INSEAD Library

Includes bibliographical references and index

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Multilevel and Longitudinal Modeling Using Stata Contents Preface 1 Linear variance-components models 1.1 xxi 1 Introduction ............................................................................................................................. 1 1.2 How reliable are expiratory flow measurements? ............................................................. 1 1.3 The variance-components model ........................................................................................... 4 1.3.1 Model specification and path diagram .................................................................... 4 1.3.2 Error components, variance components, and reliability .................................. 7 1.3.3 Intraclass correlation .................................................................................................... 8 1.4 Modeling the Mini Wright measurements ........................................................................... 8 1.4.1 Estimation using xtreg ................................................................................................. 9 1.4.2 Estimation using xtmixed ........................................................................................... 10 1.4.3 Estimation using gllamm ............................................................................................ 11 1.4.4 Relative and absolute agreement ............................................................... 14 1.5 Estimation methods ............................................................................................ 15 1.6 Assigning values to the random intercepts ......................................................................... 16 1.6.1 Maximum likelihood estimation .................................................................. 17 Implementation via OLS regression .......................................................... 17 Implementation via the mean total residual ........................................... 19 1.6.2 Empirical Bayes prediction ........................................................................................ 19 1.6.3 · Empirical Bayes variances ...................................................................... 24 1.7 Summary and further reading ............................................................................. 25 1.8 Exercises .................................................................................................................................. 26 viii 2 Linear random-intercept models 2.1 Contents 31 Introduction ........................................................................................................................ 31 2.2 Are tax preparers useful?......................................................................................................................................................................................... 31 2.3 The longitudinal data structure ....................................................................... 32 2.4 Panel data and correlated residuals ................................................................ 32 2.5 The random-intercept model ............................................................................. 34 2.5.1 Estimation using xtreg ......................................................................... 36 2.5.2 Estimation using xtmixed .................................................................... 38 2.6 Different kinds of effects in panel models ..................................................... 39 2.6.1 Between-taxpayer effects ........................................................................ 39 2.6.2 Within-taxpayer effects ........................................................................... 40 2.6.3 Relations among the estimators ........................................................... 42 2.7 Endogeneity and between-taxpayer effects ........................................................ 42 2.8 Residual diagnostics ................................................................................... 45 2.9 Summary and further reading ......................................................................... 50 2.10 Exercises ......................................................................................................... 51 3 Linear random-coefficient and growth-curve models 3.1 3.2 3.3 57 Introduction ............................................................................................... 57 How effective are different schools? ................................................................ 57 Separate linear regressions for each school ................................................. 58 3.4 The random-coefficient model ............................................................................ 61 3.4.1 Specification and interpretation of a random-coefficient model . .......... 61 3.4.2 Estimation and prediction using xtmixed ............................................ 68 Estimation of random-intercept model ............................................... 68 Estimation of random-coefficient model .............................................. 69 Empirical Bayes prediction using xtmixed .......................................... 72 3.4.3 Estimation and prediction using gllamm ............................................. 74 Estimation of random-intercept model ............................................... 75 Estimation of random-coefficient model .............................................. 75 Empirical Bayes prediction ................................................................. 77 Contents 3.5 How do children grow? ........................................................................................ 78 3.6 Growth-curve modeling ....................................................................................... 79 3.6.1 Observed growth trajectories ................................................................. 79 3.6.2 Estimation using xtmixed .................................................................... 80 Quadratic growth model with random intercept ................................. 80 Quadratic growth model with random intercept and random slope .. 80 Including a child-level covariate ........................................................................ 83 3.6.3 Estimation using gllamm ..................................................................... 84 Quadratic growth model with random intercept ................................. 84 Quadratic growth model with random intercept and random slope 85 Including a child-level covariate ........................................................... 86 3.7 Two-stage model formulation ............................................................................. 87 3.7.1 Model specification .................................................................................. 88 3.7.2 Estimation ........................................................................................... 88 3.8 3.9 Prediction of trajectories for individual children ...................................................... 91 · Complex level-1 variation or heteroskedasticity .......................................... 92 ix 3.10 Summary and further reading ....................................................................... 94 3.11 Exercises .................................................................................................................................. 95 4 Dichotomous or binary responses 4.1 101 Models for dichotomous responses .............................................................. 101 4.1.1 Generalized linear model formulation .................................................. 101 4.1.2 Latent-response formulation ............................................................... 107 Logistic regression .............................................................................. 108 Probit regression ............................................................................... 109 4.2 4.3 Which treatment is best for toenail infection? ............................................. 111 The longitudinal data structure ................................................................ 112 4.4 Population-averaged or marginal probabilities ................................................. 113 4.5 4.6 Random-intercept logistic regression........................................................... 116 Subject-specific vs. population-averaged relationships ............................... 120 x Contents 4.7 Maximum likelihood estimation using adaptive quadrature ..................... 124 4.7.1 Some practical considerations ................................................................... 128 4.8 Empirical Bayes (EB) predictions ........................................................................... 129 4.8.1 EB prediction of random effects ................................................................ 129 4.8.2 EB prediction of response probabilities ................................................ 130 4.9 Other approaches to clustered dichotomous data .......................................... 131 4.9.1 Conditional logistic regression ................................................................... 131 4.9.2 Generalized estimating equations (GEE) ............................................... 132 4.10 Summary and further reading .............................................................................. 134 4.11 Exercises ......................................................................................................................... 134 5 Ordinal responses 5.1 143 Introduction ................................................................................................................. 143 5.2 Cumulative models for ordinal responses .......................................................... 143 5.2.1 Generalized linear model formulation ..................................................... 143 5.2.2 Latent-response formulation ...................................................................... 144 5.2.3 Proportional odds ............................................................................................. 147 5.2.4 Identification ...................................................................................................... 148 5.3 Are antipsychotic drugs effective for patients with schizophrenia? . . .. 149 5.4 Longitudinal data structure and graphs ............................................................. 150 5.4.1 The longitudinal data structure ................................................................ 150 5.4.2 Plotting cumulative proportions ................................................................ 151 5.4.3 Plotting cumulative logits and transforming the time scale . . . 152 5.5 A proportional-odds model ........................................................................................ 154 5.5.1 Model specification .......................................................................................... 154 5.5.2 Estimation ........................................................................................................... 155 5.6 A random-intercept proportional-odds model .................................................. 157 5.6.1 Model specification .......................................................................................... 157 5.6.2 Estimation ........................................................................................................... 158 5.7 A random-coefficient proportional-odds model ................................................ 159 Contents 5.7.1 Model specification ................................................................................................. 159 5.7.2 Estimation ................................................................................................................ 159 5.8 Marginal and patient-specific probabilities .................................................................... 161 5.8.1 Marginal probabilities ............................................................................................ 161 5.8.2 Patient-specific cumulative response probabilities ........................................ 163 5.9 Do experts differ in their grading of student essays?................................................................................................ 165 5.10 A random-intercept model with grader bias ................................................................ 165 5.10.1 Model specification .............................................................................................. 165 5.10.2 Estimation .............................................................................................................. 166 5.11 v Including grader-specific measurement error variances ...................................... 167 5.11.1 Model specification .............................................................................................. 167 5.11.2 Estimation .............................................................................................................. 167 5.12 · Including grader-specific thresholds .......................................................................... 169 5.12.1 Model specification .............................................................................................. 169 5.12.2 Estimation .............................................................................................................. 170 5.13 Summary and further reading ........................................................................................ 175 5.14 Exercises ............................................................................................................................... 175 6 Counts 6.1 181 xi Introduction ..................................................................................................................... 181 6.2 Types of counts ...................................................................................................................... 181 6.3 Poisson models for counts .................................................................................................. 182 6.4 Did the German health-care reform reduce the number of doctor visits? 184 6.5 6.6 Longitudinal data structure ......................................................................................... 184 Poisson regression ignoring overdispersion and clustering ................................. 185 6.6.1 Model specification ................................................................................................. 185 6.6.2 Estimation ................................................................................................................ 185 6.7 Poisson regression with overdispersion but ignoring clustering ............................... 187 6.7.1 Using a level-1 random intercept ....................................................................... 187 Model specification ............................................................................................. 187 Estimation ............................................................................................................ 187 xii Contents 6.7.2 Quasilikelihood ................................................................................................. 188 Specification ..................................................................................................... 188 Estimation ......................................................................................................... 189 6.8 Random-intercept Poisson regression .................................................................. 190 6.8.1 Model specification .......................................................................................... 190 6.8.2 Estimation ........................................................................................................... 191 6.9 Random-coefficient Poisson regression ................................................................ 193 6.9.1 Model specification .......................................................................................... 193 6.9.2 Estimation ........................................................................................................... 194 6.10 Other approaches to clustered counts .............................................................. 196 6.10.1 Conditional Poisson regression ............................................................... 196 6.10.2 Generalized estimating equations (GEE) ............................................ 197 6.11 Which Scottish counties have a high risk of lip cancer? .......................... 197 6.12 Standardized mortality ratios ................................................................................ 198 6.13 Random-intercept Poisson regression ................................................................ 201 6.13.1 Model specification ....................................................................................... 201 6.13.2 Estimation ......................................................................................................... 201 6.13.3 Introducing a county-level covariate ..................................................... 202 6.13.4 Prediction .......................................................................................................... 203 6.14 · Nonparametric maximum likelihood estimation ........................................ 206 6.14.1 Specification .................................................................................................... 206 6.14.2 Estimation ......................................................................................................... 206 6.14.3 Prediction .......................................................................................................... 210 6.15 Summary and further reading .............................................................................. 211 6.16 Exercises ......................................................................................................................... 212 7 Higher level models and nested random effects 7.1 217 Introduction ................................................................................................................. 217 7.2 Which method is best for measuring expiratory flow? ................................. 218 7.3 Two-level variance-components models .............................................................. 219 Contents xiii 7.3.1 Model specification .................................................................................................. 219 7.3.2 Estimation ................................................................................................................. 219 7.4 Three-level variance-components models ...................................................................... 222 7.4.1 Model specification .................................................................................................. 222 7.4.2 Different types of intraclass correlation ............................................................ 224 7.4.3 Three-stage formulation ........................................................................................ 225 7.4.4 Estimation using xtmixed ..................................................................................... 225 7.4.5 Prediction using xtmixed ....................................................................................... 228 7.5 Did the Guatemalan immunization campaign work?.................................................... 229 7.6 A three-level logistic random-intercept model ................................................................ 230 7.6.1 Model specification .................................................................................................. 230 7.6.2 Different types of intraclass correlations for the latent responses ............. 230 7.6.3 Three-stage formulation ........................................................................................ 231 7.6.4 Estimation ................................................................................................................. 231 7.6.5 Introducing a random coefficient at level 3 ...................................................... 235 7.6.6 Prediction .................................................................................................................. 239 7.7 Summary and further reading ........................................................................................... 240 7.8 Exercises ............................................................................................................................. 241 249 8 Crossed random effects 8.1 Introduction ....................................................................................................................... 249 8.2 How does investment depend on expected profit and capital stock? . . 250 8.3 A two-way error-components model .................................................................................. 251 8.3.1 Model specification .................................................................................................. 251 8.3.2 Intraclass correlations ........................................................................................... 252 8.3.3 Estimation ................................................................................................................. 252 8.3.4 Prediction .................................................................................................................. 254 8.4 How much do primary and secondary schools affect attainment at age 16? 257 8.5 An additive crossed random-effects model .................................................................... 259 8.5.1 Specification .............................................................................................................. 259 xiv Contents 8.5.2 Estimation .............................................................................................................. 259 8.6 Including a random interaction .................................................................................... 260 8.6.1 Model specification .............................................................................................. 260 8.6.2 Intraclass correlations ........................................................................................ 261 8.6.3 Estimation .............................................................................................................. 262 8.6.4 Some diagnostics .................................................................................................. 264 8.7 · A trick requiring fewer random effects .................................................................... 266 8.8 Summary and further reading ...................................................................................... 269 8.9 Exercises .............................................................................................................................. 269 A Syntax for gllamm, eq, and gllapred B Syntax for gllamm C Syntax for gllapred D Syntax for gllasim References Author index Subject index 275 281 293 297 301 309 313