Generalized Latent Variable Modeling: Multilevel, Longitudinal, and Structural Equation Models

Generalized Latent Variable Modeling: Multilevel, Longitudinal, And Structural Equation Models......Page 1
Contents......Page 5
Preface......Page 9
Dedication......Page 11
Part I: Methodology......Page 12
1.1 Introduction......Page 13
1.2 ‘True’ Variable Measured With Error......Page 14
1.3 Hypothetical Constructs......Page 16
1.4 Unobserved Heterogeneity......Page 21
1.5 Missing Values And Counterfactuals......Page 26
1.6 Latent Responses......Page 28
1.7 Generating .exible Distributions......Page 29
1.9 Summary......Page 30
References......Page 31
2.1 Introduction......Page 32
2.2.1 Introduction......Page 33
2.2.2 Model structure......Page 35
2.2.3 Mean function and choice of link function......Page 37
2.3.1 Modeling underdispersion and overdispersion......Page 38
2.3.3 Models for polytomous responses......Page 39
2.3.4 Models for ordinal responses......Page 40
2.3.5 Composite links......Page 43
2.4.1 Grouped, interval-censored, ordinal and dichotomous responses......Page 44
2.4.2 Censored responses......Page 46
2.4.3 Comparative responses......Page 47
2.5 Modeling durations or survival......Page 50
2.5.1 Continuous time durations......Page 51
2.5.2 Discrete time durations......Page 55
2.6 Summary and further reading......Page 57
References......Page 447
3.2 Multilevel Regression Models......Page 59
3.2.1 Two-level Random Intercept Models......Page 60
3.2.2 Two-level Random Coe.cient Models......Page 63
3.2.3 Three-level Models......Page 67
3.2.5 Generalized Linear Mixed Models......Page 68
3.2.6 Models With Nonhierarchical Random E.ects......Page 70
3.3.1 Platonic True Scores: Measurement Models......Page 73
3.3.2 Classical True Scores: Common Factor Models......Page 76
3.3.3 Multidimensional Factor Models......Page 78
3.3.4 Item Response Models......Page 81
3.4 Latent Class Models......Page 84
3.5 Structural Equation Models With Latent Variables......Page 85
3.6 Longitudinal Models......Page 90
3.6.1 Models With Unit-speci.c E.ects......Page 91
3.6.2 Models With Correlated Residuals......Page 97
3.6.3 Models With Lagged Responses......Page 99
3.6.4 Other Covariance Structures......Page 100
3.6.5 Generalized Estimating Equations For Nonnormal Responses......Page 102
3.7 Summary And Further Reading......Page 103
4.1 Introduction......Page 105
4.2 Response Model......Page 106
4.2.1 Unifying Conventional Random Coe.cient And Factor Models......Page 107
4.2.2 Linear Predictor In Generalized Factor (gf) Formulation......Page 109
4.2.3 Linear Predictor In Generalized Random Coe.cient (grc) Formulation......Page 111
4.2.4 Example: Two-level Factor Model In Gf And Grc Formulation......Page 113
4.2.5 Example: Exploratory Latent Class Model In Gf And Grc Formulation......Page 115
4.2.6 Relaxing Conditional Independence......Page 117
4.3.1 Continuous Latent Variables......Page 118
4.3.2 Discrete Latent Variables......Page 121
4.4 Distribution Of The Disturbances......Page 122
4.4.1 Continuous Distributions......Page 123
4.4.4 Mixed Continuous And Discrete Distributions......Page 124
4.5 Parameter Restrictions And Fundamental Parameters......Page 127
4.6.3 Reduced Form For The Linear Predictor......Page 130
4.7.2 Moment Structure Of......Page 131
4.7.3 Example: 3-level Model......Page 132
4.8.1 Marginal Mean Structures And Population Average E.ects......Page 133
4.8.2 Marginal Covariance Structures......Page 135
4.9.1 Latent Variable Integration......Page 137
4.9.2 Latent Response Integration......Page 138
4.10 Reduced Form Parameters......Page 140
4.11 Summary And Further Reading......Page 142
5.1 Introduction......Page 144
5.2.2 Methods for analytical investigation of local identi.cation......Page 145
5.2.3 Applying the Jacobian method for local identi.cation......Page 147
5.2.4 Regular points and local identi.cation......Page 157
5.2.5 Empirical identi.cation......Page 159
5.3.1 Definitions......Page 161
5.3.2 Analytical investigation of equivalence......Page 162
5.3.3 Empirical equivalence......Page 165
5.4 Summary and further reading......Page 166
6.1 Introduction......Page 168
6.2 Maximum likelihood: Closed form marginal likelihood......Page 169
6.3 Maximum likelihood(  pproximate marginal likelihood......Page 170
6.3.1 Laplace approximation......Page 171
6.3.2 Numerical integration......Page 174
6.3.3 Monte Carlo Integration......Page 179
6.3.4 A tailored simulator: GHK......Page 180
6.4.1 EM algorithm......Page 183
6.4.2 Gradient methods......Page 189
6.5 Nonparametric maximum likelihood estimation......Page 191
6.6 Restricted/Residual maximum likelihood (REML)......Page 193
6.7 Limited information methods......Page 194
6.8.1 Iteratively reweighted least squares......Page 200
6.8.2 Iterative generalized least squares......Page 202
6.8.3 Marginal and penalized quasi-likelihood......Page 203
6.9 Generalized Estimating Equations (GEE)......Page 206
6.10.1 Joint maximum likelihood......Page 209
6.10.2 Conditional maximum likelihood......Page 210
6.11.1 Introduction......Page 213
6.11.3 Hierarchical Bayesian models......Page 214
6.11.4 Prior distributions......Page 215
6.11.5 Markov chain Monte Carlo......Page 216
6.11.6 Advantages and disadvantages of MCMC......Page 222
6.12 Summary......Page 223
Appendix: Some software and references......Page 225
7.1 Introduction......Page 229
7.2 Posterior distributions......Page 230
7.3.1 Empirical Bayes prediction......Page 233
7.3.2 Empirical Bayes variances and covariances......Page 237
7.4.1 Empirical Bayes modal prediction......Page 243
7.5.1 Latent score estimation......Page 245
7.5.2 Variances and covariances of ML estimator......Page 247
7.7 Ad hoc scoring methods......Page 249
7.7.1 Raw Sumscores......Page 250
7.8.1 Introduction......Page 251
7.8.2 Factor scoring as measurement proper......Page 252
7.8.3 Random e.ects scores as cluster-speci.c e.ects......Page 253
7.8.4 Classi.cation......Page 254
7.9 Summary and further reading......Page 255
Appendix: Some software......Page 257
8.2.1 Types of statistical model......Page 258
8.2.2 Modeling strategies......Page 261
8.3.1 Properties of the fundamental and structural parameter estimates......Page 264
8.3.2 Model-based standard errors......Page 265
8.3.3 Robust standard errors......Page 266
8.3.4 Likelihood ratio, Wald and score tests......Page 267
8.4 Model selection: Relative .t criteria......Page 269
8.4.2 Bayesian model selection......Page 270
8.4.3 The Akaike information criterion (AIC)......Page 273
8.5 Model adequacy: Global absolute .t criteria......Page 274
8.5.1 Tests for misspeci.cation......Page 275
8.5.2 Goodness of Fit Indices......Page 277
8.5.4 Cross-validation......Page 278
8.6 Model diagnostics: Local absolute .t criteria......Page 279
8.6.2 Higher-level residuals......Page 280
8.6.3 Assessing latent variable distributions......Page 281
8.6.4 Level-1 residuals......Page 282
8.6.5 Identifying outliers......Page 284
8.6.6 In.uence......Page 285
8.7 Summary and further reading......Page 287
Part II: Applications......Page 289
9.2 Respiratory infection in children: A random intercept model......Page 290
9.3 Diagnosis of myocardial infarction: A latent class model......Page 293
9.4 Arithmetic reasoning: Item response models......Page 297
9.5.1 Introduction......Page 304
9.5.2 Approaches to meta-analysis......Page 305
9.5.3 Random e.ects modeling of patient-level data......Page 307
9.6 Wives’ employment transitions: Markov models with unobserved heterogeneity......Page 312
9.7 Counting snowshoe hares: Capture-recapture models with heterogeneity......Page 316
9.8 Attitudes to abortion: A multilevel item response model......Page 320
9.9 Summary and further reading......Page 323
10.2.1 Introduction......Page 326
10.2.2 Growth curve modeling......Page 327
10.2.3 Latent growth curve modeling......Page 329
10.3.1 Introduction......Page 331
10.3.2 Factor dimensionality and reliability......Page 334
10.3.3 Item bias......Page 339
10.3.4 Conclusion......Page 342
10.4.1 Introduction......Page 343
10.4.2 Factor dimensionality......Page 345
10.4.3 Reliabilities......Page 348
10.5 Summary and further reading......Page 352
11.2.1 Introduction......Page 354
11.2.2 Modeling overdispersion......Page 355
11.3.2 Modelling repeated counts......Page 360
11.3.3 Model diagnostics......Page 363
11.4.1 Introduction......Page 366
11.4.2 Random intercept models......Page 369
11.4.3 Spatially correlated random intercepts......Page 370
11.4.4 Introducing a countyklevel covariate......Page 372
11.5 Summary and further reading......Page 375
12.2 Modeling multiple events clustered duration data......Page 378
12.3.1 Introduction......Page 380
12.3.2 Random intercept models......Page 383
12.3.3 Modeling ‘telescoping’ e.ects......Page 386
12.4.1 Introduction......Page 387
12.4.2 Cox regression......Page 388
12.4.3 Proportional hazards regression with multinormal latent variables......Page 390
12.4.4 Latent class proportional hazards models......Page 393
12.4.5 Nonparametric maximum likelihood estimation......Page 394
12.5 Summary and further reading......Page 395
13.2 Heterogeneity and ‘Independence from Irrelevant Alternatives\'......Page 397
13.3.1 Fixed part......Page 399
13.3.3 Level-2 random part......Page 400
13.4 British general elections: Multilevel models for discrete choice and rankings......Page 401
13.5 Post-materialism: A latent class model for rankings......Page 408
13.6 Consumer preferences for co.ee makers: A conjoint choice......Page 412
13.7 Summary and further reading......Page 419
14.2.1 Introduction......Page 420
14.2.2 Logistic regression with covariate measurement error......Page 421
14.3.1 Introduction......Page 425
14.3.2 Latent class logistic regression......Page 427
14.4.2 The compliance problem......Page 430
14.4.3 CACE modeling......Page 433
14.5.1 Introduction......Page 435
14.5.2 Drinking model......Page 437
14.5.3 Advice model......Page 438
14.5.4 Joint model for drinking and advice: advice as endogenous treatment......Page 439
14.6.1 Introduction......Page 441
14.6.2 Latent marker model......Page 442
14.6.3 Hazard model with latent covariate......Page 443
14.7 Summary and further reading......Page 446