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Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates

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  • Bartolucci, Francesco
  • Farcomeni, Alessio
  • Pennoni, Fulvia

Abstract

We provide a comprehensive overview of latent Markov (LM) models for the analysis of longitudinal data. The main assumption behind these models is that the response variables are conditionally independent given a latent process which follows a first-order Markov chain. We first illustrate the more general version of the LM model which includes individual covariates. We then illustrate several constrained versions of the general LM model, which make the model more parsimonious and allow us to consider and test hypotheses of interest. These constraints may be put on the conditional distribution of the response variables given the latent process (measurement model) or on the distribution of the latent process (latent model). For the general version of the model we also illustrate in detail maximum likelihood estimation through the Expectation-Maximization algorithm, which may be efficiently implemented by recursions known in the hidden Markov literature. We discuss about the model identifiability and we outline methods for obtaining standard errors for the parameter estimates. We also illustrate methods for selecting the number of states and for path prediction. Finally, we illustrate Bayesian estimation method. Models and related inference are illustrated by the description of relevant socio-economic applications available in the literature.

Suggested Citation

  • Bartolucci, Francesco & Farcomeni, Alessio & Pennoni, Fulvia, 2012. "Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates," MPRA Paper 39023, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:39023
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    References listed on IDEAS

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    1. Francesco Bartolucci & Alessio Farcomeni, 2010. "A note on the mixture transition distribution and hidden Markov models," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(2), pages 132-138, March.
    2. Congdon, Peter, 2006. "Bayesian model choice based on Monte Carlo estimates of posterior model probabilities," Computational Statistics & Data Analysis, Elsevier, vol. 50(2), pages 346-357, January.
    3. D. Oakes, 1999. "Direct calculation of the information matrix via the EM," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(2), pages 479-482.
    4. Wilfried Seidel & Hana Ševčíková, 2004. "Types of likelihood maxima in mixture models and their implication on the performance of tests," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 56(4), pages 631-654, December.
    5. Bornmann, Lutz & Mutz, Rüdiger & Daniel, Hans-Dieter, 2008. "Latent Markov modeling applied to grant peer review," Journal of Informetrics, Elsevier, vol. 2(3), pages 217-228.
    6. C. P. Robert & T. Rydén & D. M. Titterington, 2000. "Bayesian inference in hidden Markov models through the reversible jump Markov chain Monte Carlo method," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 57-75.
    7. Francesco Bartolucci & Fulvia Pennoni, 2007. "A Class of Latent Markov Models for Capture–Recapture Data Allowing for Time, Heterogeneity, and Behavior Effects," Biometrics, The International Biometric Society, vol. 63(2), pages 568-578, June.
    8. Luigi Spezia, 2010. "Bayesian analysis of multivariate Gaussian hidden Markov models with an unknown number of regimes," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(1), pages 1-11, January.
    9. Farcomeni, Alessio, 2011. "Hidden Markov partition models," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1766-1770.
    10. Francesco Bartolucci & Fulvia Pennoni & Giorgio Vittadini, 2011. "Assessment of School Performance Through a Multilevel Latent Markov Rasch Model," Journal of Educational and Behavioral Statistics, , vol. 36(4), pages 491-522, August.
    11. Altman, Rachel MacKay, 2007. "Mixed Hidden Markov Models: An Extension of the Hidden Markov Model to the Longitudinal Data Setting," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 201-210, March.
    12. Francesco Bartolucci, 2006. "Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 155-178.
    13. Bartolucci, Francesco & Farcomeni, Alessio, 2009. "A Multivariate Extension of the Dynamic Logit Model for Longitudinal Data Based on a Latent Markov Heterogeneity Structure," Journal of the American Statistical Association, American Statistical Association, vol. 104(486), pages 816-831.
    14. Antonello Maruotti, 2011. "Mixed Hidden Markov Models for Longitudinal Data: An Overview," International Statistical Review, International Statistical Institute, vol. 79(3), pages 427-454, December.
    15. C. Yau & O. Papaspiliopoulos & G. O. Roberts & C. Holmes, 2011. "Bayesian non‐parametric hidden Markov models with applications in genomics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 37-57, January.
    16. Francesco Bartolucci & Fulvia Pennoni & Brian Francis, 2007. "A latent Markov model for detecting patterns of criminal activity," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 170(1), pages 115-132.
    17. S. Bacci & S. Pandolfi & F. Pennoni, 2014. "A comparison of some criteria for states selection in the latent Markov model for longitudinal data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(2), pages 125-145, June.
    18. Chib, Siddhartha, 1996. "Calculating posterior distributions and modal estimates in Markov mixture models," Journal of Econometrics, Elsevier, vol. 75(1), pages 79-97, November.
    19. Richard McHugh, 1956. "Efficient estimation and local identification in latent class analysis," Psychometrika, Springer;The Psychometric Society, vol. 21(4), pages 331-347, December.
    20. Farcomeni Alessio & Arima Serena, 2012. "A Bayesian autoregressive three-state hidden Markov model for identifying switching monotonic regimes in Microarray time course data," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 11(4), pages 1-31, June.
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    Citations

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    Cited by:

    1. Alessio Farcomeni, 2015. "Latent class recapture models with flexible behavioural response," Statistica, Department of Statistics, University of Bologna, vol. 75(1), pages 5-17.
    2. Ulf Böckenholt & Blakeley McShane, 2014. "Comments on: Latent Markov models: a review of the general framework for the analysis of longitudinal data with covariates," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 469-472, September.
    3. Alessio Farcomeni, 2015. "Generalized Linear Mixed Models Based on Latent Markov Heterogeneity Structures," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 42(4), pages 1127-1135, December.
    4. Gordon Anderson & Alessio Farcomeni & Grazia Pittau & Roberto Zelli, 2017. "Rectangular latent Markov models for time-specific clustering," Working Papers tecipa-589, University of Toronto, Department of Economics.
    5. Ingmar Visser & Maarten Speekenbrink, 2014. "Comments on: Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(3), pages 478-483, September.
    6. Francesco Bartolucci & Fulvia Pennoni & Giorgio Vittadini, 2016. "Causal Latent Markov Model for the Comparison of Multiple Treatments in Observational Longitudinal Studies," Journal of Educational and Behavioral Statistics, , vol. 41(2), pages 146-179, April.
    7. Lekkas, Peter & Paquet, Catherine & Howard, Natasha J. & Daniel, Mark, 2017. "Illuminating the lifecourse of place in the longitudinal study of neighbourhoods and health," Social Science & Medicine, Elsevier, vol. 177(C), pages 239-247.
    8. repec:oup:jconrs:v:44:y:2017:i:3:p:519-535. is not listed on IDEAS
    9. Maria Marino & Alessio Farcomeni, 2015. "Linear quantile regression models for longitudinal experiments: an overview," METRON, Springer;Sapienza Università di Roma, vol. 73(2), pages 229-247, August.

    More about this item

    Keywords

    EM algorithm; Bayesian framework; Forward-Backward recursions; Hidden Markov models; Measurement errors; Panel data; Unobserved heterogeneity;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • C33 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Models with Panel Data; Spatio-temporal Models

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