Bayesian Smoothing and Regression for Longitudinal, Spatial and Event History Data

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ISBN: 9780199533022 | 0199533024
Cover: Hardcover
Copyright: 6/24/2011

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Several recent advances in smoothing and semiparametric regression are presented in this book from a unifying, Bayesian perspective. Simulation-based full Bayesian Markov chain Monte Carlo (MCMC) inference, as well as empirical Bayes procedures closely related to penalized likelihood estimation and mixed models, are considered here. Throughout, the focus is on semiparametric regression and smoothing based on basis expansions of unknown functions and effects in combination with smoothness priors for the basis coefficients. Beginning with a review of basic methods for smoothing and mixed models, longitudinal data, spatial data and event history data are treated in separate chapters. Worked examples from various fields such as forestry, development economics, medicine and marketing are used to illustrate the statistical methods covered in this book. Most of these examples have been analyzed using implementations in the Bayesian software, BayesX, and some with R Codes.

List of Algorithms	p. XV
List of Examples	p. xvi
Introduction: Scope of the Book and Applications	p. 1
Semiparametric regression	p. 1
Applications	p. 4
Basic Concepts for Smoothing and Semiparametric Regression	p. 18
Time series smoothing	p. 19
Gaussian observation models	p. 19
Penalized least-squares smoothing	p. 20
Bayesian smoothing	p. 23
Some modifications and extensions	p. 27
Estimation of smoothing parameters and variances	p. 27
Other model components	p. 27
Correlated errors	p. 28
Locally adaptive smoothing	p. 29
Unequally spaced time-series observations	p. 30
Non-Gaussian observation models	p. 30
Semiparametric regression based on penalized splines	p. 34
Gaussian observation models	p. 34
Polynomial splines	p. 35
Truncated power series and B-splines	p. 36
Nonparametric regression based on polynomial splines	p. 39
Characteristics of a spline fit	p. 41
P-splines	p. 42
Customized penalties	p. 47
Bayesian P-splines	p. 48
Bayesian inference	p. 52
Degrees of freedom of a P-spline	p. 55
Univariate non-Gaussian observation models	p. 57
Penalized likelihood estimation	p. 58
Bayesian inference	p. 60
Latent variable representations	p. 63
Categorical observation models	p. 67
Nominal response models	p. 68
Cumulative models for ordinal responses	p. 73
Sequential models for ordinal responses	p. 77
Penalised likelihood inference	p. 79
Bayesian inference	p. 81
Related smoothing approaches	p. 82
Integral penalties	p. 82
Smoothing splines	p. 83
Bayesian interpretation of smoothing splines	p. 85
Reproducing kernel Hilbert spaces	p. 87
Other types of basis functions	p. 89
Generalized additive models	p. 90
Gaussian additive models	p. 90
Simultaneous penalized least-squares (PLS) smoothing	p. 92
Backfitting	p. 93
Bayesian backfitting: the Gibbs sampler	p. 94
Non-Gaussian additive models	p. 100
Notes and further reading	p. 104
Generalized Linear Mixed Models	p. 107
Linear mixed models with Gaussian random effects	p. 108
Linear mixed models for longitudinal data	p. 108
Advantages of mixed models	p. 111
Marginal and conditional formulation	p. 114
Multilevel models	p. 115
General linear mixed models	p. 116
Bayesian linear mixed models	p. 117
Likelihood-based inference	p. 119
Estimation and prediction	p. 119
Estimation of regression coefficients for given variance components	p. 121
Maximum likelihood (ML) estimation of variance components	p. 122
REML estimation for the variance parameters	p. 122
Details on REML estimation for variance parameters	p. 124
Bayesian interpretation of ML and REML estimation	p. 125
Testing hypotheses	p. 126
Bayesian inference	p. 127
Empirical Bayes inference	p. 127
Full Bayes inference	p. 128
Full Bayes inference for longitudinal data	p. 129
Linear mixed models with flexible random effects priors	p. 138
Finite mixture models	p. 139
Finite mixture of normals prior: the heterogeneity model	p. 140
MCMC inference for finite mixture models	p. 141
Penalized mixture of normals priors	p. 142
Dirichlet processes	p. 143
Dirichlet process: descriptive definition	p. 145
Stick breaking representation	p. 145
Posterior Dirichlet process	p. 147
Predictive distribution and clustering property	p. 148
Dirichlet process mixtures	p. 149
LMM with DP-based random effects priors	p. 150
Longitudinal data LMMs with DP random effects priors	p. 150
Longitudinal data LMMs with DPM priors	p. 153
Generalized linear mixed models	p. 155
Generalized linear mixed models for longitudinal data	p. 157
GLMMs for univariate responses	p. 157
Marginal and conditional models	p. 159
Interpretation of regression parameters	p. 160
GLMMs for categorical responses	p. 161
General mixed models for non-Gaussian responses	p. 164
Likelihood-based and empirical Bayes inference	p. 164
Full Bayesian inference for longitudinal data	p. 171
Longitudinal data GLMMs with flexible random effects priors	p. 174
GLMMs with DP random effects priors	p. 174
GLMMs with DPM random effects priors	p. 175
Notes and further reading	p. 176
Semiparametric Mixed Models for Longitudinal Data	p. 178
Semiparametric mixed models based on Gaussian priors	p. 179
Observation models for univariate responses from exponential families	p. 179
Generalized additive mixed models	p. 179
Varying coefficient mixed models	p. 181
ANOVA type interactions	p. 182
Generic representation	p. 182
Observation models for categorical responses	p. 183
Gaussian priors for regression parameters and functions	p. 185
Estimation	p. 187
Empirical Bayes inference	p. 187
Intuitive example	p. 187
Mixed model representation for P-splines	p. 190
Mixed model representation for general penalized smoothers	p. 192
Mixed model-based estimation of SPMMs	p. 196
Identifiability	p. 200
Constrained smoothness priors	p. 205
Credible intervals and bands	p. 206
Tests on the functional form	p. 207
Full Bayes estimation with Gaussian priors for regression parameters	p. 212
Gaussian SPMMs	p. 213
Exponential family SPMMs	p. 219
Categorical SPMMs	p. 222
Credible intervals and bands	p. 225
Smoothing and correlation	p. 229
Correlations induced by penalized smoothing approaches	p. 230
Identifiability problems	p. 233
Radial basis functions and correlated errors	p. 237
Summary	p. 238
Extensions based on non-Gaussian priors	p. 238
SPMMs with DP-based random effects priors	p. 238
Shrinkage priors for high-dimensional regression parameters	p. 245
Ridge prior	p. 245
Lasso prior	p. 247
L_q priors	p. 254
Further examples	p. 255
Locally adaptive priors for functions	p. 256
Locally adaptive penalties	p. 256
Knot selection strategies	p. 261
Model choice and model checking	p. 263
Bayes factors and model selection criteria	p. 264
Bayes factors and marginal likelihoods	p. 264
Methods for estimating Bayes factors and marginal likelihoods	p. 266
Information criteria: AIC, BIC and DIC	p. 269
BIC	p. 270
AIC	p. 270
DIC	p. 275
Predictive methods for model assessment	p. 276
Alternative predictive distributions	p. 277
Assessing calibration: Pit and Bot	p. 278
Proper scoring rules	p. 281
Custom summary statistics	p. 282
Monte Carlo estimation of predictive measures	p. 283
Posterior predictive goodness-of-fit assessment	p. 284
Exact cross validatory predictive assessment	p. 286
Approximate cross validatory predictive assessment	p. 287
Predictor selection using spike and slab priors	p. 290
Variable selection	p. 291
Function selection	p. 297
Covariance matrix selection for random effects	p. 302
Notes and further reading	p. 303
Individual-specific curves and functional mixed models	p. 303
Approximate Bayesian inference	p. 304
Variational Bayes approaches	p. 304
Integrated nested laplace approximation (INLA)	p. 305
Further comments	p. 306
Spatial Smoothing, Interactions and Geoadditive Regression	p. 307
Spatial data structures	p. 309
Point-referenced data: Continuous spatial information	p. 309
Interaction surfaces	p. 314
Areal data: Discrete spatial information	p. 318
Continuous vs. discrete spatial information	p. 321
Spatial regression models	p. 322
Other types of spatial data	p. 325
Discrete spatial data: Markov random fields	p. 325
A heuristic spatial smoothness prior	p. 326
Markov random fields	p. 328
Definition of Markov random fields	p. 328
Brook's lemma	p. 330
Negpotential function and Hammersley-Clifford theorem	p. 332
Auto-models	p. 334
Gaussian Markov random fields/auto-normal models	p. 336
Basis function representation of Gmrfs	p. 340
Direct vs. latent autoregressive models	p. 344
Extended Markov random field models	p. 346
Spatial smoothing approaches and interactions	p. 347
Tensor product penalized splines	p. 347
Tensor product bases	p. 348
Kronecker product penalties for tensor product bases	p. 352
Generalized penalty concepts	p. 358
Null spaces of bivariate penalties	p. 362
Higher-order Markov random fields on regular grids	p. 364
Higher-order interactions	p. 364
Radial bases	p. 366
Space filling algorithm	p. 370
Penalties for radial bases	p. 371
Tensor products vs. radial bases	p. 373
Continuous spatial data: Stationary Gaussian random fields	p. 374
Model formulation	p. 375
Correlation functions	p. 378
Parametric classes of correlations functions	p. 381
Bochner's theorem	p. 385
Range anisotropic correlation functions	p. 386
Variogram	p. 389
Estimation of covariance and correlation parameters	p. 391
Nonparametric covariogram and variogram estimation	p. 392
Gaussian random fields as radial basis function smoothers	p. 395
Identifiability in geostatistical models	p. 396
Classical geostatistics	p. 398
Geoadditive regression	p. 399
Full Bayes inference	p. 400
Empirical Bayes inference	p. 407
Notes and further reading	p. 413
Event History Data	p. 415
Survival data	p. 419
Basic notions for continuous survival times	p. 419
Censoring and truncation	p. 420
Likelihood contributions for different types of censoring	p. 424
Discrete-time survival data	p. 425
Continuous-time hazard regression	p. 428
Observation models, priors and likelihoods	p. 430
Observation models	p. 430
Priors	p. 431
Likelihoods	p. 432
Full Bayes inference for right-censored observations	p. 433
Piecewise exponential model	p. 433
Models with general structured additive predictor	p. 434
Empirical Bayes (EB) inference for right-censored observations	p. 444
Inference for interval-censored observations	p. 451
Discrete-time hazard regression	p. 455
Accelerated failure time models	p. 464
Observation models, likelihoods and priors	p. 466
Penalized Gaussian mixture (PGM)	p. 467
Aft models with DP(M) priors	p. 470
Multi-state models	p. 470
Continuous-time transition rate models	p. 471
Counting process representation and likelihood Contributions	p. 472
Empirical and full Bayes inference	p. 474
Model checking based on martingale residuals	p. 481
Discrete-time multi-state models	p. 485
Notes and further reading	p. 488
Models for correlated survival data	p. 490
Joint modelling of longitudinal and event history data	p. 492
Bibliography	p. 495
Index	p. 519
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Bayesian Smoothing and Regression for Longitudinal, Spatial and Event History Data

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