Multilevel and Longitudinal Modeling Using Stata, Third Edition

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Volume I: Continuous Responses

Volume II: Categorical Responses, Counts, and Survival


Authors: Sophia Rabe-Hesketh and Anders Skrondal
Publisher: Stata Press
Copyright: 2012
ISBN-13: 978-1-59718-108-2
Pages: 974; paperback

Volume I: Continuous Responses

ISBN-13: 978-1-59718-103-7
Pages: 497; paperback

Author index for Volume I (PDF) 
Subject index for Volume I (PDF) 
Preface (PDF) 
gllamm companion for Volume I (PDF) 


Volume II: Categorical Responses, Counts, and Survival

ISBN-13: 978-1-59718-104-4
Pages: 477; paperback

Author index for Volume II (PDF) 
Subject index for Volume II (PDF) 
Preface (PDF) 
Chapter 10—Dichotomous or binary responses(PDF) 

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Multilevel and Longitudinal Modeling Using Stata, Third Edition, by Sophia Rabe-Hesketh and Anders Skrondal, looks specifically at Stata’s treatment of generalized linear mixed models, also known as multilevel or hierarchical models. These models are “mixed” because they allow fixed and random effects, and they are “generalized” because they are appropriate for continuous Gaussian responses as well as binary, count, and other types of limited dependent variables.


The material in the third edition consists of two volumes, a result of the substantial expansion of material from the second edition, and has much to offer readers of the earlier editions. The text has almost doubled in length from the second edition and almost quadrupled in length from the original version to almost 1,000 pages across the two volumes. Fully updated for Stata 12, the book has 5 new chapters and many new exercises and datasets.


The two volumes comprise 16 chapters organized into eight parts.


Volume I is devoted to continuous Gaussian linear mixed models and has nine chapters organized into four parts. The first part reviews the methods of linear regression. The second part provides in-depth coverage of two-level models, the simplest extensions of a linear regression model.


Rabe-Hesketh and Skrondal 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.


The third part of volume I describes models for longitudinal and panel data, including dynamic models, marginal models (a new chapter), and growth-curve models (a new chapter). The fourth and final part covers models with nested and crossed random effects, including a new chapter describing in more detail higher-level nested models for continuous outcomes.


The mixed-model foundation and the in-depth coverage of the mixed-model principles provided in volume I for continuous outcomes make it straightforward to transition to generalized linear mixed models for noncontinuous outcomes, which are described in volume II.


Volume II is devoted to generalized linear mixed models for binary, categorical, count, and survival outcomes. The second volume has seven chapters also organized into four parts. The first three parts in volume II cover models for categorical responses, including binary, ordinal, and nominal (a new chapter); models for count data; and models for survival data, including discrete-time and continuous-time (a new chapter) survival responses. The fourth and final part in volume II describes models with nested and crossed-random effects with an emphasis on binary outcomes.


The book has extensive 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 xtmixedxtmelogit, and xtmepoisson commands for fitting multilevel models, in addition to other xt commands for fitting standard random-intercept models. The types 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.


Sophia Rabe-Hesketh is a professor of educational statistics and biostatistics at the University of California at Berkeley and a chair of social statistics at the Institute of Education, University of London.


Anders Skrondal is a senior biostatistician at the Division of Epidemiology, Norwegian Institute of Public Health. He was previously a professor of statistics and director of the Methodology Institute at the London School of Economics and a professor of biostatistics at the University of Oslo.


List of Tables
List of Figures
Preface (PDF)
Multilevel and longitudinal models: When and why?
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 Dummy variables 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 Causal and noncausal interpretations of regression coefficients 
1.13.1 Regression as conditional expectation 
1.13.2 Regression as structural model 
1.14 Summary and further reading 
1.15 Exercises 
II Two-level models
2 Variance-components models
2.1 Introduction 
2.2 How reliable are peak-expiratory-flow measurements? 
2.3 Inspecting within-subject dependence 
2.4 The variance-components model 
2.4.1 Model specification 
2.4.2 Path diagram 
2.4.3 Between-subject heterogeneity 
2.4.4 Within-subject dependence 
Intraclass correlation 
Intraclass correlation versus Pearson correlation 
2.5 Estimation using Stata 
2.5.1 Data preparation: Reshaping to long form 
2.5.2 Using xtreg 
2.5.3 Using xtmixed 
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 
Likelihood-ratio test 
F test 
Confidence intervals 
2.7 Model as data-generating mechanism 
2.8 Fixed versus random effects 
2.9 Crossed versus nested effects 
2.10 Parameter estimation 
2.10.1 Model assumptions 
Mean structure and covariance structure 
Distributional assumptions 
2.10.2 Different estimation methods 
2.10.3 Inference for β
Estimate and standard error: Balanced case 
Estimate: Unbalanced case 
2.11 Assigning values to the random intercepts 
2.11.1 Maximum “likelihood” estimation 
Implementation via OLS regression 
Implementation via the mean total residual 
2.11.2 Empirical Bayes prediction 
2.11.3 Empirical Bayes standard errors 
Comparative standard errors 
Diagnostic standard errors 
2.12 Summary and further reading 
2.13 Exercises 
3 Random-intercept models with covariates
3.1 Introduction 
3.2 Does smoking during pregnancy affect birthweight? 
3.2.1 Data structure and descriptive statistics 
3.3 The linear random-intercept model with covariates 
3.3.1 Model specification 
3.3.2 Model assumptions 
3.3.3 Mean structure 
3.3.4 Residual variance and intraclass correlation 
3.3.5 Graphical illustration of random-intercept model 
3.4 Estimation using Stata 
3.4.1 Using xtreg 
3.4.2 Using xtmixed 
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 random-intercept variance 
3.7 Between and within effects of level-1 covariates 
3.7.1 Between-mother effects 
3.7.2 Within-mother effects 
3.7.3 Relations among estimators 
3.7.4 Level-2 endogeneity and cluster-level confounding 
3.7.5 Allowing for different within and between effects 
3.7.6 Hausman endogeneity test 
3.8 Fixed versus random effects revisited 
3.9 Assigning values to random effects: Residual diagnostics 
3.10 More on statistical inference 
3.10.1 Overview of estimation methods 
3.10.2 Consequences of using standard regression modeling for clustered data 
3.10.3 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 a random-coefficient model 
4.4.2 Interpretation of the random-effects variances and covariances 
4.5 Estimation using xtmixed 
4.5.1 Random-intercept model 
4.5.2 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 
4.10.1 Meaningful specification 
4.10.2 Many random coefficients 
4.10.3 Convergence problems 
4.10.4 Lack of identification 
4.11 Summary and further reading 
4.12 Exercises 
III Models for longitudinal and panel data
Introduction to models for longitudinal and panel data (part III)
5 Subject-specific effects and dynamic models
5.1 Introduction 
5.2 Conventional random-intercept model 
5.3 Random-intercept models accommodating endogenous covariates 
5.3.1 Consistent estimation of effects of endogenous time-varying covariates 
5.3.2 Consistent estimation of effects of endogenous time-varying and endogenous time-constant covariates 
5.4 Fixed-intercept model 
5.4.1 Using xtreg or regress with a differencing operator 
5.4.2 Using anova 
5.5 Random-coefficient model 
5.6 Fixed-coefficient model 
5.7 Lagged-response or dynamic models 
5.7.1 Conventional lagged-response model 
5.7.2 Lagged-response model with subject-specific intercepts 
5.8 Missing data and dropout 
5.8.1 Maximum likelihood estimation under MAR: A simulation 
5.9 Summary and further reading 
5.10 Exercises 
6 Marginal models
6.1 Introduction 
6.2 Mean structure 
6.3 Covariance structures 
6.3.1 Unstructured covariance matrix 
6.3.2 Random-intercept or compound symmetric/exchangeable structure 
6.3.3 Random-coefficient structure 
6.3.4 Autoregressive and exponential structures 
6.3.5 Moving-average residual structure 
6.3.6 Banded and Toeplitz structures 
6.4 Hybrid and complex marginal models 
6.4.1 Random effects and correlated level-1 residuals 
6.4.2 Heteroskedastic level-1 residuals over occasions 
6.4.3 Heteroskedastic level-1 residuals over groups 
6.4.4 Different covariance matrices over groups 
6.5 Comparing the fit of marginal models 
6.6 Generalized estimating equations (GEE) 
6.7 Marginal modeling with few units and many occasions 
6.7.1 Is a highly organized labor market beneficial for economic growth? 
6.7.2 Marginal modeling for long panels 
6.7.3 Fitting marginal models for long panels in Stata 
6.8 Summary and further reading 
6.9 Exercises 
7 Growth-curve models
7.1 Introduction 
7.2 How do children grow? 
7.2.1 Observed growth trajectories 
7.3 Models for nonlinear growth 
7.3.1 Polynomial models 
Fitting the models 
Predicting the mean trajectory 
Predicting trajectories for individual children 
7.3.2 Piecewise linear models 
Fitting the models 
Predicting the mean trajectory 
7.4 Two-stage model formulation 
7.5 Heteroskedasticity 
7.5.1 Heteroskedasticity at level 1 
7.5.2 Heteroskedasticity at level 2 
7.6 How does reading improve from kindergarten through third grade? 
7.7 Growth-curve model as a structural equation model 
7.7.1 Estimation using sem 
7.7.2 Estimation using xtmixed 
7.8 Summary and further reading 
7.9 Exercises 
IV Models with nested and crossed random effects
8 Higher-level models with nested random effects
8.1 Introduction 
8.2 Do peak-expiratory-flow measurements vary between methods within subjects? 
8.3 Inspecting sources of variability 
8.4 Three-level variance-components models 
8.5 Different types of intraclass correlation 
8.6 Estimation using xtmixed 
8.7 Empirical Bayes prediction 
8.8 Testing variance components 
8.9 Crossed versus nested random effects revisited 
8.10 Does nutrition affect cognitive development of Kenyan children? 
8.11 Describing and plotting three-level data 
8.11.1 Data structure and missing data 
8.11.2 Level-1 variables 
8.11.3 Level-2 variables 
8.11.4 Level-3 variables 
8.11.5 Plotting growth trajectories 
8.12 Three-level random-intercept model 
8.12.1 Model specification: Reduced form 
8.12.2 Model specification: Three-stage formulation 
8.12.3 Estimation using xtmixed 
8.13 Three-level random-coefficient models 
8.13.1 Random coefficient at the child level 
8.13.2 Random coefficient at the child and school levels 
8.14 Residual diagnostics and predictions 
8.15 Summary and further reading 
8.16 Exercises 
9 Crossed random effects
9.1 Introduction 
9.2 How does investment depend on expected profit and capital stock? 
9.3 A two-way error-components model 
9.3.1 Model specification 
9.3.2 Residual variances, covariances, and intraclass correlations 
Longitudinal correlations 
Cross-sectional correlations 
9.3.3 Estimation using xtmixed 
9.3.4 Prediction 
9.4 How much do primary and secondary schools affect attainment at age 16? 
9.5 Data structure 
9.6 Additive crossed random-effects model 
9.6.1 Specification 
9.6.2 Estimation using xtmixed 
9.7 Crossed random-effects model with random interaction 
9.7.1 Model specification 
9.7.2 Intraclass correlations 
9.7.3 Estimation using xtmixed 
9.7.4 Testing variance components 
9.7.5 Some diagnostics 
9.8 A trick requiring fewer random effects 
9.9 Summary and further reading 
9.10 Exercises 
A Useful Stata commands
List of Tables
List of Figures
V Models for categorical responses
10.1 Introduction 
10.2 Single-level logit and probit regression models for dichotomous responses 
10.2.1 Generalized linear model formulation 
10.2.2 Latent-response formulation 
Logistic regression 
Probit regression 
10.3 Which treatment is best for toenail infection? 
10.4 Longitudinal data structure 
10.5 Proportions and fitted population-averaged or marginal probabilities 
10.6 Random-intercept logistic regression 
10.6.1 Model specification 
Reduced-form specification 
Two-stage formulation 
10.7 Estimation of random-intercept logistic models 
10.7.1 Using xtlogit 
10.7.2 Using xtmelogit 
10.7.3 Using gllamm 
10.8 Subject-specific or conditional vs. population-averaged or marginal relationships 
10.9 Measures of dependence and heterogeneity 
10.9.1 Conditional or residual intraclass correlation of the latent responses 
10.9.2 Median odds ratio 
10.9.3 Measures of association for observed responses at median fixed part of the model 
10.10 Inference for random-intercept logistic models 
10.10.1 Tests and confidence intervals for odds ratios 
10.10.2 Tests of variance components 
10.11 Maximum likelihood estimation 
10.11.1 Adaptive quadrature 
10.11.2 Some speed and accuracy considerations 
Advice for speeding up estimation in gllamm 
10.12 Assigning values to random effects 
10.12.1 Maximum “likelihood” estimation 
10.12.2 Empirical Bayes prediction 
10.12.3 Empirical Bayes modal prediction 
10.13 Different kinds of predicted probabilities 
10.13.1 Predicted population-averaged or marginal probabilities 
10.13.2 Predicted subject-specific probabilities 
Predictions for hypothetical subjects: Conditional probabilities 
Predictions for the subjects in the sample: Posterior mean probabilities 
10.14 Other approaches to clustered dichotomous data 
10.14.1 Conditional logistic regression 
10.14.2 Generalized estimating equations (GEE) 
10.15 Summary and further reading 
10.16 Exercises 
11 Ordinal responses
11.1 Introduction 
11.2 Single-level cumulative models for ordinal responses 
11.2.1 Generalized linear model formulation 
11.2.2 Latent-response formulation 
11.2.3 Proportional odds 
11.2.4 Identification 
11.3 Are antipsychotic drugs effective for patients with schizophrenia? 
11.4 Longitudinal data structure and graphs 
11.4.1 Longitudinal data structure 
11.4.2 Plotting cumulative proportions 
11.4.3 Plotting cumulative sample logits and transforming the time scale 
11.5 A single-level proportional odds model 
11.5.1 Model specification 
11.5.2 Estimation using Stata 
11.6 A random-intercept proportional odds model 
11.6.1 Model specification 
11.6.2 Estimation using Stata 
11.6.3 Measures of dependence and heterogeneity 
Residual intraclass correlation of latent responses 
Median odds ratio 
11.7 A random-coefficient proportional odds model 
11.7.1 Model specification 
11.7.2 Estimation using gllamm 
11.8 Different kinds of predicted probabilities 
11.8.1 Predicted population-averaged or marginal probabilities 
11.8.2 Predicted subject-specific probabilities: Posterior mean 
11.9 Do experts differ in their grading of student essays? 
11.10 A random-intercept probit model with grader bias 
11.10.1 Model specification 
11.10.2 Estimation using gllamm 
11.11 Including grader-specific measurement error variances 
11.11.1 Model specification 
11.11.2 Estimation using gllamm 
11.12 Including grader-specific thresholds 
11.12.1 Model specification 
11.12.2 Estimation using gllamm 
11.13 Other link functions 
Cumulative complementary log-log model 
Continuation-ratio logit model 
Adjacent-category logit model 
Baseline-category logit and stereotype models 
11.14 Summary and further reading 
11.15 Exercises 
12 Nominal responses and discrete choice
12.1 Introduction 
12.2 Single-level models for nominal responses 
12.2.1 Multinomial logit models 
12.2.2 Conditional logit models 
Classical conditional logit models 
Conditional logit models also including covariates that vary only over units 
12.3 Independence from irrelevant alternatives 
12.4 Utility-maximization formulation 
12.5 Does marketing affect choice of yogurt? 
12.6 Single-level conditional logit models 
12.6.1 Conditional logit models with alternative-specific intercepts 
12.7 Multilevel conditional logit models 
12.7.1 Preference heterogeneity: Brand-specific random intercepts 
12.7.2 Response heterogeneity: Marketing variables with random coefficients 
12.7.3 Preference and response heterogeneity 
Estimation using gllamm 
Estimation using mixlogit 
12.8 Prediction of random effects and response probabilities 
12.9 Summary and further reading 
12.10 Exercises 
VI Models for counts
13 Counts
13.1 Introduction 
13.2 What are counts? 
13.2.1 Counts versus proportions 
13.2.2 Counts as aggregated event-history data 
13.3 Single-level Poisson models for counts 
13.4 Did the German health-care reform reduce the number of doctor visits? 
13.5 Longitudinal data structure 
13.6 Single-level Poisson regression 
13.6.1 Model specification 
13.6.2 Estimation using Stata 
13.7 Random-intercept Poisson regression 
13.7.1 Model specification 
13.7.2 Measures of dependence and heterogeneity 
13.7.3 Estimation using Stata 
Using xtpoisson 
Using xtmepoisson 
Using gllamm 
13.8 Random-coefficient Poisson regression 
13.8.1 Model specification 
13.8.2 Estimation using Stata 
Using xtmepoisson 
Using gllamm 
13.8.3 Interpretation of estimates 
13.9 Overdispersion in single-level models 
13.9.1 Normally distributed random intercept 
13.9.2 Negative binomial models 
Mean dispersion or NB2 
Constant dispersion or NB1 
13.9.3 Quasilikelihood 
13.10 Level-1 overdispersion in two-level models 
13.11 Other approaches to two-level count data 
13.11.1 Conditional Poisson regression 
13.11.2 Conditional negative binomial regression 
13.11.3 Generalized estimating equations 
13.12 Marginal and conditional effects when responses are MAR 
13.13 Which Scottish counties have a high risk of lip cancer? 
13.14 Standardized mortality ratios 
13.15 Random-intercept Poisson regression 
13.15.1 Model specification 
13.15.2 Estimation using gllamm 
13.15.3 Prediction of standardized mortality ratios 
13.16 Nonparametric maximum likelihood estimation 
13.16.1 Specification 
13.16.2 Estimation using gllamm 
13.16.3 Prediction 
13.17 Summary and further reading 
13.18 Exercises 
VII Models for survival or duration data
Introduction to models for survival or duration data (part VII)
14 Discrete-time survival
14.1 Introduction 
14.2 Single-level models for discrete-time survival data 
14.2.1 Discrete-time hazard and discrete-time survival 
14.2.2 Data expansion for discrete-time survival analysis 
14.2.3 Estimation via regression models for dichotomous responses 
14.2.4 Including covariates 
Time-constant covariates 
Time-varying covariates 
14.2.5 Multiple absorbing events and competing risks 
14.2.6 Handling left-truncated data 
14.3 How does birth history affect child mortality? 
14.4 Data expansion 
14.5 Proportional hazards and interval-censoring 
14.6 Complementary log-log models 
14.7 A random-intercept complementary log-log model 
14.7.1 Model specification 
14.7.2 Estimation using Stata 
14.8 Population-averaged or marginal vs. subject-specific or conditional survival probabilities 
14.9 Summary and further reading 
14.10 Exercises 
15 Continuous-time survival
15.1 Introduction 
15.2 What makes marriages fail? 
15.3 Hazards and survival 
15.4 Proportional hazards models 
15.4.1 Piecewise exponential model 
15.4.2 Cox regression model 
15.4.3 Poisson regression with smooth baseline hazard 
15.5 Accelerated failure-time models 
15.5.1 Log-normal model 
15.6 Time-varying covariates 
15.7 Does nitrate reduce the risk of angina pectoris? 
15.8 Marginal modeling 
15.8.1 Cox regression 
15.8.2 Poisson regression with smooth baseline hazard 
15.9 Multilevel proportional hazards models 
15.9.1 Cox regression with gamma shared frailty 
15.9.2 Poisson regression with normal random intercepts 
15.9.3 Poisson regression with normal random intercept and random coefficient 
15.10 Multilevel accelerated failure-time models 
15.10.1 Log-normal model with gamma shared frailty 
15.10.2 Log-normal model with log-normal shared frailty 
15.11 A fixed-effects approach 
15.11.1 Cox regression with subject-specific baseline hazards 
15.12 Different approaches to recurrent-event data 
15.12.1 Total time 
15.12.2 Counting process 
15.12.3 Gap time 
15.13 Summary and further reading 
15.14 Exercises 
VIII Models with nested and crossed random effects
16 Models with nested and crossed random effects
16.1 Introduction 
16.2 Did the Guatemalan immunization campaign work? 
16.3 A three-level random-intercept logistic regression model 
16.3.1 Model specification 
16.3.2 Measures of dependence and heterogeneity 
Types of residual intraclass correlations of the latent responses 
Types of median odds ratios 
16.3.3 Three-stage formulation 
16.4 Estimation of three-level random-intercept logistic regression models 
16.4.1 Using gllamm 
16.4.2 Using xtmelogit 
16.5 A three-level random-coefficient logistic regression model 
16.6 Estimation of three-level random-coefficient logistic regression models 
16.6.1 Using gllamm 
16.6.2 Using xtmelogit 
16.7 Prediction of random effects 
16.7.1 Empirical Bayes prediction 
16.7.2 Empirical Bayes modal prediction 
16.8 Different kinds of predicted probabilities 
16.8.1 Predicted population-averaged or marginal probabilities: New clusters 
16.8.2 Predicted median or conditional probabilities 
16.8.3 Predicted posterior mean probabilities: Existing clusters 
16.9 Do salamanders from different populations mate successfully? 
16.10 Crossed random-effects logistic regression 
16.11 Summary and further reading 
16.12 Exercises 
A Syntax for gllamm, eq, and gllapred: The bare essentials
B Syntax for gllamm
C Syntax for gllapred
D Syntax for gllasim