Latent Markov Models for Longitudinal Data

248 Pages·2012·2.406 MB·English

by  Francesco Bartolucci, Alessio Farcomeni, Fulvia Pennoni

Latent
Markov Models
for Longitudinal
Data
© 2012 by Taylor & Francis Group, LLC
Chapman & Hall/CRC
Statistics in the Social and Behavioral Sciences Series
Series Editors
A. Colin Cameron J. Scott Long
University of California, Davis, USA Indiana University, USA
Andrew Gelman Sophia Rabe-Hesketh
Columbia University, USA University of California, Berkeley, USA
Anders Skrondal
Norwegian Institute of Public Health, Norway
Aims and scope
Large and complex datasets are becoming prevalent in the social and behavioral
sciences and statistical methods are crucial for the analysis and interpretation of such
data. This series aims to capture new developments in statistical methodology with
particular relevance to applications in the social and behavioral sciences. It seeks to
promote appropriate use of statistical, econometric and psychometric methods in
these applied sciences by publishing a broad range of reference works, textbooks and
handbooks.
The scope of the series is wide, including applications of statistical methodology in
sociology, psychology, economics, education, marketing research, political science,
criminology, public policy, demography, survey methodology and official statistics. The
titles included in the series are designed to appeal to applied statisticians, as well as
students, researchers and practitioners from the above disciplines. The inclusion of real
examples and case studies is therefore essential.
Published Titles
Analysis of Multivariate Social Science Data, Second Edition
David J. Bartholomew, Fiona Steele, Irini Moustaki, and Jane I. Galbraith
Applied Survey Data Analysis
Steven G. Heeringa, Brady T. West, and Patricia A. Berglund
Bayesian Methods: A Social and Behavioral Sciences Approach, Second Edition
Jeff Gill
Foundations of Factor Analysis, Second Edition
Stanley A. Mulaik
Informative Hypotheses: Theory and Practice for Behavioral and Social Scientists
Herbert Hoijtink
Latent Markov Models for Longitudinal Data
Francesco Bartolucci, Alessio Farcomeni, and Fulvia Pennoni
Linear Causal Modeling with Structural Equations
Stanley A. Mulaik
Multiple Correspondence Analysis and Related Methods
Michael Greenacre and Jorg Blasius
Multivariable Modeling and Multivariate Analysis for the Behavioral Sciences
Brian S. Everitt
Statistical Test Theory for the Behavioral Sciences
Dato N. M. de Gruijter and Leo J. Th. van der Kamp
© 2012 by Taylor & Francis Group, LLC
Chapman & Hall/CRC
Statistics in the Social and Behavioral Sciences Series
Latent
Markov Models
for Longitudinal
Data
Francesco Bartolucci
Alessio Farcomeni
Fulvia Pennoni
© 2012 by Taylor & Francis Group, LLC
CRC Press
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Contents
List of Figures xi
List of Tables xiii
Preface xvii
1 Overview on latent Markov modeling 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature review on latent Markov models . . . . . . . 4
1.3 Alternative approaches . . . . . . . . . . . . . . . . . . 7
1.4 Example datasets . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Marijuana consumption dataset . . . . . . . . . . 8
1.4.2 Criminal conviction history dataset . . . . . . . . 9
1.4.3 Labor market dataset . . . . . . . . . . . . . . . 11
1.4.4 Student math achievement dataset . . . . . . . . 13
2 BackgroundonlatentvariableandMarkovchainmodels 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Latent variable models . . . . . . . . . . . . . . . . . . 17
2.3 Expectation-Maximization algorithm . . . . . . . . . . 21
2.4 Standard errors . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Latent class model . . . . . . . . . . . . . . . . . . . . . 26
2.5.1 Basic version . . . . . . . . . . . . . . . . . . . . 27
2.5.2 Advanced versions . . . . . . . . . . . . . . . . . 28
2.5.3 Maximum likelihood estimation . . . . . . . . . . 32
2.5.4 Selection of the number of latent classes . . . . . 33
2.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.1 Marijuana consumption dataset . . . . . . . . . . 35
2.6.2 Criminal conviction history dataset . . . . . . . . 38
2.7 Markov chain model for longitudinal data . . . . . . . . 41
2.7.1 Basic version . . . . . . . . . . . . . . . . . . . . 41
2.7.2 Advanced versions . . . . . . . . . . . . . . . . . 43
2.7.3 Likelihood inference . . . . . . . . . . . . . . . . 44
2.7.4 Maximum likelihood estimation . . . . . . . . . . 45
v
© 2012 by Taylor & Francis Group, LLC
vi Contents
2.7.5 Model selection . . . . . . . . . . . . . . . . . . . 46
2.8 Applications . . . . . . . . . . . . . . . . . . . . . . . . 46
2.8.1 Marijuana consumption dataset . . . . . . . . . . 46
2.8.2 Criminal conviction history dataset . . . . . . . . 48
3 Basic latent Markov model 51
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Univariate formulation . . . . . . . . . . . . . . . . . . 51
3.3 Multivariate formulation . . . . . . . . . . . . . . . . . 56
3.4 Model identifiability . . . . . . . . . . . . . . . . . . . . 58
3.5 Maximum likelihood estimation . . . . . . . . . . . . . 59
3.5.1 Expectation-Maximizationalgorithm . . . . . . . 60
3.5.1.1 Univariate formulation . . . . . . . . . 60
3.5.1.2 Multivariate formulation . . . . . . . . 63
3.5.1.3 Initializationofthealgorithmandmodel
identifiability . . . . . . . . . . . . . . . 64
3.5.2 Alternativealgorithmsformaximumlikelihoodes-
timation . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.3 Standard errors . . . . . . . . . . . . . . . . . . . 67
3.6 Selection of the number of latent states . . . . . . . . . 67
3.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . 68
3.7.1 Marijuana consumption dataset . . . . . . . . . . 69
3.7.2 Criminal conviction history dataset . . . . . . . . 74
Appendix 1: Efficient implementation of recursions . . . . . . 77
4 Constrained latent Markov models 79
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Constraints on the measurement model . . . . . . . . . 80
4.2.1 Univariate formulation . . . . . . . . . . . . . . . 80
4.2.1.1 Binary response variables . . . . . . . . 81
4.2.1.2 Categoricalresponse variables . . . . . 83
4.2.2 Multivariate formulation . . . . . . . . . . . . . . 85
4.3 Constraints on the latent model . . . . . . . . . . . . . 86
4.3.1 Linear model on the transition probabilities . . . 87
4.3.2 Generalized linear model on the transition proba-
bilities . . . . . . . . . . . . . . . . . . . . . . . . 88
4.4 Maximum likelihood estimation . . . . . . . . . . . . . 90
4.4.1 Expectation-Maximizationalgorithm . . . . . . . 91
4.4.1.1 Univariate formulation . . . . . . . . . 91
4.4.1.2 Multivariate formulation . . . . . . . . 93
4.4.1.3 Initializationofthealgorithmandmodel
identifiability . . . . . . . . . . . . . . . 93
© 2012 by Taylor & Francis Group, LLC
Contents vii
4.5 Model selection and hypothesis testing . . . . . . . . . 94
4.5.1 Model selection . . . . . . . . . . . . . . . . . . . 94
4.5.2 Hypothesis testing . . . . . . . . . . . . . . . . . 95
4.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . 96
4.6.1 Marijuana consumption dataset . . . . . . . . . . 96
4.6.2 Criminal conviction history dataset . . . . . . . . 100
Appendix 1: Marginal parametrization . . . . . . . . . . . . 102
Appendix 2: Implementation of the M-step . . . . . . . . . . 105
5 Including individual covariates and relaxing basic model
assumptions 109
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 109
5.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Covariates in the measurement model . . . . . . . . . . 112
5.3.1 Univariate formulation . . . . . . . . . . . . . . . 112
5.3.2 Multivariate formulation . . . . . . . . . . . . . . 114
5.4 Covariates in the latent model . . . . . . . . . . . . . . 115
5.5 Interpretation of the resulting models . . . . . . . . . . 116
5.6 Maximum likelihood estimation . . . . . . . . . . . . . 117
5.6.1 Expectation-Maximizationalgorithm . . . . . . . 118
5.7 Observed information matrix, identifiability, and stan-
dard errors . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.8 Relaxing local independence . . . . . . . . . . . . . . . 121
5.8.1 Conditional serial dependence . . . . . . . . . . . 121
5.8.2 Conditional contemporary dependence . . . . . . 123
5.9 Higher order extensions . . . . . . . . . . . . . . . . . . 126
5.10 Applications . . . . . . . . . . . . . . . . . . . . . . . . 130
5.10.1 Criminal conviction history dataset . . . . . . . . 130
5.10.2 Labor market dataset . . . . . . . . . . . . . . . 134
Appendix 1: Multivariate link function . . . . . . . . . . . . 137

6 Including random effects and extension to multilevel
data 139
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2 Random-effects formulation . . . . . . . . . . . . . . . . 139
6.2.1 Model assumptions . . . . . . . . . . . . . . . . . 140
6.2.1.1 Random effects in the measurement
model . . . . . . . . . . . . . . . . . . . 140
6.2.1.2 Random effects in the latent model . . 142
6.2.2 Manifest distribution . . . . . . . . . . . . . . . . 143
6.3 Maximum likelihood estimation . . . . . . . . . . . . . 145
6.4 Multilevel formulation . . . . . . . . . . . . . . . . . . . 148
© 2012 by Taylor & Francis Group, LLC
viii Contents
6.4.1 Model assumptions . . . . . . . . . . . . . . . . . 148
6.4.2 Manifestdistributionandmaximumlikelihoodes-
timation . . . . . . . . . . . . . . . . . . . . . . . 151
6.5 Application to the student math achievement dataset . 152
7 Advanced topics about latent Markov modeling 157
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 157
7.2 Dealing with continuous response variables . . . . . . . 157
7.2.1 Linear regression . . . . . . . . . . . . . . . . . . 158
7.2.2 Quantile regression . . . . . . . . . . . . . . . . . 159
7.2.3 Estimation . . . . . . . . . . . . . . . . . . . . . 160
7.3 Dealing with missing responses . . . . . . . . . . . . . . 162
7.4 Additional computational issues . . . . . . . . . . . . . 164
7.4.1 MaximizationofthelikelihoodthroughtheNewton-
Raphson algorithm . . . . . . . . . . . . . . . . . 164
7.4.1.1 A general description of the algorithm . 164
7.4.1.2 Use for latent Markov models. . . . . . 165
7.4.2 Parametric bootstrap . . . . . . . . . . . . . . . 166
7.5 Decoding and forecasting . . . . . . . . . . . . . . . . . 168
7.5.1 Local decoding . . . . . . . . . . . . . . . . . . . 169
7.5.2 Global decoding . . . . . . . . . . . . . . . . . . 170
7.5.3 Forecasting . . . . . . . . . . . . . . . . . . . . . 171
7.6 Selection of the number of latent states . . . . . . . . . 172
8 Bayesian latent Markov models 177
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 177
8.2 Prior distributions . . . . . . . . . . . . . . . . . . . . . 178
8.2.1 Basic latent Markov model . . . . . . . . . . . . 178
8.2.2 Constrained and extended latent Markov models 180
8.3 Bayesian inference via Reversible Jump . . . . . . . . . 181
8.3.1 Reversible Jump algorithm . . . . . . . . . . . . 181
8.3.2 Post-processingthe Reversible Jump output . . . 186
8.3.3 Inference based on the simulated posterior distri-
bution . . . . . . . . . . . . . . . . . . . . . . . . 187
8.4 Alternative sampling strategy . . . . . . . . . . . . . . 188
8.4.1 Continuousbirthanddeathprocessbasedondata
augmentation . . . . . . . . . . . . . . . . . . . . 188
8.4.2 Parallelsampling . . . . . . . . . . . . . . . . . . 190
8.5 Application to the labor market dataset . . . . . . . . . 190
© 2012 by Taylor & Francis Group, LLC
Contents ix
A Software 197
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 197
A.2 Package LMest . . . . . . . . . . . . . . . . . . . . . . . 197
List of Main Symbols 209
Bibliography 215
Index 231

© 2012 by Taylor & Francis Group, LLC
© 2012 by Taylor & Francis Group, LLC