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Multilevel and Longitudinal Modeling Using Stata, Second Edition - by Sophia Rabe-Hesketh and Anders Skrondal


Multilevel and Longitudinal Modeling Using Stata, Second 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 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 reference to the first edition, a reviewer for American Statistician commends Rabe-Hesketh and Skrondal for promoting the appropriate use of multilevel and longitudinal modeling. The reviewer writes in the August 2006 issue, “All too often computer manuals leave off ... important aspects of an analysis, but the authors have been careful to provide a well-rounded and complete approach to model fitting and interpretation.”

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.

Table of contents

List of Tables
List of Figures
Preface

I Preliminaries

  1. Review of linear regression
    1. Introduction
    2. Is there genderdiscrimination in faculty salaries?
    3. Independent-samples t test
    4. One-way analysis of variance
    5. Simple linear regression
    6. Dummy variables
    7. Multiple linear regression
    8. Interactions
    9. Dummies for more than two groups
    10. Other types of interactions
      1. Interaction between dummy variables
      2. Interaction between continuous covariates
    11. Nonlinear effects
    12. Residual diagnostics
    13. Summary and further reading
    14. Exercises

II Two-level linear models

  1. Variance-components models
    1. Introduction
    2. How reliable are peak-expiratory-flow measurements
    3. The variance-components model
      1. Model specification and path diagram
      2. Error components, variance components, and reliability
      3. Intraclass correlation
    4. Fixed versus random effects
    5. Estimation using Stata
      1. Data preparation
      2. Using xtreg
      3. Using xtmixed
      4. Using gllamm
    6. Hypothesis tests and confidence intervals
      1. Hypothesis test and confidence interval for the population mean
      2. Hypothesis test and confidence interval for the between-cluster variance
    7. More on statistical inference
      1. Different estimation models
      2. Inference for B
        1. Estimate and standard error: Balanced case
        2. Estimate: Unbalanced case
    8. Crossed versus nested effects
    9. Assigning values to the random intercepts
      1. Maximum likelihood estimation
        1. Implementation via OLS regression
        2. Implementation via the mean total residual
      2. Empirical Bayes prediction
      3. Empirical Bayes variances
    10. Summary and further reading
    11. Exercises
  1. Random-intercept models with covariates
    1. Introduction
    2. Does smoking during pregnancy affect birthweight?
    3. The linear random-intercept model with covariates
      1. Model specification
      2. Residual variance and intraclass correlation
    4. Estimation using Stata
      1. Using xtreg
      2. Using xtmixed
      3. Using gllamm
    5. Coefficients of determination or variance explained
    6. Hypothesis tests and confidence intervals
      1. Hypothesis tests for regression coefficients
        1. Hypothesis tests for individual regression coefficients
        2. Joint hypothesis tests for several regression coefficients
      2. Predicted means and confidence intervals
      3. Hypothesis test for between-cluster variance
    7. Between and within effects
      1. Between-mother effects
      2. Within-mother effects
      3. Relations among estimators
      4. Endogeneity and different within- and between-mother effects
      5. Hausman endogeneity test
    8. Fixed versus random effects revisited
    9. Residual diagnostics
    10. More on statistical inference for regression coefficients
      1. Consequences of using ordinary regression for clustered data
      2. Power and sample-size determination
    11. Summary and further reading
    12. Exercises
  1. Random coefficient models
    1. Introduction
    2. How effective are different schools
    3. Separate linear regressions for each school
    4. Specification and interpretation of a random-coefficient model
      1. Specification of random-coefficient model
      2. Interpretation of the random-effects variances and covariances
    5. Estimation using Stata
      1. Using xtmixed
        1. Random-intercept model
        2. Random-coefficient model
      2. Using gllamm
        1. Random-intercept model
        2. Random-coefficient model
    6. Testing the slope variance
    7. Interpretation of estimates
    8. Assigning values to the random intercepts and slopes
      1. Maximum likelihood estimation
      2. Empirical Bayes prediction
      3. Model visualization
      4. Residual diagnostics
      5. Inferences for individual schools
    9. Two-stage model formulation
    10. Some warnings about random-coefficient models
    11. Summary and further reading
    12. Exercises
  1. Longitudinal, panel, and growth-curve models
    1. Introduction
    2. How and why do wages change over time?
    3. Data structure
      1. Missing data
      2. Time-varying and time-constant variables
    4. Time scales in longitudinal data
    5. Random- and fixed-effects approaches
      1. Correlated residuals
      2. Fixed-intercept model
        1. Using xtreg
        2. Using anova
      3. Random-intercept model
      4. Random-coefficient model
      5. Marginal mean and covariance structure induced by random effects
        1. Marginal mean and covariance structure for random-intercept models
        2. Marginal mean and covariance structure for random-coefficient models
    6. Marginal modeling
      1. Covariance structures
        1. Compound symmetric or exchangeable structure
        2. Random-coefficient structure
        3. Autoregressive residual structure
        4. Unstructured covariance matrix
      2. Marginal modeling using Stata
    7. Autoregressive- or lagged-response models
    8. Hybrid approaches
      1. Autoregressive response and random effects
      2. Autoregressive responses and autoregressive residuals
      3. Autoregressive residuals and random or fixed effects
    9. Missing data
      1. Maximum likelihood estimation under MAR: A simulation
    10. How do children grow?
      1. Observed growth trajectories
    11. Growth-curve modeling
      1. Random-intercept model
      2. Random-coefficient model
      3. Two-stage model formulation
    12. Prediction of trajectories for individual children
    13. Prediction of mean growth trajectory and 95% band
    14. Complex level-1 variation or heteroskedasticity
    15. Summary and further reading
    16. Exercises

III Two-level generalized linear models

  1. Dichotomous or binary responses
    1. Introduction
    2. Single-level models for dichotomous responses
      1. Generalized linear model formulation
      2. Latent-response formulation
        1. Logistic regression
        2. Probit regression
    3. Which treatment is best for toenail infection?
    4. Longitudinal data structure
    5. Population-averaged or marginal probabilities
    6. Random-intercept logistic regression
    7. Estimation of logistic random-intercept models
      1. Using xtlogit
      2. Using xtmelogit
      3. Using gllamm
    8. Inference for logistic random-intercept models
    9. Subject-specific vs. population-averaged relationships
    10. Measures of dependence and heterogeneity
      1. Conditional or residual intraclass correlation of the latent responses
      2. Median odds ratio
    11. Maximum likelihood estimation
      1. Adaptive quadrature
      2. Some speed considerations
        1. Advice for speeding up gllamm
    12. Assigning values to random effects
      1. Maximum likelihood estimation
      2. Empirical Bayes prediction
      3. Empirical Bayes modal prediction
    13. Different kinds of predicted probabilities
      1. Predicted population-averaged probabilities
      2. Predicted subject-specific probabilities
        1. Predictions for hypothetical subjects
        2. Predictions for the subjects in the sample
    14. Other approaches to clustered dichotomous data
      1. Conditional logistic regression
      2. Generalized estimating equations (GEE)
    15. Summary and further reading
    16. Exercises
  1. Ordinal responses
    1. Introduction
    2. Single-level cumulative models for ordinal responses
      1. Generalized linear model formulation
      2. Latent-response formulation
      3. Proportional odds
      4. Identification
    3. Are antipsychotic drugs effective for patients with schizophrenia?
    4. Longitudinal data structure and graphs
      1. Longitudinal data structure
      2. Plotting cumulative proportions
      3. Plotting estimated cumulative logits and transforming the time scale
    5. A single-level proportional odds model
      1. Model specification
      2. Estimation using Stata
    6. A random-intercept proportional odds model
      1. Model specification
      2. Estimation using Stata
    7. A random-intercept proportional odds model
      1. Model specification
      2. Estimation using gllamm
    8. Different kinds of predicted probabilities
      1. Predicted population-averaged probabilities
      2. Predicted patient-specific probabilities
    9. Do experts differ in the grading of student essays?
    10. A random-intercept probit model with grader bias
      1. Model specification
      2. Estimation
    11. Including grader-specific measurement error variances
      1. Model specification
      2. Estimation
    12. Including grader-specific thresholds
      1. Model specification
      2. Estimation
    13. Summary and further reading
    14. Exercises
  1. Discrete-time survival
    1. Introduction
      1. Censoring and truncation
      2. Time-varying covariates and different time-scales
      3. Discrete- versus continuous-time survival data
    2. Single-level models for discrete-time survival data
      1. Discrete-time hazard and discrete-time survival
      2. Data expansion for discrete-time survival analysis
      3. Estimation via regression models for dichotomous responses
      4. Including covariates
        1. Time-constant covariates
        2. Time-varying covariates
      5. Handling left-truncated data
    3. How does birth history affect child mortality?
    4. Data expansion
    5. Proportional hazards and interval censoring
    6. Complementary log-log models
    7. A random-intercept complementary log-log model
      1. Model specification
      2. Estimation using Stata
    8. Marginal and conditional survival probabilities
    9. Summary and further reading
    10. Exercises
  1. Counts
    1. Introduction
    2. What are counts?
      1. Counts versus proportions
      2. Counts as aggregated event-history data
    3. Single-level Poisson models for counts
    4. Did the German health-care reform reduce the number of doctor visits?
    5. Longitudinal data structure
    6. Single-level Poisson regression
      1. Model specification
      2. Estimation using Stata
    7. Random-intercept Poisson regression
      1. Model specification
      2. Estimation using Stata
        1. Using xtpoisson
        2. Using xtmepoisson
        3. Using gllamm
    8. Random-coefficient Poisson regression
      1. Model specification
      2. Estimation using Stata
        1. Using xtmepoisson
        2. Using gllamm
      3. Interpretation of estimates
    9. Overdispersion in single-level models
      1. Normally distributed random intercept
      2. Negative binomial models
        1. Mean dispersion or NB2
        2. Constant dispersion or NB1
      3. Quasilikelihood or robust standard errors
    10. Level-1 overdispersion in two-level models
    11. Other approaches to two-level count data
      1. Conditional Poisson regression
      2. Conditional negative binomial regression
      3. Generalized estimating equations
      4. Marginal and conditional estimates when responses are MAR
      5. How does birth history affect child mortality?
      6. Simple piecewise exponential survival model
      7. Piecewise exponential survival model with covariates and frailty
      8. Which Scottish counties have a high risk of lip cancer?
      9. Standardized mortality ratios
      10. Random-intercept Poisson regression
        1. Model specification
        2. Estimation using gllamm
        3. Prediction of standardized mortality ratios
      11. Nonparametric maximum likelihood estimation
        1. Specification
        2. Estimation using gllamm
        3. Prediction
      12. Summary and further reading
      13. Exercises

    IV Models with nested and crossed random effects

    1. Higher-level models with nested random effects
      1. Introduction
      2. Do peak-expiratory-flow measurements vary between methods?
      3. Two-level variance-components models
        1. Model specification
        2. Estimation using xtmixed
      4. Three-level variance-components models
        1. Model specification
        2. Different types of intraclass correlation
        3. Three-stage formulation
        4. Estimating using xtmixed
        5. Empirical Bayes prediction using xtmixed
      5. Did the Guatemalan immunization campaign work?
      6. A three-level logistic random-intercept model
        1. Model specification
        2. Different types of intraclass correlations for the latent responses
        3. Different kinds of median odds ratios
        4. Three-stage formulation
      7. Estimation of three-level logistic random-intercept models using Stata
        1. Using gllamm
        2. Using xtmelogit
      8. A three-level logistic random-coefficient model
      9. Estimation of three-level logistic random-coefficient models using Stata
        1. Using gllamm
        2. Using xtmelogit
      10. Prediction of random effects
        1. Empirical Bayes prediction
        2. Empirical Bayes modal prediction
      11. Different kinds of predicted probabilities
        1. Predicted marginal probabilities
        2. Predicted median or conditional probabilities
        3. Predicted posterior mean probabilities
      12. Summary and further reading
      13. Exercises
  1. Crossed random effects
    1. Introduction
    2. How does investment depend on expected profit and capital stock?
    3. A two-way error-components model
      1. Models specification
      2. Residual intraclass correlations
      3. Estimation
      4. Prediction
    4. How much do primary and secondary schools affect attainment at age 16?
    5. An additive crossed random-effects model
      1. Specification
      2. Estimation using xtmixed
    6. Including a random interaction
      1. Model specification
      2. Intraclass correlations
      3. Estimation using xtmixed
      4. Some diagnostics
    7. A trick requiring fewer random effects
    8. Do salamanders from different populations mate successfully?
    9. Crossed random-effects logistic regression
    10. Summary and further reading
    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
Authors Index
Subject Index


 
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