Latent Markov Models for Longitudinal Data: 1st Edition (Hardback) book cover

Latent Markov Models for Longitudinal Data

1st Edition

By Francesco Bartolucci, Alessio Farcomeni, Fulvia Pennoni

Chapman and Hall/CRC

252 pages | 9 B/W Illus.

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Drawing on the authors’ extensive research in the analysis of categorical longitudinal data, Latent Markov Models for Longitudinal Data focuses on the formulation of latent Markov models and the practical use of these models. Numerous examples illustrate how latent Markov models are used in economics, education, sociology, and other fields. The R and MATLAB® routines used for the examples are available on the authors’ website.

The book provides you with the essential background on latent variable models, particularly the latent class model. It discusses how the Markov chain model and the latent class model represent a useful paradigm for latent Markov models. The authors illustrate the assumptions of the basic version of the latent Markov model and introduce maximum likelihood estimation through the Expectation-Maximization algorithm. They also cover constrained versions of the basic latent Markov model, describe the inclusion of the individual covariates, and address the random effects and multilevel extensions of the model. After covering advanced topics, the book concludes with a discussion on Bayesian inference as an alternative to maximum likelihood inference.

As longitudinal data become increasingly relevant in many fields, researchers must rely on specific statistical and econometric models tailored to their application. A complete overview of latent Markov models, this book demonstrates how to use the models in three types of analysis: transition analysis with measurement errors, analyses that consider unobserved heterogeneity, and finding clusters of units and studying the transition between the clusters.


"I enjoyed reading this book very much: the writing style is clear and concise, and the mathematical presentation is easy to follow. Notations are well thought out and the technical derivations are thorough. The book is a valuable resource on latent Markov models to students, researchers, and practitioners."

—Alexander R. De Leon, Technometrics, February 2015

"… a useful contribution to the literature. … The exposition is easy to follow for anyone who has encountered random effects models for longitudinal data. … The overall structure is well thought out. … The authors clearly have considerable practical experience in the application of this technique, and they have made important contributions to its literature."

—Geoff Jones, Australian & New Zealand Journal of Statistics, 56, 2014

"The book gives an excellent introduction as well as coverage of theoretical basics of latent Markov model analysis and their practical applications. … I enjoyed reading the book, its clarity of exposition, its fairly compact format, and carefully worked out examples that did a good job in illustrating the background theory."

—Seppo Pynnönen, International Statistical Review, 2014

Table of Contents

Overview on Latent Markov Modeling


Literature review on latent Markov models

Alternative approaches

Example datasets

Background on Latent Variable and Markov Chain Models


Latent variable models

Expectation-Maximization algorithm

Standard errors

Latent class model

Selection of the number of latent classes


Markov chain model for longitudinal data


Basic Latent Markov Model


Univariate formulation

Multivariate formulation

Model identifiability

Maximum likelihood estimation

Selection of the number of latent states


Constrained Latent Markov Models


Constraints on the measurement model

Constraints on the latent model

Maximum likelihood estimation

Model selection and hypothesis testing


Including Individual Covariates and Relaxing Basic Model Assumptions



Covariates in the measurement model

Covariates in the latent model

Interpretation of the resulting models

Maximum likelihood estimation

Observed information matrix, identifiability, and standard errors

Relaxing local independence

Higher order extensions


Including Random Effects and Extension to Multilevel Data


Random-effects formulation

Maximum likelihood estimation

Multilevel formulation

Application to the student math achievement dataset

Advanced Topics about Latent Markov Modeling


Dealing with continuous response variables

Dealing with missing responses

Additional computational issues

Decoding and forecasting

Selection of the number of latent states

Bayesian Latent Markov Models


Prior distributions

Bayesian inference via reversible jump

Alternative sampling

Application to the labor market dataset

Appendix: Software

List of Main Symbols



About the Authors

Francesco Bartolucci is a professor of statistics in the Department of Economics, Finance and Statistics at the University of Perugia, where he also coordinates the Ph.D. program in mathematical and statistical methods for the economic and social sciences. His main research interests include latent variable models for cross-sectional and longitudinal categorical data, with applications ranging from educational and psychometric contexts to the analysis of labor market data.

Alessio Farcomeni is a researcher at the University of Rome "La Sapienza". His interests range from analysis of panel data and categorical time series to multiple testing, multivariate analysis and clustering, and model selection.

Fulvia Pennoni is an assistant professor of statistics in the Department of Statistics at the University of Milano-Bicocca. Her main expertise encompasses latent variable modeling. She is currently carrying out research in methods and statistics with intensive statistical programming applications.

About the Series

Chapman & Hall/CRC Statistics in the Social and Behavioral Sciences

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General