The second edition has much to offer for readers of the first edition,
reading more like a sequel than an update. The text has almost doubled in
length from the original, coming in at 562 pages. This second edition
incorporates three new chapters: a chapter on standard linear regression, a
chapter on discrete-time survival analysis, and a chapter on longitudinal
and panel data containing an expanded discussion of random-coefficient and
growth-curve models. The authors have updated this edition for Stata 10,
expanding on discussions in the original edition and adding new in-text
examples and end-of-chapter exercises. In particular, the authors have
thoroughly covered the new Stata commands xtmelogit and
xtmepoisson.
The first chapter provides a review of the methods of linear regression.
Rabe-Hesketh and Skrondal then begin with the comparatively simple
random-intercept linear model without covariates, developing the mixed model
from principles and 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 a
discussion of the various estimators (between, within, and random-effects).
The authors then discuss models with random coefficients, followed by models
for growth curves. 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 text continues with a discussion of how to use multilevel methods in
discrete-time survival analysis, for example, using complimentary log-log
regression to fit the proportional hazards model. 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. As of version 10, Stata contains the
xtmixed, xtmelogit, and xtmepoisson commands for
fitting multilevel models, in addition to other xt commands for
fitting standard random-intercept models. 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,
available predictions, and available postestimation statistics.
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.
List of Tables
List of Figures
I Preliminaries
1 Review of linear regression
1.1 Introduction
1.2 Is there gender discrimination in faculty salaries?
1.3 Independent-samples t test
1.4 One-way analysis of variance
1.5 Simple linear regression
1.6 Dummy variables
1.7 Multiple linear regression
1.8 Interactions
1.9 Dummies for more than two groups
1.10 Other types of interactions
1.10.1 Interaction between dummy variables
1.10.2 Interaction between continuous covariates
1.11 Nonlinear effects
1.12 Residual diagnostics
1.13 Summary and further reading
1.14 Exercises
II Two-level linear models
2 Variance-components models
2.1 Introduction
2.2 How reliable are peak-expiratory-flow measurements
2.3 The variance-components model
2.3.1 Model specification and path diagram
2.3.2 Error components, variance components, and reliability
2.3.3 Intraclass correlation
2.4 Fixed versus random effects
2.5 Estimation using Stata
2.5.1 Data preparation
2.5.2 Using xtreg
2.5.3 Using xtmixed
2.5.4 Using gllamm
2.6 Hypothesis tests and confidence intervals
2.6.1 Hypothesis test and confidence interval for the population mean
2.6.2 Hypothesis test and confidence interval for the between-cluster variance
2.7 More on statistical inference
2.7.1 Different estimation models
2.7.2 Inference for Β
Estimate and standard error: Balanced case
Estimate: Unbalanced case
2.8 Crossed versus nested effects
2.9 Assigning values to the random intercepts
2.9.1 Maximum likelihood estimation
Implementation via OLS regression
Implementation via the mean total residual
2.9.2 Empirical Bayes prediction
2.9.3 Empirical Bayes variances
2.10 Summary and further reading
2.11 Exercises
3 Random-intercept models with covariates
3.1 Introduction
3.2 Does smoking during pregnancy affect birthweight?
3.3 The linear random-intercept model with covariates
3.3.1 Model specification
3.3.2 Residual variance and intraclass correlation
3.4 Estimation using Stata
3.4.1 Using xtreg
3.4.2 Using xtmixed
3.4.3 Using gllamm
3.5 Coefficients of determination or variance explained
3.6 Hypothesis tests and confidence intervals
3.6.1 Hypothesis tests for regression coefficients
Hypothesis tests for individual regression coefficients
Joint hypothesis tests for several regression coefficients
3.6.2 Predicted means and confidence intervals
3.6.3 Hypothesis test for between-cluster variance
3.7 Between and within effects
3.7.1 Between-mother effects
3.7.2 Within-mother effects
3.7.3 Relations among estimators
3.7.4 Endogeneity and different within- and between-mother effects
3.7.5 Hausman endogeneity test
3.8 Fixed versus random effects revisited
3.9 Residual diagnostics
3.10 More on statistical inference for regression coefficients
3.10.1 Consequences of using ordinary regression for clustered data
3.10.2 Power and sample-size determination
3.11 Summary and further reading
3.12 Exercises
4 Random-coefficient models
4.1 Introduction
4.2 How effective are different schools
4.3 Separate linear regressions for each school
4.4 Specification and interpretation of a random-coefficient model
4.4.1 Specification of random-coefficient model
4.4.2 Interpretation of the random-effects variances and covariances
4.5 Estimation using Stata
4.5.1 Using xtmixed
Random-intercept model
Random-coefficient model
4.5.2 Using gllamm
Random-intercept model
Random-coefficient model
4.6 Testing the slope variance
4.7 Interpretation of estimates
4.8 Assigning values to the random intercepts and slopes
4.8.1 Maximum likelihood estimation
4.8.2 Empirical Bayes prediction
4.8.3 Model visualization
4.8.4 Residual diagnostics
4.8.5 Inferences for individual schools
4.9 Two-stage model formulation
4.10 Some warnings about random-coefficient models
4.11 Summary and further reading
4.12 Exercises
5 Longitudinal, panel, and growth-curve models
5.1 Introduction
5.2 How and why do wages change over time?
5.3 Data structure
5.3.1 Missing data
5.3.2 Time-varying and time-constant variables
5.4 Time scales in longitudinal data
5.5 Random- and fixed-effects approaches
5.5.1 Correlated residuals
5.5.2 Fixed-intercept model
Using xtreg
Using anova
5.5.3 Random-intercept model
5.5.4 Random-coefficient model
5.5.5 Marginal mean and covariance structure induced by random effects
Marginal mean and covariance structure for random-intercept models
Marginal mean and covariance structure for random-coefficient models
5.6 Marginal modeling
5.6.1 Covariance structures
Compound symmetric or exchangeable structure
Random-coefficient structure
Autoregressive residual structure
Unstructured covariance matrix
5.6.2 Marginal modeling using Stata
5.7 Autoregressive- or lagged-response models
5.8 Hybrid approaches
5.8.1 Autoregressive response and random effects
5.8.2 Autoregressive responses and autoregressive residuals
5.8.3 Autoregressive residuals and random or fixed effects
5.9 Missing data
5.9.1 Maximum likelihood estimation under MAR: A simulation
5.10 How do children grow?
5.10.1 Observed growth trajectories
5.11 Growth-curve modeling
5.11.1 Random-intercept model
5.11.2 Random-coefficient model
5.11.3 Two-stage model formulation
5.12 Prediction of trajectories for individual children
5.13 Prediction of mean growth trajectory and 95% band
5.14 Complex level-1 variation or heteroskedasticity
5.15 Summary and further reading
5.16 Exercises
III Two-level generalized linear models
6 Dichotomous or binary responses
6.1 Introduction
6.2 Single-level models for dichotomous responses
6.2.1 Generalized linear model formulation
6.2.2 Latent-response formulation
Logistic regression
Probit regression
6.3 Which treatment is best for toenail infection?
6.4 Longitudinal data structure
6.5 Population-averaged or marginal probabilities
6.6 Random-intercept logistic regression
6.7 Estimation of logistic random-intercept models
6.7.1 Using xtlogit
6.7.2 Using xtmelogit
6.7.3 Using gllamm
6.8 Inference for logistic random-intercept models
6.9 Subject-specific vs. population-averaged relationships
6.10 Measures of dependence and heterogeneity
6.10.1 Conditional or residual intraclass correlation of the latent responses
6.10.2 Median odds ratio
6.11 Maximum likelihood estimation
6.11.1 Adaptive quadrature
6.11.2 Some speed considerations
Advice for speeding up gllamm
6.12 Assigning values to random effects
6.12.1 Maximum likelihood estimation
6.12.2 Empirical Bayes prediction
6.12.3 Empirical Bayes modal prediction
6.13 Different kinds of predicted probabilities
6.13.1 Predicted population-averaged probabilities
6.13.2 Predicted subject-specific probabilities
Predictions for hypothetical subjects
Predictions for the subjects in the sample
6.14 Other approaches to clustered dichotomous data
6.14.1 Conditional logistic regression
6.14.2 Generalized estimating equations (GEE)
6.15 Summary and further reading
6.16 Exercises
7 Ordinal responses
7.1 Introduction
7.2 Single-level cumulative models for ordinal responses
7.2.1 Generalized linear model formulation
7.2.2 Latent-response formulation
7.2.3 Proportional odds
7.2.4 Identification
7.3 Are antipsychotic drugs effective for patients with schizophrenia?
7.4 Longitudinal data structure and graphs
7.4.1 Longitudinal data structure
7.4.2 Plotting cumulative proportions
7.4.3 Plotting estimated cumulative logits and transforming the time scale
7.5 A single-level proportional odds model
7.5.1 Model specification
7.5.2 Estimation using Stata
7.6 A random-intercept proportional odds model
7.6.1 Model specification
7.6.2 Estimation using Stata
7.7 A random-intercept proportional odds model
7.7.1 Model specification
7.7.2 Estimation using gllamm
7.8 Different kinds of predicted probabilities
7.8.1 Predicted population-averaged probabilities
7.8.2 Predicted patient-specific probabilities
7.9 Do experts differ in the grading of student essays?
7.10 A random-intercept probit model with grader bias
7.10.1 Model specification
7.10.2 Estimation
7.11 Including grader-specific measurement error variances
7.11.1 Model specification
7.11.2 Estimation
7.12 Including grader-specific thresholds
7.12.1 Model specification
7.12.2 Estimation
7.13 Summary and further reading
7.14 Exercises
8 Discrete-time survival
8.1 Introduction
8.1.1 Censoring and truncation
8.1.2 Time-varying covariates and different time-scales
8.1.3 Discrete- versus continuous-time survival data
8.2 Single-level models for discrete-time survival data
8.2.1 Discrete-time hazard and discrete-time survival
8.2.2 Data expansion for discrete-time survival analysis
8.2.3 Estimation via regression models for dichotomous responses
8.2.4 Including covariates
Time-constant covariates
Time-varying covariates
8.2.5 Handling left-truncated data
8.3 How does birth history affect child mortality?
8.4 Data expansion
8.5 Proportional hazards and interval censoring
8.6 Complementary log-log models
8.7 A random-intercept complementary log-log model
8.7.1 Model specification
8.7.2 Estimation using Stata
8.8 Marginal and conditional survival probabilities
8.9 Summary and further reading
8.10 Exercises
9 Counts
9.1 Introduction
9.2 What are counts?
9.2.1 Counts versus proportions
9.2.2 Counts as aggregated event-history data
9.3 Single-level Poisson models for counts
9.4 Did the German health-care reform reduce the number of doctor visits?
9.5 Longitudinal data structure
9.6 Single-level Poisson regression
9.6.1 Model specification
9.6.2 Estimation using Stata
9.7 Random-intercept Poisson regression
9.7.1 Model specification
9.7.2 Estimation using Stata
Using xtpoisson
Using xtmepoisson
Using gllamm
9.8 Random-coefficient Poisson regression
9.8.1 Model specification
9.8.2 Estimation using Stata
Using xtmepoisson
Using gllamm
9.8.3 Interpretation of estimates
9.9 Overdispersion in single-level models
9.9.1 Normally distributed random intercept
9.9.2 Negative binomial models
Mean dispersion or NB2
Constant dispersion or NB1
9.9.3 Quasilikelihood or robust standard errors
9.10 Level-1 overdispersion in two-level models
9.11 Other approaches to two-level count data
9.11.1 Conditional Poisson regression
9.11.2 Conditional negative binomial regression
9.11.3 Generalized estimating equations
9.11.4 Marginal and conditional estimates when responses are MAR
Simulation
9.12 How does birth history affect child mortality?
9.12.1 Simple piecewise exponential survival model
9.12.2 Piecewise exponential survival model with covariates and frailty
9.13 Which Scottish counties have a high risk of lip cancer?
9.14 Standardized mortality ratios
9.15 Random-intercept Poisson regression
9.15.1 Model specification
9.15.2 Estimation using gllamm
9.15.3 Prediction of standardized mortality ratios
9.16 Nonparametric maximum likelihood estimation
9.16.1 Specification
9.16.2 Estimation using gllamm
9.16.3 Prediction
9.17 Summary and further reading
9.18 Exercises
IV Models with nested and crossed random effects
10 Higher-level models with nested random effects
10.1 Introduction
10.2 Do peak-expiratory-flow measurements vary between methods?
10.3 Two-level variance-components models
10.3.1 Model specification
10.3.2 Estimation using xtmixed
10.4 Three-level variance-components models
10.4.1 Model specification
10.4.2 Different types of intraclass correlation
10.4.3 Three-stage formulation
10.4.4 Estimating using xtmixed
10.4.5 Empirical Bayes prediction using xtmixed
10.5 Did the Guatemalan immunization campaign work?
10.6 A three-level logistic random-intercept model
10.6.1 Model specification
10.6.2 Different types of intraclass correlations for the latent responses
10.6.3 Different kinds of median odds ratios
10.6.4 Three-stage formulation
10.7 Estimation of three-level logistic random-intercept models using Stata
10.7.1 Using gllamm
10.7.2 Using xtmelogit
10.8 A three-level logistic random-coefficient model
10.9 Estimation of three-level logistic random-coefficient models using Stata
10.9.1 Using gllamm
10.9.2 Using xtmelogit
10.10 Prediction of random effects
10.10.1 Empirical Bayes prediction
10.10.2 Empirical Bayes modal prediction
10.11 Different kinds of predicted probabilities
10.11.1 Predicted marginal probabilities
10.11.2 Predicted median or conditional probabilities
10.11.e Predicted posterior mean probabilities
10.12 Summary and further reading
10.13 Exercises
11 Crossed random effects
11.1 Introduction
11.2 How does investment depend on expected profit and capital stock?
11.3 A two-way error-components model
11.3.1 Models specification
11.3.2 Residual intraclass correlations
11.3.3 Estimation
11.3.4 Prediction
11.4 How much do primary and secondary schools affect attainment at age 16?
11.5 An additive crossed random-effects model
11.5.1 Specification
11.5.2 Estimation using xtmixed
11.6 Including a random interaction
11.6.1 Model specification
11.6.2 Intraclass correlations
11.6.3 Estimation using xtmixed
11.6.4 Some diagnostics
11.7 A trick requiring fewer random effects
11.8 Do salamanders from different populations mate successfully?
11.9 Crossed random-effects logistic regression
11.10 Summary and further reading
11.11 Exercises
A Syntax for gllamm, eq, and gllapred: The bare essentials
B Syntax for gllamm
C Syntax for gllapred
D Syntax for gllasim
References