# Multilevel and Longitudinal Modeling Using Stata Volume I: Continuous Responses

Multilevel and Longitudinal Modeling Using Stata, Fourth Edition, by Sophia Rabe-Hesketh and Anders Skrondal, is a complete resource for learning to model data in which observations are grouped—whether those groups are formed by a nesting structure, such as children nested in classrooms, or formed by repeated observations on the same individuals. This text introduces random-effects models, fixed-effects models, mixed-effects models, marginal models, dynamic models, and growth-curve models, all of which account for the grouped nature of these types of data. As Rabe-Hesketh and Skrondal introduce each model, they explain when the model is useful, its assumptions, how to fit and evaluate the model using Stata, and how to interpret the results. With this comprehensive coverage, researchers who need to apply multilevel models will find this book to be the perfect companion. It is also the ideal text for courses in multilevel modeling because it provides examples from a variety of disciplines as well as end-of-chapter exercises that allow students to practice newly learned material.

The book comprises two volumes. Volume I focuses on linear models for continuous outcomes, while volume II focuses on generalized linear models for binary, ordinal, count, and other types of outcomes.

Volume I begins with a review of linear regression and then builds on this review to introduce two-level models, the simplest extensions of linear regression to models for multilevel and longitudinal/panel data. Rabe-Hesketh and Skrondal introduce the random-intercept model without covariates, developing the 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 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 also discuss models with random coefficients. The text then turns to models specifically designed for longitudinal and panel data—dynamic models, marginal models, and growth-curve models. The last portion of volume I covers models with more than two levels and models with crossed random effects.

The foundation and in-depth coverage of linear-model principles provided in volume I allow for a straightforward transition to generalized linear models for noncontinuous outcomes, which are described in volume II. This second volume begins with chapters introducing multilevel and longitudinal models for binary, ordinal, nominal, and count data. Focus then turns to survival analysis, introducing multilevel models for both discrete-time survival data and continuous-time survival data. The volume concludes by extending the two-level generalized linear models introduced in previous chapters to models with three or more levels and to models with crossed random effects.

In both volumes, readers will find extensive applications of multilevel and longitudinal models. Using many datasets that appeal to a broad audience, Rabe-Hesketh and Skrondal provide worked examples in each chapter. They also show the breadth of Stata’s commands for fitting the models discussed. They demonstrate Stata’s xt suite of commands (xtregxtlogitxtpoisson, etc.), which is designed for two-level random-intercept models for longitudinal/panel data. They demonstrate the me suite of commands (mixedmelogitmepoisson, etc.), which is designed for multilevel models, including those with random coefficients and those with three or more levels. In volume 2, they discuss gllamm, a community-contributed Stata command developed by Rabe-Hesketh and Skrondal that can fit many latent-variable models, of which the generalized linear mixed-effects model is a special case.The types of models fit by the xt commands, the me commands, and gllamm sometimes overlap; when this happens, the authors highlight the differences in syntax, data organization, and output for the commands. The authors also point out the strengths and weaknesses of these commands, based on considerations such as computational speed, accuracy, available predictions, and available postestimation statistics.

The fourth edition of Multilevel and Longitudinal Modeling Using Stata has been thoroughly revised and updated. In it, you will find new material on Kenward–Roger degrees-of-freedom adjustments for small sample sizes, difference-in-differences estimation for natural experiments, instrumental-variables estimation to account for level-one endogeneity, and Bayesian estimation for crossed-effects models. In addition, you will find new discussions of meologitcmxtmixlogitmestregmenbreg, and other commands introduced in Stata since the third edition of the book.

In summary, Multilevel and Longitudinal Modeling Using Stata, Fourth Edition is the most complete, up-to-date depiction of Stata’s capacity for fitting models to multilevel and longitudinal data. Readers will also find thorough explanations of the methods and practical advice for using these techniques. This text is a great introduction for researchers and students wanting to learn about these powerful data analysis tools.

List of tables
List of figures

List of displays
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

1.15 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 from wide form to long form
2.5.2 Using xtreg
2.5.3 Using mixed

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
Score test
F test
Confidence interval

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

Posterior and comparative standard errors
Diagnostic standard errors
Accounting for uncertainty in β̂ β^

2.11.4 Bayesian interpretation of REML estimation and prediction

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 covariance structure
3.3.5 Graphical illustration of random-intercept model

3.4 Estimation using Stata

3.4.1 Using xtreg
3.4.2 Using mixed

3.5 Coefficients of determination or variance explained
3.6 Hypothesis tests and confidence intervals

3.6.1 Hypothesis tests for individual regression coefficients
3.6.2 Joint hypothesis tests for several regression coefficients
3.6.3 Predicted means and confidence intervals
3.6.4 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 Robust Hausman 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

Pooled OLS
Feasible generalized least squares (FGLS)
ML by iterative GLS (IGLS)
ML by Newton–Raphson and Fisher scoring
ML by the expectation-maximization (EM) algorithm
REML

3.10.2 Consequences of using standard regression modeling for clustered data

Purely between-cluster covariate
Purely within-cluster covariate

3.10.3 Power and sample-size determination

Purely between-cluster covariate
Purely within-cluster covariate

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 mixed

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

4.12 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 Random-effects approach: No endogeneity
5.3 Fixed-effects approach: Level-2 endogeneity

5.3.1 De-meaning and subject dummies

De-meaning
Subject dummies

5.3.2 Hausman test
5.3.3 Mundlak approach and robust Hausman test
5.3.4 First-differencing

5.4 Difference-in-differences and repeated-measures ANOVA

5.4.1 Does raising the minimum wage reduce employment?
5.4.2 Repeated-measures ANOVA

5.5 Subject-specific coefficients

5.5.1 Random-coefficient model: No endogeneity
5.5.2 Fixed-coefficient model: Level-2 endogeneity

5.6 Hausman–Taylor: Level-2 endogeneity for level-1 and level-2 covariates
5.7 Instrumental-variable methods: Level-1 (and level-2) endogeneity

5.7.1 Do deterrents decrease crime rates?
5.7.2 Conventional fixed-effects approach
5.7.3 Fixed-effects IV estimator
5.7.4 Random-effects IV estimator
5.7.5 More Hausman tests

5.8 Dynamic models

5.8.1 Dynamic model without subject-specific intercepts
5.8.2 Dynamic model with subject-specific intercepts

5.9 Missing data and dropout

5.9.1 Maximum likelihood estimation under MAR: A simulation

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

6.9 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

Estimation using mixed
Predicting the mean trajectory
Predicting trajectories for individual children

7.3.2 Piecewise linear models

Estimation using mixed
Predicting the mean trajectory

7.4 Two-stage model formulation and cross-level interaction
7.5 Heteroskedasticity

7.5.1 Heteroskedasticity at level 1
7.5.2 Heteroskedasticity at level 2

7.7 Growth-curve model as a structural equation model

7.7.1 Estimation using sem
7.7.2 Estimation using mixed

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 mixed
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 mixed

8.13 Three-level random-coefficient models

8.13.1 Random coefficient at the child level

Estimation using mixed

8.13.2 Random coefficient at the child and school levels

Estimation using mixed

8.14 Residual diagnostics and predictions
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 mixed
9.3.4 Prediction

9.4 How much do primary and secondary schools affect attainment at age 16?
9.5 Data structure

9.6.1 Specification
9.6.2 Intraclass correlations
9.6.3 Estimation using mixed

9.7 Crossed random-effects model with random interaction

9.7.1 Model specification
9.7.2 Intraclass correlations
9.7.3 Estimation using mixed
9.7.4 Testing variance components
9.7.5 Some diagnostics

9.8 A trick requiring fewer random effects
9.10 Exercises

A Useful Stata commands
References

Subject index (PDF) 