Beginning with the comparatively simple random-intercept linear model
without covariates, Rabe-Hesketh and Skrondal develop the mixed model from
principles, thereby familiarizing the reader with terminology, summarizing
and relating the widely used estimating strategies, and providing historical
perspective.
Once the authors have established the mixed-model foundation, they smoothly
generalize to random-intercept models with covariates and then to
random-coefficient models. The middle chapters of the book apply the
concepts for Gaussian models to models for binary responses (e.g., logit and
probit), ordinal responses (e.g., ordered logit and ordered probit), and
count responses (e.g., Poisson). The authors then consider models with
multiple levels of random variation and models with crossed (nonnested)
random effects. In its examples and end-of-chapter exercises, the book
contains real datasets and data from the medical, social, and behavioral
sciences literature.
The book has several applications of generalized mixed models performed in
Stata. Rabe-Hesketh and Skrondal developed gllamm, a Stata program
that can fit many latent-variable models, of which the generalized linear
mixed model is a special case. With the release of version 9, Stata
introduced the xtmixed command for fitting linear (Gaussian) mixed
models. Stata users can use gllamm and xtmixed in conjunction
with the rest of the xt suite of commands to perform comparative
mixed-model analyses for various response families. The type of models fit
by these commands sometimes overlap; when this happens, the authors
highlight the differences in syntax, data organization, and output for the
two (or more) commands that can be used to fit the same model. The authors
also point out the relative strengths and weaknesses of each command when
used to fit the same model, based on considerations such as computational
speed, accuracy, and available predictions and postestimation statistics.
The book delineates the relationship between gllamm and
xtmixed, clearly showing how they complement one another.
In summary, this book is the most complete, up-to-date depiction of Stata's
capacity for fitting generalized linear mixed models. The authors provide an
ideal introduction for Stata users wishing to learn about this powerful
data-analysis tool.
1 Linear variance-components models
1.1 Introduction
1.2 How reliable are expiratory flow measurements?
1.3 The variance-components model
1.3.1 Model specification and path diagram
1.3.2 Error components, variance components, and reliability
1.3.3 Intraclass correlation
1.4 Modeling the Mini Wright measurements
1.4.1 Estimation using xtreg
1.4.2 Estimation using xtmixed
1.4.3 Estimation using gllamm
1.4.4 Relative and absolute agreement
1.5 Estimation methods
1.6 Assigning values to the random intercepts
1.6.1 Maximum likelihood estimation
Implementation via OLS regression
Implementation via the mean total residual
1.6.2 Empirical Bayes prediction
1.6.3 Empirical Bayes variances
1.7 Summary and further reading
1.8 Exercises
2 Linear random-intercept models
2.1 Introduction
2.2 Are tax preparers useful?
2.3 The longitudinal data structure
2.4 Panel data and correlated residuals
2.5 The random-intercept model
2.5.1 Estimation using xtreg
2.5.2 Estimation using xtmixed
2.6 Different kinds of effects in panel models
2.6.1 Between-taxpayer effects
2.6.2 Within-taxpayer effects
2.6.3 Relations among the estimators
2.7 Endogeneity and between-taxpayer effects
2.8 Residual diagnostics
2.9 Summary and further reading
2.10 Exercises
3 Linear random-coefficient and growth-curve models
3.1 Introduction
3.2 How effective are different schools?
3.3 Separate linear regressions for each school
3.4 The random-coefficient model
3.4.1 Specification and interpretation of a random-coefficient model
3.4.2 Estimation and prediction using xtmixed
Estimation of random-intercept model
Estimation of random-coefficient model
Empirical Bayes prediction using xtmixed
3.4.3 Estimation and prediction using gllamm
Estimation of random-intercept model
Estimation of random-coefficient model
Empirical Bayes prediction
3.5 How do children grow?
3.6 Growth-curve modeling
3.6.1 Observed growth trajectories
3.6.2 Estimation using xtmixed
Quadratic growth model with random intercept
Quadratic growth model with random intercept and random slope
Including a child-level covariate
3.6.3 Estimation using gllamm
Quadratic growth model with random intercept
Quadratic growth model with random intercept and random slope
Including a child-level covariate
3.7 Two-stage model formulation
3.7.1 Model specification
3.7.2 Estimation
3.8 Prediction of trajectories for individual children
3.9 Complex level-1 variation or heteroskedasticity
3.10 Summary and further reading
3.11 Exercises
4 Dichotomous or binary responses
4.1 Models for dichotomous responses
4.1.1 Generalized linear model formulation
4.1.2 Latent-response formulation
Logistic regression
Probit regression
4.2 Which treatment is best for toenail infection?
4.3 The longitudinal data structure
4.4 Population-averaged or marginal probabilities
4.5 Random-intercept logistic regression
4.6 Subject-specific vs. population-averaged relationships
4.7 Maximum likelihood estimation using adaptive quadrature
4.7.1 Some practical considerations
4.8 Empirical Bayes (EB) predictions
4.8.1 EB prediction of random effects
4.8.2 EB prediction of response probabilities
4.9 Other approaches to clustered dichotomous data
4.9.1 Conditional logistic regression
4.9.2 Generalized estimating equations (GEE)
4.10 Summary and further reading
4.11 Exercises
5 Ordinal responses
5.1 Introduction
5.2 Cumulative models for ordinal responses
5.2.1 Generalized linear model formulation
5.2.2 Latent-response formulation
5.2.3 Proportional odds
5.2.4 Identification
5.3 Are antipsychotic drugs effective for patients with schizophrenia?
5.4 Longitudinal data structure and graphs
5.4.1 The longitudinal data structure
5.4.2 Plotting cumulative proportions
5.4.3 Plotting cumulative logits and transforming the time scale
5.5 A proportional-odds model
5.5.1 Model specification
5.5.2 Estimation
5.6 A random-intercept proportional-odds model
5.6.1 Model specification
5.6.2 Estimation
5.7 A random-coefficient proportional-odds model
5.7.1 Model specification
5.7.2 Estimation
5.8 Marginal and patient-specific probabilities
5.8.1 Marginal probabilities
5.8.2 Patient-specific cumulative response probabilities
5.9 Do experts differ in their grading of student essays?
5.10 A random-intercept model with grader bias
5.10.1 Model specification
5.10.2 Estimation
5.11 Including grader-specific measurement error variances
5.11.1 Model specification
5.11.2 Estimation
5.12 Including grader-specific thresholds
5.12.1 Model specification
5.12.2 Estimation
5.13 Summary and further reading
5.14 Exercises
6 Counts
6.1 Introduction
6.2 Types of counts
6.3 Poisson model for counts
6.4 Did the German health-care reform reduce the number of doctor visits?
6.5 Longitudinal data structure
6.6 Poisson regression ignoring overdispersion and clustering
6.6.1 Model specification
6.6.2 Estimation
6.7 Poisson regression with overdispersion but ignoring clustering
6.7.1 Using a level-1 random intercept
Model specification
Estimation
6.7.2 Quasilikelihood
Specification
Estimation
6.8 Random-intercept Poisson regression
6.8.1 Model specification
6.8.2 Estimation
6.9 Random-coefficient Poisson regression
6.9.1 Model specification
6.9.2 Estimation
6.10 Other approaches to clustered counts
6.10.1 Conditional Poisson regression
6.10.2 Generalized estimating equations (GEE)
6.11 Which Scottish countries have a high risk of lip cancer?
6.12 Standardized mortality ratios
6.13 Random-intercept Poisson regression
6.13.1 Model specification
6.13.2 Estimation
6.13.3 Introducing a county-level covariate
6.13.4 Prediction
6.14 Nonparametric maximum likelihood estimation
6.14.1 Specification
6.14.2 Estimation
6.14.3 Prediction
6.15 Summary and further reading
6.16 Exercises
7 Higher level models and nested random effects
7.1 Introduction
7.2 Which method is best for measuring expiratory flow?
7.3 Two-level variance-components models
7.3.1 Model specification
7.3.2 Estimation
7.4 Three-level variance-components models
7.4.1 Model specification
7.4.2 Different types of intraclass correlation
7.4.3 Three-stage formulation
7.4.4 Estimation using xtmixed
7.4.5 Prediction using xtmixed
7.5 Did the Guatemalan immunization campaign work?
7.6 A three-level logistic random-intercept model
7.6.1 Model specification
7.6.2 Different types of intraclass correlations for the latent responses
7.6.3 Three-stage formulation
7.6.4 Estimation
7.6.5 Introducing a random coefficient at level 3
7.6.6 Prediction
7.7 Summary and further reading
7.8 Exercises
8 Crossed random effects
8.1 Introduction
8.2 How does investment depend on expected profit and capital stock?
8.3 A two-way error-components model
8.3.1 Model specification
8.3.2 Intraclass correlations
8.3.3 Estimation
8.3.4 Prediction
8.4 How much do primary and secondary schools affect attainment at age 16?
8.5 An additive crossed random-effects model
8.5.1 Specification
8.5.2 Estimation
8.6 Including a random interaction
8.6.1 Model specification
8.6.2 Intraclass correlations
8.6.3 Estimation
8.6.4 Some diagnostics
8.7 A trick requiring fewer random effects
8.8 Summary and further reading
8.9 Exercises
A Syntax for gllamm, eq, and gllapred
B Syntax for gllamm
C Syntax for gllapred
D Syntax for gllasim
References