoulli - Logistic Regression
Logistic regression model
Likelihood
ML equations
Large-sample tests and confidence intervals
Bernoulli - Logistic with Random Intercepts
Model
Likelihood
Large-sample tests and confidence intervals
Prediction
Conditional Inference
Exercises
Linear Models (LMs)
A General Model
A Linear Model for Fixed Effects
Mle Under Normality
Sufficient Statistics
Many Apparent Estimators
General result
Mean and variance
Invariance properties
Distributions
Estimable Functions
Introduction
Definition
Properties
Estimation
A Numerical Example
Estimating Residual Variance
Estimation
Distribution of estimators
The One- and Two-Way Classifications
The one-way classification
The two-way classification
Testing Linear Hypotheses
Likelihood ratio test
Wald test
t-Tests and Confidence Intervals
Unique Estimation Using Restrictions
Exercises
Generalized Linear Models (GLMs)
Introduction
Structure of the Model
Distribution of y
Link function
Predictors
Linear models
Transforming Versus Linking
Estimation by Maximum Likelihood
Likelihood
Some useful identities
Likelihood equations
Large-sample variances
Solving the ML equations
Example: Potato flour dilutions
Tests of Hypotheses
Likelihood ratio tests
Wald tests
Illustration of tests
Confidence intervals
Illustration of confidence intervals
Maximum Quasi-Likelihood
Introduction
Definition
Exercises
Linear Mixed Models (LMMs)
A General Model
Introduction
Basic properties
Attributing Structure to Var(y)
Example
Taking covariances between factors as zero
The traditional variance components model
An LMM for longitudinal data
Estimating Fixed Effects for V Known
Estimating Fixed Effects for V Unknown
Estimation
Sampling variance
Bias in the variance
Approximate F-statistics
Predicting Random Effects for V Known
Predicting Random Effects for V Unknown
Estimation
Sampling variance
Bias in the variance
Anova Estimation of Variance Components
Balanced data
Unbalanced data
Maximum Likelihood (ML) Estimation
Estimators
Information matrix
Asymptotic sampling variances
Restricted Maximum Likelihood (REML)
Estimation
Sampling variances
Notes and Extensions
ML or REML?
Other methods for estimating variances
Appendix for Chapter 6
Differentiating a log likelihood
Differentiating a generalized inverse
Differentiation for the variance components model
Exercises
Generalized Linear Mixed Models
Introduction
Structure of the Model
Conditional distribution of y
Consequences of Having Random Effects
Marginal versus conditional distribution
Mean of y
Variances
Covariances and correlations
Estimation by Maximum Likelihood
Likelihood
Likelihood equations
Other Methods of Estimation
Penalized quasi-likelihood
Conditional likelihood
Simpler models
Tests of Hypotheses
Likelihood ratio tests
Asymptotic variances
Wald tests
Score tests
Illustration: Chestnut Leaf Blight
A random effects probit model
Exercises
Models for Longitudinal Data
Introduction
A Model for Balanced Data
Prescription
Estimating the mean
Estimating V[subscript 0]
A Mixed Model Approach
Fixed and random effects
Variances
Random Intercept and Slope Models
Variances
Within-subject correlations
Predicting Random Effects
Uncorrelated subjects
Uncorrelated between, and within, subjects
Uncorrelated between, and autocorrelated within
Random intercepts and slopes
Estimating Parameters
The general case
Uncorrelated subjects
Uncorrelated between, and autocorrelated within, subjects
Unbalanced Data
Example and model
Uncorrelated subjects
Models for Non-Normal Responses
Covariances and correlations
Estimation
Prediction of random effects
Binary responses, random intercepts and slopes
A Summary of Results
Balanced data
Unbalanced data
Appendix
For Section 8.4a
For Section 8.4b
Exercises
Marginal Models
Introduction
Examples of Marginal Regression Models
Generalized Estimating Equations
Models with marginal and conditional interpretations
Contrasting Marginal and Conditional Models
Exercises
Multivariate Models
Introduction
Multivariate Normal Outcomes
Non-Normally Distributed Outcomes
A multivariate binary model
A binary/normal example
A Poisson/Normal Example
Correlated Random Effects
Likelihood-Based Analysis
Example: Osteoarthritis Initiative
Notes and Extensions
Missing data
Efficiency
Exercises
Nonlinear Models
Introduction
Example: Corn Photosynthesis
Pharmacokinetic Models
Computations for Nonlinear Mixed Models
Exercises
Departures from Assumptions
Introduction
Incorrect Model for Response
Omitted covariates
Misspecified link functions
Misclassified binary outcomes
Informative cluster sizes
Incorrect Random Effects Distribution
Incorrect distributional family
Correlation of covariates and random effects
Covariate-dependent random effects variance
Diagnosing Misspecification
Conditional likelihood methods
Between/within cluster covariate decompositions
Specification tests
Nonparametric maximum likelihood
A Summary of Results
Exercises
Prediction
Introduction
Best Prediction (BP)
The best predictor
Mean and variance properties
A correlation property
Maximizing a mean
Normality
Best Linear Prediction (BLP)
BLP(u)
Example
Derivation
Ranking
Linear Mixed Model Prediction (BLUP)
BLUE(X[beta])
BLUP(t'X[beta] + s'u)
Two variances
Other derivations
Required Assumptions
Estimated Best Prediction
Henderson's Mixed Model Equations
Origin
Solutions
Use in ML estimation of variance components
Appendix
Verification of (13.5)
Verification of (13.7) and (13.8)
Exercises
Computing
Introduction
Computing ML Estimates for LMMs
The EM algorithm
Using E[uy]
Newton-Raphson method
Computing ML Estimates for GLMMs
Numerical quadrature
EM algorithm
Markov chain Monte Carlo algorithms
Stochastic approximation algorithms
Simulated maximum likelihood
Penalized Quasi-Likelihood and Laplace
Iterative Bootstrap Bias Correction
Exercises
Some Matrix Results
Vectors and Matrices of Ones
Kronecker (or Direct) Products
A Matrix Notation in Terms of Elements
Generalized Inverses
Definition
Generalized inverses of X'X
Two results involving X(X'V[superscript -1]X)[superscript -]X'V[superscript -1]
Solving linear equations
Rank results
Vectors orthogonal to columns of X
A theorem for K' with K'X being null
Differential Calculus
Definition
Scalars
Vectors
Inner products
Quadratic forms
Inverse matrices
Determinants
Some Statistical Results
Moments
Conditional moments
Mean of a quadratic form
Moment generating function
Normal Distributions
Univariate
Multivariate
Quadratic forms in normal variables
Exponential Families
Maximum Likelihood
The likelihood function
Maximum likelihood estimation
Asymptotic variance-covariance matrix
Asymptotic distribution of MLEs
Likelihood Ratio Tests
MLE Under Normality
Estimation of [beta]
Estimation of variance components
Asymptotic variance-covariance matrix
Restricted maximum likelihood (REML)
References
Index

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