# Multilevel and Longitudinal Modeling Using Stata Volume II: Categorical Responses, Counts, and Survival

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
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V Models for categorical responses

10 Dichotomous or binary responses
10.1 Introduction
10.2 Single-level logit and probit regression models for dichotomous responses

10.2.1 Generalized linear model formulation

Labor-participation data
Estimation using logit
Estimation using glm

10.2.2 Latent-response formulation

Logistic regression
Probit regression
Estimation using probit

10.3 Which treatment is best for toenail infection?
10.4 Longitudinal data structure
10.5 Proportions and fitted population-averaged or marginal probabilities

Estimation using logit

10.6 Random-intercept logistic regression

10.6.1 Model specification

Reduced-form specification
Two-stage formulation

10.6.2 Model assumptions
10.6.3 Estimation

Using xtlogit
Using melogit
Using gllamm

10.7 Subject-specific or conditional versus population-averaged or marginal relationships
10.8 Measures of dependence and heterogeneity

10.8.1 Conditional or residual intraclass correlation of the latent responses
10.8.2 Median odds ratio
10.8.3 Measures of association for observed responses at median fixed part of the model

10.9 Inference for random-intercept logistic models

10.9.1 Tests and confidence intervals for odds ratios
10.9.2 Tests of variance components

10.10 Maximum likelihood estimation

10.10.2 Some speed and accuracy considerations

Integration methods and number of quadrature points
Starting values
Using melogit and gllamm for collapsible data

10.11 Assigning values to random effects

10.11.1 Maximum “likelihood” estimation
10.11.2 Empirical Bayes prediction
10.11.3 Empirical Bayes modal prediction

10.12 Different kinds of predicted probabilities

10.12.1 Predicted population-averaged or marginal probabilities
10.12.2 Predicted subject-specific probabilities

Predictions for hypothetical subjects: Conditional probabilities
Predictions for the subjects in the sample: Posterior mean probabilities

10.13 Other approaches to clustered dichotomous data

10.13.1 Conditional logistic regression

Estimation using clogit

10.13.2 Generalized estimating equations (GEE)

Estimation using xtgee

10.15 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 Single-level proportional-odds model

11.5.1 Model specification

Estimation using ologit

11.6 Random-intercept proportional-odds model

11.6.1 Model specification

Estimation using meologit
Estimation using gllamm

11.6.2 Measures of dependence and heterogeneity

Residual intraclass correlation of latent responses
Median odds ratio

11.7 Random-coefficient proportional-odds model

11.7.1 Model specification

Estimation using meologit
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

Estimation using gllamm

11.11.1 Model specification

Estimation using gllamm

11.12.1 Model specification

Estimation using gllamm

Cumulative complementary log–log model
Continuation-ratio logit model
Baseline-category logit and stereotype models

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

Transport data version 1
Estimation using mlogit

12.2.2 Conditional logit models with alternative-specific covariates

Transport data version 2: Expanded form
Estimation using clogit
Estimation using cmclogit

12.2.3 Conditional logit models with alternative- and unit-specific covariates

Estimation using clogit
Estimation using cmclogit

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

Estimation using clogit
Estimation using cmclogit

12.7 Multilevel conditional logit models

12.7.1 Preference heterogeneity: Brand-specific random intercepts

Estimation using cmxtmixlogit
Estimation using gllamm

12.7.2 Response heterogeneity: Marketing variables with random coefficients

Estimation using cmxtmixlogit
Estimation using gllamm

12.7.3 Preference and response heterogeneity

Estimation using cmxtmixlogit
Estimation using gllamm

12.8 Prediction of marginal choice probabilities
12.9 Prediction of random effects and household-specific choice probabilities
12.11 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 healthcare reform reduce the number of doctor visits?
13.5 Longitudinal data structure
13.6 Single-level Poisson regression

13.6.1 Model specification

Estimation using poisson
Estimation using glm

13.7 Random-intercept Poisson regression

13.7.1 Model specification
13.7.2 Measures of dependence and heterogeneity
13.7.3 Estimation

Using xtpoisson
Using mepoisson
Using gllamm

13.8 Random-coefficient Poisson regression

13.8.1 Model specification

Estimation using mepoisson
Estimation using gllamm

13.9 Overdispersion in single-level models

13.9.1 Normally distributed random intercept

Estimation using xtpoisson

13.9.2 Negative binomial models

Mean dispersion or NB2
Constant dispersion or NB1

13.9.3 Quasilikelihood

Estimation using glm

13.10 Level-1 overdispersion in two-level models

13.10.1 Random-intercept Poisson model with robust standard errors

Estimation using mepoisson

13.10.2 Three-level random-intercept model
13.10.3 Negative binomial models with random intercepts

Estimation using menbreg

13.10.4 The HHG model

13.11 Other approaches to two-level count data

13.11.1 Conditional Poisson regression

Estimation using xtpoisson, fe
Estimation using Poisson regression with dummy variables for clusters

13.11.2 Conditional negative binomial regression
13.11.3 Generalized estimating equations

Estimation using xtgee

13.12 Marginal and conditional effects when responses are MAR

Simulation

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

Estimation using gllamm

13.15.2 Prediction of standardized mortality ratios

13.16 Nonparametric maximum likelihood estimation

13.16.1 Specification

Estimation using gllamm

13.16.2 Prediction

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

Promotions data

14.2.2 Data expansion for discrete-time survival analysis
14.2.3 Estimation via regression models for dichotomous responses

Estimation using logit

14.2.4 Including time-constant covariates

Estimation using logit

14.2.5 Including time-varying covariates

Estimation using logit

14.2.6 Multiple absorbing events and competing risks

Estimation using mlogit

14.2.7 Handling left-truncated data

14.3 How does mother’s birth history affect child mortality?
14.4 Data expansion
14.5 Proportional hazards and interval-censoring
14.6 Complementary log–log models

14.6.1 Marginal baseline hazard

Estimation using cloglog

14.6.2 Including covariates

Estimation using cloglog

14.7 Random-intercept complementary log-log model

14.7.1 Model specification

Estimation using mecloglog
14.8 Population-averaged or marginal vs. cluster-specific or conditional survival probabilities
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

Estimation using streg
Estimation using poisson

15.4.2 Cox regression model

Estimation using stcox

15.4.3 Cox regression via Poisson regression for expanded data

Estimation using xtpoisson, fe

15.4.4 Approximate Cox regression: Poisson regression, smooth baseline hazard

Estimation using poisson

15.5 Accelerated failure-time models

15.5.1 Log-normal model

Estimation using streg
Estimation using stintreg

15.6 Time-varying covariates

Estimation using streg

15.7 Does nitrate reduce the risk of angina pectoris?
15.8 Marginal modeling

15.8.1 Cox regression with occasion-specific dummy variables

Estimation using stcox

15.8.2 Cox regression with occasion-specific baseline hazards

Estimation using stcox, strata

15.8.3 Approximate Cox regression

Estimation using poisson

15.9 Multilevel proportional hazards models

15.9.1 Cox regression with gamma shared frailty

Estimation using stcox, shared

15.9.2 Approximate Cox regression with log-normal shared frailty

Estimation using mepoisson

15.9.3 Approximate Cox regression with normal random intercept and coefficient

Estimation using mepoisson

15.10 Multilevel accelerated failure-time models

15.10.1 Log-normal model with gamma shared frailty

Estimation using streg

15.10.2 Log-normal model with log-normal shared frailty

Estimation using mestreg

15.10.3 Log-normal model with normal random intercept and random coefficient

Estimation using mestreg

15.11 Fixed-effects approach

15.11.1 Stratified Cox regression with subject-specific baseline hazards

Estimation using stcox, strata

15.12 Different approaches to recurrent-event data

15.12.1 Total time risk interval
15.12.2 Counting process risk interval
15.12.3 Gap-time risk interval

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
16.3.4 Estimation

Using melogit
Using gllamm

16.4 A three-level random-coefficient logistic regression model

16.4.1 Estimation

Using melogit
Using gllamm

16.5 Prediction of random effects

16.5.1 Empirical Bayes prediction
16.5.2 Empirical Bayes modal prediction

16.6 Different kinds of predicted probabilities

16.6.1 Predicted population-averaged or marginal probabilities: New clusters
16.6.2 Predicted median or conditional probabilities
16.6.3 Predicted posterior mean probabilities: Existing clusters

16.7 Do salamanders from different populations mate successfully
16.8 Crossed random-effects logistic regression

16.8.1 Setup for estimating crossed random-effects model using melogit
16.8.2 Approximate maximum likelihood estimation

Estimation using melogit

16.8.3 Bayesian estimation

Brief introduction to Bayesian inference
Priors for the salamander data
Estimation using bayes: melogit

16.8.4 Estimates compared
16.8.5 Fully Bayesian versus empirical Bayesian inference for random effects

16.10 Exercises

A Syntax for gllamm, eq, and gllapred: The bare essentials

B Syntax for gllamm

C Syntax for gllapred

D Syntax for gllasim

References

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Subject index (PDF) 