List of Tables

List of Figures

Preface

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

References

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

References