Multilevel Statistical Models, 4th Edition

Multilevel Statistical Models......Page 5
Contents......Page 9
Preface......Page 17
Acknowledgements......Page 19
Notation......Page 21
A general classification notation and diagram......Page 22
Glossary......Page 25
1.1 Hierarchically structured data......Page 27
1.2 School effectiveness......Page 29
1.4 Repeated measures data......Page 31
1.5 Event history and survival models......Page 32
1.7 Multivariate models......Page 33
1.8 Nonlinear models......Page 34
1.10 Cross classifications and multiple membership structures......Page 35
1.12 Levels of aggregation and ecological fallacies......Page 36
1.13 Causality......Page 37
1.14 The latent normal transformation and missing data......Page 39
1.16 A caveat......Page 40
2.1 Introduction......Page 41
2.2 The 2-level model......Page 43
2.3.1 The variance components model......Page 45
2.3.2 The general 2-level model with random coefficients......Page 47
2.4 Maximum likelihood estimation using iterative generalised least squares (IGLS)......Page 48
2.6 Residuals......Page 51
2.7 The adequacy of ordinary least squares estimates......Page 53
2.8 A 2-level example using longitudinal educational achievement data......Page 54
2.8.1 Checking for outlying units......Page 56
2.8.2 Model checking using estimated residuals......Page 57
2.9 General model diagnostics......Page 58
2.10 Higher level explanatory variables and compositional effects......Page 60
2.11 Transforming to normality......Page 62
2.12.1 Fixed parameters......Page 65
2.12.2 Random parameters......Page 67
2.12.3 Hypothesis testing for non-nested models......Page 68
2.12.4 Inferences for residual estimates......Page 69
2.13 Bayesian estimation using Markov Chain Monte Carlo (MCMC)......Page 71
2.13.1 Gibbs sampling......Page 73
2.13.3 Convergence of MCMC chains......Page 74
2.13.4 Making inferences......Page 75
2.13.5 An example......Page 76
2.14 Data augmentation......Page 81
Appendix 2.1 The general structure and maximum likelihood estimation for a multilevel model......Page 83
2.2.1 Shrunken estimates......Page 86
2.2.2 Delta method estimators for the covariance matrix of residuals......Page 87
Appendix 2.3 Estimation using profile and extended likelihood......Page 89
Appendix 2.4 The EM algorithm......Page 91
2.5.1 Gibbs sampling......Page 93
2.5.2 Metropolis-Hastings (MH) sampling......Page 96
2.5.3 Hierarchical centring......Page 97
2.5.4 Orthogonalisation of the explanatory variables and parameter expansion......Page 98
3.1 Complex variance structures......Page 99
3.1.1 Partitioning the variance and intra-unit correlation......Page 105
3.1.2 Variances for subgroups defined at level 1......Page 106
3.1.3 Variance as a function of predicted value......Page 109
3.2 A 3-level complex variation model example......Page 111
3.3 Parameter constraints......Page 114
3.4 Weighting units......Page 116
3.4.1 Maximum likelihood estimation with weights......Page 117
3.4.2 Weighted MCMC estimation......Page 118
3.5 Robust (sandwich) estimators and jacknifing......Page 119
3.6.2 The fully parametric bootstrap......Page 121
3.6.3 The iterated parametric bootstrap and bias correction......Page 122
3.6.4 The residuals bootstrap......Page 124
3.7 Aggregate level analyses......Page 127
3.7.1 Inferences about residuals from aggregate level analyses......Page 129
3.8 Meta analysis......Page 130
3.8.1 Aggregate and mixed level analysis......Page 132
3.8.3 An example: meta analysis of class size data......Page 133
3.8.4 Practical issues in meta analysis......Page 134
3.9 Design issues......Page 135
4.1 Generalised linear models......Page 137
4.2 Proportions as responses......Page 138
4.3.1 A study of contraceptive use......Page 141
4.3.2 Modelling school segregation......Page 143
4.4 Models for multiple response categories......Page 145
4.5 Models for counts......Page 148
4.6 Ordered responses......Page 149
4.7 Mixed discrete-continuous response models......Page 150
4.8 A latent normal model for binary responses......Page 152
4.9 Partitioning variation in discrete response models......Page 153
4.9.2 Simulation (Method B)......Page 154
4.9.4 A latent variable approach (Method D)......Page 155
4.9.5 An example of VPC calculations......Page 156
4.1.1 Approximate quasilikelihood estimates......Page 158
4.1.2 Differentials for some discrete response models......Page 160
4.2.1 Simulated maximum likelihood estimation......Page 161
4.2.2 Residuals......Page 164
4.2.4 Computing issues......Page 165
4.2.5 Maximum likelihood estimation via quadrature......Page 166
4.3.2 Latent variable models for binary data......Page 168
4.3.3 Proportions as responses......Page 170
4.4.1 The iterated bootstrap......Page 171
5.1 Repeated measures data......Page 173
5.2 A 2-level repeated measures model......Page 174
5.3 A polynomial model example for adolescent growth and the prediction of adult height......Page 175
5.4 Modelling an autocorrelation structure at level 1......Page 179
5.5 A growth model with autocorrelated residuals......Page 180
5.7 Scaling across time......Page 182
5.9 Missing data......Page 183
5.10 Longitudinal discrete response data......Page 185
6.1 Introduction......Page 187
6.2 The basic 2-level multivariate model......Page 188
6.4 A rotation design example using Science Survey test scores......Page 190
6.5 Informative response selection: subject choice in examinations......Page 193
6.6 Multivariate structures at higher levels and future predictions......Page 194
6.7 Multivariate responses at several levels......Page 196
6.7.1 Fitting responses at several levels using random data augmentation......Page 197
6.8 Principal components analysis......Page 198
6.9 Multiple discriminant analysis......Page 199
Appendix 6.1 MCMC algorithm for a multivariate normal response model with constraints......Page 201
6.1.1 Constraints among parameters......Page 202
7.1 The normal multilevel multivariate model......Page 205
7.3 Sampling ordered categorical responses......Page 206
7.5 Sampling count data......Page 208
7.7 Sampling the level 1 and level 2 covariance matrices......Page 209
7.8 Model fit......Page 210
7.10 Hybrid normal/ordered variables......Page 211
7.10.1 Ordered data with known thresholds......Page 212
7.11 Discussion......Page 213
8.1 A 2-stage 2-level factor model......Page 215
8.2 A general multilevel factor model......Page 217
8.3 MCMC estimation for the factor model......Page 218
8.3.1 A 2-level factor example......Page 219
8.4 Structural equation models......Page 221
8.5 Discrete response multilevel structural equation models......Page 223
8.7 Multilevel mixture models......Page 224
9.2 Nonlinear functions of linear components......Page 227
9.3 Estimating population means......Page 228
9.4 Nonlinear functions for variances and covariances......Page 229
9.5 Examples of nonlinear growth and nonlinear level 1 variance......Page 230
Appendix 9.1 Nonlinear model estimation......Page 233
9.1.1 Modelling variances and covariances as nonlinear functions......Page 234
10.1 Sample survey structures......Page 237
10.2.1 Superpopulations......Page 238
10.2.2 Finite population inference......Page 239
10.3 Small area estimation......Page 240
10.3.1 Information at domain level only......Page 241
10.3.3 Multivariate responses......Page 242
11.1 Introduction......Page 243
11.3 Hazard and survival functions......Page 244
11.4 Parametric proportional hazard models......Page 245
11.5 The semiparametric Cox model......Page 246
11.6 Tied observations......Page 247
11.8 Example using birth interval data......Page 248
11.9 Log duration models......Page 249
11.9.1 Censored data......Page 251
11.9.2 Infinite durations......Page 252
11.10 Examples with birth interval data and children’s activity episodes......Page 253
11.11 The grouped discrete time hazards model......Page 255
11.11.1 A 2-level discrete time event history model for repeated events......Page 258
11.11.3 General discrete time event history models......Page 260
11.12 Discrete time latent normal event history models......Page 263
11.12.3 Information about the timing of events in an interval......Page 264
11.12.4 Modelling the threshold parameters and time varying covariates......Page 265
11.12.5 An example using partnership durations......Page 266
12.1 Random cross classifications......Page 269
12.2 A basic cross-classified model......Page 272
12.3 Examination results for a cross classification of schools......Page 273
12.5 Cross classifications with one unit per cell......Page 274
12.7 A general notation for cross classification......Page 275
12.8 MCMC estimation in cross-classified models......Page 276
12.1.1 An efficient IGLS algorithm......Page 278
12.1.2 Computational considerations......Page 279
13.1 Multiple membership structures......Page 281
13.2 Notation and classifications for multiple membership structures......Page 282
13.3 An example of salmonella infection......Page 283
13.4 A repeated measures multiple membership model......Page 284
13.5 Individuals as higher level units......Page 285
13.6 Spatial models......Page 287
13.7 Missing identification models......Page 289
Appendix 13.1 MCMC estimation for multiple membership models......Page 291
14.1 A basic measurement error model......Page 293
14.2.1 Measurement errors in level 1 variables......Page 294
14.2.2 Measurement errors in higher level variables......Page 295
14.3 A 2-level example with measurement error at both......Page 297
14.4 Multivariate responses......Page 299
14.6 Measurement errors for discrete explanatory variables......Page 300
14.7 MCMC estimation for measurement error models......Page 301
14.1.1 Moment based estimators for a basic 2-level model......Page 303
14.1.2 Parameter estimation......Page 304
14.1.4 Nonlinear models......Page 305
14.1.5 MCMC estimation for measurement error models: continuous variables......Page 306
14.1.6 MCMC estimation for measurement error models: discrete variables......Page 307
14.1.7 A latent normal model for discrete and continuous variables with measurement errors......Page 310
15.2.1 Regression splines......Page 311
15.3 Smoothing splines......Page 312
15.4 Semiparametric smoothing models......Page 313
15.5 Multilevel smoothing models......Page 316
15.6 General multilevel semiparametric smoothing models......Page 318
15.8 An example......Page 319
15.9 Conclusions......Page 325
16.1 Introduction......Page 327
16.2 Creating a completed dataset......Page 328
16.3 Joint modelling for missing data......Page 330
16.4.1 Sampling level 1 non-normal responses......Page 331
16.5 Multiple imputation......Page 332
16.6 A simulation example of multiple imputation for missing data......Page 333
16.7 Longitudinal data with attrition......Page 334
16.8 Partially known data values......Page 335
16.8.1 An application to record linkage......Page 336
16.8.2 Estimating a probability distribution using record linkage weights......Page 337
16.9 Conclusions......Page 338
17.2 Non-independence of level 2 residuals......Page 341
17.3 MCMC estimation for non-independent level 2 residuals......Page 343
17.3.1 Sampling the level 2 covariance matrix......Page 344
17.4 Adaptive proposal distributions in MCMC estimation......Page 345
17.5 MCMC estimation for non-independent level 1 residuals......Page 346
17.6 Modelling the level 1 variance as a function of explanatory variables with random effects......Page 347
17.7 Discrete responses with correlated random effects......Page 348
17.7.1 The probit ordered response model where the variance is a function of explanatory variables with random effects......Page 349
17.8 Calculating the DIC statistic......Page 350
17.9 A growth dataset......Page 351
17.10 Conclusions......Page 352
18.1 Software packages......Page 355
References......Page 359
Author index......Page 373
Subject index......Page 377