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Multilevel and Longitudinal Modeling Using Stata Volume I: Continuous Responses
Third Edition
SOPHIA RABE-HESKETH University of California-Berkeley Institute of Education. University of London
ANDERS SKR.ONDAL Norwegian Institute of Public Health
A Stata Press Publication StataCorp LP College Station, Texas
Contents
List of Tables xvii
List of Figures xix
Preface xxv
Multilevel and longitudinal models: When and why? 1
I Preliminaries 9 1 Review of linear regression 11
1.1 Introduction 11
1.2 Is there gender discrimination in faculty salaries? 11
1.3 Independent-samples t test 12
1.4 One-way analysis of variance 17
1.5 Simple linear regression 19
1.6 Dummy variables 27
1.7 Multiple linear regression 30
1.8 Interactions 36
1.9 Dummy variables for more than two groups 42
1.10 Other types of interactions 48
1.10.1 Interaction between dummy variables 48
1.10.2 Interaction between continuous covariates 50
1.11 Nonlinear effects 52
1.12 Residual diagnostics 54
1.13 ••• Causal and noncausal interpretations of regression coefficients . . 56
1.13.1 Regression as conditional expectation 56
1.13.2 Regression as structural model 57
viii Contents
1.14 Summary and further reading 59
1.15 Exercises 60
II Two-level models 71 2 Variance-components models 73
2.1 Introduction 73
2.2 How reliable are peak-expiratory-flow measurements? 74
2.3 Inspecting within-subject dependence 75
2.4 The variance-components model 77
2.4.1 Model specification 77
2.4.2 Path diagram 78
2.4.3 Between-subject heterogeneity 79
2.4.4 Within-subject dependence 80
Intraclass correlation 80
Intraclass correlation versus Pearson correlation 81
2.5 Estimation using Stata 82
2.5.1 Data preparation: Reshaping to long form 83
2.5.2 Using xtreg 84
2.5.3 Using xtmixed 85
2.6 Hypothesis tests and confidence intervals 87
2.6.1 Hypothesis test and confidence interval for the population mean 87
2.6.2 Hypothesis test and confidence interval for the between-cluster variance 88
Likelihood-ratio test 88
••• Score test 89
F test 92
Confidence intervals 92
2.7 Model as data-generating mechanism 93
2.8 Fixed versus random effects 95
2.9 Crossed versus nested effects 97
Contents ix
2.10 Parameter estimation 99
2.10.1 Model assumptions 99
Mean structure and covariance structure 100
Distributional assumptions 101
2.10.2 Different estimation methods 101
2.10.3 Inference for (3 103
Estimate and standard error: Balanced case 103
Estimate: Unbalanced case 1.05
2.11 Assigning values to the random intercepts 106
2.11.1 Maximum "likelihood'' estimation 106
Implementation via OLS regression 107
Implementation via the mean total residual 108
2.11.2 Empirical Bayes prediction 109
2.11.3 Empirical Bayes standard errors 113
Comparative standard errors 113
Diagnostic standard errors 114
2.12 Summary and further reading 115
2.13 Exercises 116
3 Random-intercept models with covariates 123
3.1 Introduction 123
3.2 Does smoking during pregnancy affect birthweight? 123
3.2.1 Data structure and descriptive statistics 125
3.3 The linear random-intercept model with covariates 127
3.3.1 Model specification 127
3.3.2 Model assumptions 128
3.3.3 Mean structure 130
3.3.4 Residual variance and intraclass correlation 130
3.3.5 Graphical illustration of random-intercept model 131
3.4 Estimation using Stata 131
3.4.1 Using xtreg 132
X Contents
3.4.2 Using xtmixed 133
3.5 Coefficients of determination or variance explained 134
3.6 Hypothesis tests and confidence intervals 138
3.6.1 Hypothesis tests for regression coefficients 138
Hypothesis tests for individual regression coefficients . . . 138
Joint hypothesis tests for several regression coefficients . . 139
3.6.2 Predicted means and confidence intervals 140
3.6.3 Hypothesis test for random-intercept variance 142
3.7 Between and within effects of level-1 covariates 142
3.7.1 Between-mother effects 143
3.7.2 Within-mother effects 145
3.7.3 Relations among estimators 147
3.7.4 Level-2 endogeneity and cluster-level confounding 149
3.7.5 Allowing for different within and between effects 152
3.7.6 Hausman endogeneity test 157
3.8 Fixed versus random effects revisited 158
3.9 Assigning values to random effects: Residual diagnostics 160
3.10 More on statistical inference 164
3.10.1 ••• Overview of estimation methods 164
3.10.2 Consequences of using standard regression modeling for clustered data 167
3.10.3 ••• Power and sample-size determination 168
3.11 Summary and further reading 171
3.12 Exercises 172
4 Random-coefficient models 181
4.1 Introduction 181
4.2 How effective are different schools? 181
4.3 Separate linear regressions for each school 182
4.4 Specification and interpretation of a random-coefficient model . . . 188
4.4.1 Specification of a random-coefficient model 188
Contents xi
4.4.2 Interpretation of the random-effects variances and co-
variances 191
4.5 Estimation using xtmixed 194
4.5.1 Random-intercept model 194
4.5.2 Random-coefficient model 196
4.6 Testing the slope variance 197
4.7 Interpretation of estimates 198
4.8 Assigning values to the random intercepts and slopes 200
4.8.1 Maximum "likelihood" estimation 200
4.8.2 Empirical Bayes prediction 201
4.8.3 Model visualization 203
4.8.4 Residual diagnostics 204
4.8.5 Inferences for individual schools 207
4.9 Two-stage model formulation 210
4.10 Some warnings about random-coefficient models 213
4.10.1 Meaningful specification 213
4.10.2 Many random coefficients 213
4.10.3 Convergence problems 214
4.10.4 Lack of identification 214
4.11 Summary and further reading 215
4.12 Exercises 216
III Models for longitudinal and panel data 225 Introduction to models for longitudinal and panel data (part III) 227
5 Subject-specific effects and dynamic models 247
5.1 Introduction 247
5.2 Conventional random-intercept model 248
5.3 Random-intercept models accommodating endogenous covariates . . 250
5.3.1 Consistent estimation of effects of endogenous time-varying covariates 250
xii Contents
5.3.2 Consistent estimation of effects of endogenous
time-varying and endogenous time-constant covariates . . . 253
5.4 Fixed-intercept model 2-57
5.4.1 Using xtreg or regress with a differencing operator 259
5.4.2 ••• Using anova 262
5.5 Random-coefficient model 265
5.6 Fixed-coefficient model 267
5.7 Lagged-response or dynamic models 269
5.7.1 Conventional lagged-response model 269
5.7.2 ••• Lagged-response model with subject-specific intercepts . 273
5.8 Missing data and dropout 278
5.8.1 ••• Maximum likelihood estimation under MAR:
A simulation 279
5.9 Summary and further reading 282
5.10 Exercises 283
6 Marginal models 293
6.1 Introduction 293
6.2 Mean structure 293
6.3 Covariance structures 294
6.3.1 Unstructured covariance matrix 298 6.3.2 Random-intercept or compound
symmetric/exchangeable structure 303
6.3.3 Random-coefficient structure 305
6.3.4 Autoregressive and exponential structures 308
6.3.5 Moving-average residual structure 311
6.3.6 Banded and Toeplitz structures 313
6.4 Hybrid and complex marginal models 316
6.4.1 Random effects and correlated level-1 residuals 316
6.4.2 Heteroskedastic level-1 residuals over occasions 317
6.4.3 Heteroskedastic level-1 residuals over groups 318
6.4.4 Different covariance matrices over groups 32f
Contents ' xiii
6.5 Comparing the fit of marginal models 322
6.6 Generalized estimating equations (GEE) 325
6.7 Marginal modeling with few units and many occasions 327
6.7.1 Is a highly organized labor market beneficial for economic growth? 328
6.7.2 Marginal modeling for long panels 329
6.7.3 Fitting marginal models for long panels in Stata 329
6.8 Summary and further reading 332
6.9 Exercises 333
7 Growth-curve models 343
7.1 Introduction 343
7.2 How do children grow? 343
7.2.1 Observed growth trajectories 344
7.3 Models for nonlinear growth 345
7.3.1 Polynomial models 345
Fitting the models 346
Predicting the mean trajectory 349
Predicting trajectories for individual children 351
7.3.2 Piecewise linear models 353
Fitting the models 354
Predicting the mean trajectory 357
7.4 Two-stage model formulation 358
7.5 Heteroskedasticity 360
7.5.1 Heteroskedasticity at level 1 360
7.5.2 Heteroskedasticity at level 2 362
7.6 How does reading improve from kindergarten through third grade? 364
7.7 Growth-curve model as a structural equation model 364
7.7.1 Estimation using sem 366
7.7.2 Estimation using xtmixed 371
7.8 Summary and further reading 375
xiv Contents
7.9 Exercises 376
IV Models with nested and crossed random effects 383 8 Higher-level models with nested random effects 385
8.1 Introduction 385
8.2 Do peak-expiratory-flow measurements vary between methods within subjects? 386
8.3 Inspecting sources of variability 388
8.4 Three-level variance-components models 389
8.5 Different types of intraclass correlation 392
8.6 Estimation using xtmixed 393
8.7 Empirical Bayes prediction 394
8.8 Testing variance components 395
8.9 Crossed versus nested random effects revisited 397
8.10 Does nutrition affect cognitive development of Kenyan children? . . 399
8.11 Describing and plotting three-level data 400
8.11.1 Data structure and missing data 400
8.11.2 Level-1 variables 401
8.11.3 Level-2 variables • 402
8.11.4 Level-3 variables 403
8.11.5 Plotting growth trajectories 404
8.12 Three-level random-intercept model 405
8.12.1 Model specification: Reduced form 405
8.12.2 Model specification: Three-stage formulation 405
8.12.3 Estimation using xtmixed 406
8.13 Three-level random-coefficient models 409
8.13.1 Random coefficient at the child level 409
8.13.2 Random coefficient at the child and school levels 411
8.14 Residual diagnostics and predictions 413
8.15 Summary and further reading 418
8.16 Exercises 419
Contents xv
9 Crossed random effects 433
9.1 Introduction 433
9.2 How does investment depend on expected profit and capital stock? 434
9.3 A two-way error-components model 435
9.3.1 Model specification 435
9.3.2 Residual variances, covariances. and intraclass correlations 436
Longitudinal correlations 436
Cross-sectional correlations 436
9.3.3 Estimation using xtmixed 437
9.3.4 Prediction 441
9.4 How much do primary and secondary schools affect attainment at
age 16? 443
9.5 Data structure 444
9.6 Additive crossed random-effects model 446
9.6.1 Specification 446
9.6.2 Estimation using xtmixed 447
9.7 Crossed random-effects model with random interaction 448
9.7.1 Model specification 448
9.7.2 Intraclass correlations 448
9.7.3 Estimation using xtmixed 449
9.7.4 Testing variance components 451
9.7.5 Some diagnostics 453
9.8 A trick requiring fewer random effects 456
9.9 Summary and further reading 459
9.10 Exercises 460
A Useful Stata commands 471
References 473
Author index 485
Subject index 491
Multilevel and Longitudinal Modeling Using Stata Volume II: Categorical Responses, Counts, and Survival
Third Edition
SOPHIA RABE-HESKETH University of California. Berkeley Institute of Education. University of London
ANDERS SKRONDAL Norwegian Institute of Public Health
A Stata Press Publication StataCorp LP College Station, Texas
O
Contents
List of Tables xvii
List of Figures xix
V Models for categorical responses 499 10 Dichotomous or binary responses 501
10.1 Introduction 501
10.2 Single-level logit and probit regression models for dichotomous responses 501
10.2.1 Generalized linear model formulation 502
10.2.2 Latent-response formulation 510
Logistic regression 512
Probit regression 512
10.3 Which treatment is best for toenail infection? 515
10.4 Longitudinal data structure 515
10.5 Proportions and fitted population-averaged or marginal probabilities 517
10.6 Random-intercept logistic regression 520
10.6.1 Model specification 520
Reduced-form specification 520
Two-stage formulation 522
10.7 Estimation of random-intercept logistic models 523
10.7.1 Using xtlogit 523
10.7.2 Using xtmelogit 527
10.7.3 Using gllamrn 527
10.8 Subject-specific or conditional vs. population-averaged or marginal relationships 529
viii Contents
10.9 Measures of dependence and heterogeneity 532
10.9.1 Conditional or residual intraclass correlation of the latent responses 532
10.9.2 Median odds ratio 533
10.9.3 *•* Measures of association for observed responses at median fixed part of the model 533
10.10 Inference for random-intercept logistic models 535
10.10.1 Tests and confidence intervals for odds ratios 535
10.10.2 Tests of variance components 536
10.11 Maximum likelihood estimation 537
10.11.1 *•* Adaptive quadrature 537
10.11.2 Some speed and accuracy considerations 540
Advice for speeding up estimation in gllannn 542
10.12 Assigning values to random effects 543
10.12.1 Maximum "likelihood" estimation 544
1.0.12.2 Empirical Bayes prediction 545
10.12.3 Empirical Bayes modal prediction 546
10.13 Different kinds of predicted probabilities 548
10.13.1 Predicted population-averaged or marginal probabilities . . 548
10.13.2 Predicted subject-specific probabilities 549
Predictions for hypothetical subjects: Conditional probabilities 549
Predictions for the subjects in the sample: Posterior
mean probabilities 551
10.14 Other approaches to clustered dichotomous data -557
10.14.1 Conditional logistic regression 557
10.14.2 Generalized estimating equations (GEE) 559
10.15 Summary and further reading 562
10.16 Exercises 563
11 Ordinal responses 575
11.1 Introduction 575
Contents ix
11.2 Single-level cumulative models for ordinal responses . . . : 57-5
11.2.1 Generalized linear model formulation 57-5
11.2.2 Latent-response formulation 576
11.2.3 Proportional odds 580
11.2.4 ••• Identification 582
11.3 Are antipsychotic drugs effective for patients with schizophrenia? . 585
11.4 Longitudinal data structure and graphs 585
11.4.1 Longitudinal data structure 586
11.4.2 Plotting cumulative proportions 587
11.4.3 Plotting cumulative sample logits and transforming the
time scale 588
11.5 A single-level proportional odds model 590
11.5.1 Model specification 590
11.5.2 Estimation using Stata 591
11.6 A random-intercept proportional odds model 594
11.6.1 Model specification 594
11.6.2 Estimation using Stata 594
11.6.3 Measures of dependence and heterogeneity 595
Residual intraclass correlation of latent responses 595
Median odds ratio 596
11.7 A random-coefficient proportional odds model 596
11.7.1 Model specification 596
11.7.2 Estimation using gllamrn 596
11.8 Different kinds of predicted probabilities 599
11.8.1 Predicted population-averaged or marginal probabilities . . 599
11.8.2 Predicted subject-specific probabilities: Posterior mean . . 602
11.9 Do experts differ in their grading of student essays? 606
11.10 A random-intercept probit model with grader bias 606
11.10.1 Model specification 606
11.10.2 Estimation using gllamrn 607
x Contents
11.11 Including grader-specific measurement error variances 608
11.11.1 Model specification 608
11.11.2 Estimation using gllamrn 609
11.12 Including grader-specific thresholds 611
11.12.1 Model specification 611
11.12.2 Estimation using gllamm 611
11.13 *•* Other link functions 616
Cumulative complementary log-log model 616
Continuation-ratio logit model 616
Adjacent-category logit model 618
Baseline-category logit and stereotype models 618
11.14 Summary and further reading 619
11.15 Exercises 620
12 Nominal responses and discrete choice 629
12.1 Introduction 629
12.2 Single-level models for nominal responses 630
12.2.1 Multinomial logit models 630
12.2.2 Conditional logit models 638
Classical conditional logit models 639
Conditional logit models also including covariates that
vary only over units 645
12.3 Independence from irrelevant alternatives 648
12.4 Utility-maximization formulation 649
12.5 Does marketing affect choice of yogurt? 651
12.6 Single-level conditional logit models 653
12.6.1 Conditional logit models with alternative-specific
intercepts 654
12.7 Multilevel conditional logit models 659
12.7.1 Preference heterogeneity: Brand-specific random intercepts 659
Contents xi
12.7.2 Response heterogeneity: Marketing variables with random coefficients 663
12.7.3 ••• Preference and response heterogeneity 666
Estimation using gllamrn 667
Estimation using mixlogit 669
12.8 Prediction of random effects and response probabilities 672
12.9 Summary and further reading 676
12.10 Exercises 677
VI Models for counts 685 13 Counts 687
13.1 Introduction 687
13.2 What are counts? 687
13.2.1 Counts versus proportions 687
• 13.2.2 Counts as aggregated event-history data 688
13.3 Single-level Poisson models for counts 689
13.4 Did the German health-care reform reduce the number of doctor visits? 691
13.5 Longitudinal data structure 691
13.6 Single-level Poisson regression 692
13.6.1 Model specification 692
13.6.2 Estimation using Stata 693
13.7 Random-intercept Poisson regression 696
13.7.1 Model specification 696
13.7.2 Measures of dependence and heterogeneity 697
13.7.3 Estimation using Stata 697
Using xtpoisson 697
Using xtmepoisson 699
Using gllamrn 700
13.8 Random-coefficient Poisson regression 701
13.8.1 Model specification 701
xii Contents
13.8.2 Estimation using Stata 702
Using xtmepoisson 702
Using gllamrn 704
13.8.3 Interpretation of estimates 705
13.9 Overdispersion in single-level models 706
13.9.1 Normally distributed random intercept 706
13.9.2 Negative binomial models 707
Mean dispersion or NB2 708
Constant dispersion or NB1 709
13.9.3 Quasilikelihood 709
13.10 Level-1 overdispersion in two-level models 711
13.11 Other approaches to two-level count data 713
13.11.1 Conditional Poisson regression 713
13.11.2 Conditional negative binomial regression 715
13.11.3 Generalized estimating equations 715
13.12 Marginal a.nd conditional effects when responses are MAR 716
Simulation 717
13.13 Which Scottish counties have a high risk of
lip cancer? 720
13.14 Standardized mortality ratios 721
13.15 Random-intercept Poisson regression 723
13.15.1 Model specification 723
13.15.2 Estimation using gllamrn 724
13.15.3 Prediction of standardized mortality ratios 725
13.16 Nonparametric maximum likelihood estimation 727
13.16.1 Specification 727
13.16.2 Estimation using gllamrn 727
13.16.3 Prediction 732
13.17 Summary and further reading • 732
13.18 Exercises 733
Contents xiii
VII Models for survival or duration data 741 Introduction to models for survival or duration data (part VII) 743
14 Discrete-time survival 749
14.1 Introduction 749
14.2 Single-level models for discrete-time survival data 749
14.2.1 Discrete-time hazard and discrete-time survival 749
14.2.2 Data expansion for discrete-time survival analysis 752
14.2.3 Estimation via regression models for dichotomous
responses 754
14.2.4 Including covariates 758
Time-constant covariates 758
Time-varying covariates 762
14.2.5 Multiple absorbing events and competing risks 767
14.2.6 Handling left-truncated data 772
14.3 How does birth history affect child mortality? 773
14.4 Data expansion 774
14.5 Proportional hazards and interval-censoring 776
14.6 Complementary log-log models 777
14.7 A random-intercept complementary log-log model 781
14.7.1 Model specification 781
14.7.2 Estimation using Stata 782 14.8 '•* Population-averaged or marginal vs. subject-specific or condi
tional survival probabilities 784
14.9 Summary and further reading 788
14.10 Exercises 789
15 Continuous-time survival 797
15.1 Introduction 797
15.2 What makes marriages fail? 797
15.3 Hazards and survival 799
15.4 Proportional hazards models 805
15.4.1 Piecewise exponential model 807
xiv Contents
15.4.2 Cox regression model 815
15.4.3 Poisson regression with smooth baseline hazard 819
15.5 Accelerated failure-time models 823
15.5.1 Log-normal model 824
15.6 Time-varying covariates 829
15.7 Does nitrate reduce the risk of angina pectoris? 832
15.8 Marginal modeling 835
15.8.1 Cox regression 835
15.8.2 Poisson regression with smooth baseline hazard 838
15.9 Multilevel proportional hazards models 841
15.9.1 Cox regression with gamma shared frailty 841
15.9.2 Poisson regression with normal random intercepts 845
15.9.3 Poisson regression with normal random intercept and random coefficient 847
15.10 Multilevel accelerated failure-time models 849
15.10.1 Log-normal model with gamma shared frailty 849
15.10.2 Log-normal model with log-normal shared frailty 850
15.11 A fixed-effects approach 851
15.11.1 Cox regression with subject-specific baseline hazards .... 851
15.12 Different approaches to recurrent-event data 853
15.12.1 Total time 854
15.12.2 Counting process 858
15.12.3 Gap time 859
15.13 Summary and further reading 861
15.14 Exercises 862
VIII Models with nested and crossed random effects 871 16 Models with nested and crossed random effects 873
16.1 Introduction 873
16.2 Did the Guatemalan immunization campaign work? 873
16.3 A three-level random-intercept logistic regression model 875
Contents xv
16.3.1 Model specification 876
16.3.2 Measures of dependence and heterogeneity 876
Types of residual intraclass correlations of the latent responses 876
Types of median odds ratios 877
16.3.3 Three-stage formulation 877
16.4 Estimation of three-level random-intercept logistic regression
models 878
16.4.1 Using gllamrn 878
16.4.2 Using xtmelogit 883
16.5 A three-level random-coefficient logistic regression model 886
16.6 Estimation of three-level random-coefficient logistic regression
models 887
16.6.1 Using gllamrn 887
16.6.2 Using xtmelogit 890
16.7 Prediction of random effects 892
16.7.1 Empirical Bayes prediction 892
16.7.2 Empirical Bayes modal prediction 893
16.8 Different kinds of predicted probabilities 894
16.8.1 Predicted population-averaged or marginal probabilities:
New clusters 894
16.8.2 Predicted median or conditional probabilities 895
16.8.3 Predicted posterior mean probabilities: Existing clusters . 896
16.9 Do salamanders from different populations mate successfully? . . . 897
16.10 Crossed random-effects logistic regression 900
16.11 Summary and further reading 907
16.12 Exercises 908
A Syntax for gllamm, eq, and gllapred: The bare essentials 915
B Syntax for gllamm 921
C Syntax for gllapred 933
D Syntax for gllasim 937
References
Author index
Subject index
Contents
941
955
963