Pennoni, Fulvia and Romeo, Isabella
(2016):
*Latent Markov and growth mixture models for ordinal individual responses with covariates: a comparison.*

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## Abstract

We propose a short review between two alternative ways of modeling stability and change of longitudinal data when time-fixed and time-varying covariates referred to the observed individuals are available. They both build on the foundation of the finite mixture models and are commonly applied in many fields. They look at the data by a different perspective and in the literature they have not been compared when the ordinal nature of the response variable is of interest. The latent Markov model is based on time-varying latent variables to explain the observable behavior of the individuals. The model is proposed in a semi-parametric formulation as the latent Markov process has a discrete distribution and it is characterized by a Markov structure. The growth mixture model is based on a latent categorical variable that accounts for the unobserved heterogeneity in the observed trajectories and on a mixture of normally distributed random variable to account for the variability of growth rates. To illustrate the main differences among them we refer to a real data example on the self reported health status.

Item Type: | MPRA Paper |
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Original Title: | Latent Markov and growth mixture models for ordinal individual responses with covariates: a comparison |

English Title: | Latent Markov and growth mixture models for ordinal individual responses with covariates: a comparison |

Language: | English |

Keywords: | Dynamic factor model, Expectation-Maximization algorithm, Forward-Backward recursions, Latent trajectories, Maximum likelihood, Monte Carlo methods. |

Subjects: | C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C14 - Semiparametric and Nonparametric Methods: General C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C18 - Methodological Issues: General C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C33 - Panel Data Models ; Spatio-temporal Models C - Mathematical and Quantitative Methods > C3 - Multiple or Simultaneous Equation Models ; Multiple Variables > C38 - Classification Methods ; Cluster Analysis ; Principal Components ; Factor Models C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C63 - Computational Techniques ; Simulation Modeling I - Health, Education, and Welfare > I1 - Health > I11 - Analysis of Health Care Markets I - Health, Education, and Welfare > I1 - Health > I12 - Health Behavior |

Item ID: | 72939 |

Depositing User: | Prof. Fulvia Pennoni |

Date Deposited: | 11 Aug 2016 10:19 |

Last Modified: | 27 Sep 2019 09:19 |

References: | Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle Booktitle Second International symposium on information theory in Petrov, B. N. and Csaki F., 267-281, Akademiai Kiado Budapest. Andruff, H. and Carraro, N. and Thompson, A. and Gaudreau, P. and Louvet, B. (2009). Latent class growth modelling: a tutorial Journal Tutorials in Quantitative Methods for Psychology, 5, 11--24. Asparouhov, T. and Muth\'{e}n, B. O. (2009). Exploratory structural equation modeling Journal Structural Equation Modeling: A Multidisciplinary Journal, 16, 397--438 Publisher Taylor \& Francis. Bacci, S. and Pandolfi, S. and Pennoni, F. (2014). A comparison of some criteria for states selection in the latent {M}arkov model for longitudinal data Journal Advances in Data Analysis and Classification, 8, 125-145. Factor analysis of relative growth Baker, G. A. (1954). Journal Growth, 18, 137. Bartolucci, F. and Bacci, S. and Pennoni, F. (2014). Longitudinal analysis of self-reported health status by mixture latent auto-regressive models Journal Journal of the Royal Statistical Society C, 63, 267-288. Visser, I. and Speekenbrink, M. (2014). Comment on: Latent Markov models: a review of a general framework for the analysis longitudinal data with Test, 23, 478-483. Bartolucci, F. and Farcomeni, A., G. and Pennoni, F. (2014). Latent Markov models: a review of a general framework for the analysis of longitudinal data with covariates, Journal Test 23, 433-465. Bartolucci, F. and Farcomeni, A. and Pennoni, F. (2013). Latent Markov models for longitudinal data Chapman and Hall/CRC press, Boca Raton Menard, S. (2008) Handbook of longitudinal research: Design, measurement, and analysis Elsevier, San Diego Bauer, D. J. and Curran, P. J. (2003). Distributional assumptions of growth mixture models: implications for overextraction of latent trajectory classes. Journal Psychological methods 8, 338-363. Bauer, D. J. and Curran, P. J. (2004). The integration of continuous and discrete latent variable models: Potential problems and promising opportunities. Journal Psychological methods, 9, 3--29. Baum, L.E. and Petrie, T. and Soules, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains Journal Annals of Mathematical Statistics, 41, 164-171. Bollen, K. A. and Curran, P. J. (2006). Latent curve models: a structural equation perspective. Wiley, New Jersey. Boucheron, S. and Gassiat, E. (2007). An information-theoretic perspective on order estimation In: Inference in Hidden {M}arkov Models Editors O. Capp\'{e}, E. Moulines, T. Ryd\'{e}n, 565--602. Cheng, R. C. H. and Liu, W. B. (2001). The consistency of estimators in finite mixture models Journal Scandinavian Journal of Statistics, 28, 603-616. Dempster, A. P. and Laird, N. M. and Rubin, D. B.(1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion) Journal Journal of the Royal Statistical Society B, 39, 1-38. Duncan, T. E. and Duncan, S. C. and Strycker, L. A. and Li, F. and Alpert, A. (1999). An introduction to latent variable growth curve modeling: Concepts, issues, and application Erlbaum Associates, London Feng, Z. and McCulloch, C. E. (1996). Using bootstrap likelihood ratios in finite mixture models Journal J. R. Statist. Soc., series B, 58, 609-617. Green, P. J. (1995). Reversible jump Markov chain Monte Carlo computation and bayesian model determination Biometrika, 82, 711--732. Hipp, J. R. and Bauer, D. J. (2006). Local solutions in the estimation of growth mixture models Journal Psychological Methods, 11, 36--53 Author Hoyle, R. H. (2012). Handbook of structural equation modeling Guilford Publication Juang, B. H. and Rabiner, L. R. (1991). Hidden Markov models for speech recognition Journal Technometrics, 33, 251-272. Kreuter, F. and Muthén, B. O. (2008). Analyzing criminal trajectory profiles: Bridging multilevel and group-based approaches using growth mixture modeling Journal Journal of Quantitative Criminology, 24, 1--31. Lazarsfeld, P. F. and Henry, N. W. (1968). Latent structure analysis Houghton Mifflin, Boston. Louis, T. A. (1982). Finding the observed information matrix when using the EM-algorithm, Journal Journal of the Royal Statistical Society, Series B, 44, 226-233 Lu, T. and Poon, W. and Tsang, Y. (2011). Latent growth curve modeling for longitudinal ordinal responses with applications Journal Computational Statistics \& Data Analysis, 55, 1488--1497. Magidson, J. and Vermunt, J. K. (2001). Latent class factor and cluster models, bi-plots and related graphical displays Journal Sociological Methodology, 31, 223-264. Masyn, K. E. and Petras, H. and Liu, W. (2014). Growth curve models with categorical outcomes. Encyclopedia of Criminology and Criminal Justice, 2013--2025. McArdle, J. J. (1986). Latent variable growth within behavior genetic models Journal Behavior Genetics, 16, 163--200. McArdle, J. J. and Epstein, D. (1987). Latent growth curves within developmental structural equation models Author Journal Child development, 58, 110--133. McCullagh, P. (1980). Regression models for ordinal data (with discussion), Journal Journal of the Royal Statistical Society, Series B, 42, 109-142. McLachlan, G. J. and Peel, D. (2000). Finite mixture models. Wiley Meredith, W. and Tisak, J. (1990). Latent curve analysis, Journal Psychometrika, 55, 107--122. Meredith, W. and Tisak, J. (1984). ``Tuckerizing'' curves Proceedings annual meeting of the Psychometric Society, Santa Barbara, CA Miller, G. A. (1952). Finite Markov processes in psychology Journal Psychometrika, 17, 149-167. Muthén, B. O. (2003). Statistical and substantive checking in growth mixture modeling: comment on Bauer and Curran Journal Psychological Methods, 8, 369--377. Muthén, B.O. (2001). Second generation structural equation modeling with a combination of categorical and continuous latent variables: New opportunities for latent class latent growth modeling, In: New methods for the analysis of change. Washington, D.C., APA Editors L. M. Collins, A. Sayer 291--322. Muthén, B. O. and Muthén, L. K. (2000). Integrating person-centered and variable-centered analyses: Growth mixture modeling with latent trajectory classes, Journal Alcoholism: Clinical and experimental research, 24, 882--891. Muthén, B. O. and Shedden, K. (1999). Finite mixture modeling with mixture outcomes using the EM algorithm Journal Biometrics 55, 463--469. Mplus (2012). Muthén, L. K. and Muthén, B. O. The comprehensive modelling program for applied researchers: user's guide Volume 5. Muthén, B. O. (2002). Beyond SEM: general latent variable modeling, Journal Behaviormetrika, 29, 81--118. Muthén, B. O. (2001). Latent variable mixture modeling In: New developments and techniques in structural equation modeling Editors G. A. Marcoulides, R. E. Schumacker, Lawrence Erlbaum Associates 1--33. Nagin, D. S. (1999). Analyzing developmental trajectories: A semiparametric, group-based approach. Journal Psychological methods, 4, 139--157. Nagin, D. S. and Tremblay, R. E. (2005). Developmental trajectory groups: fact or a useful statistical fiction? Journal Criminology, 43, 873--904. Nagin, D. S. and Tremblay, R. E. (2001). Analyzing developmental trajectories of distinct but related behaviors: A group-based method. Journal Psychological methods, 6, 18--34. Nylund, K. L. and Masyn, K. E. (2008). Covariates and latent class analysis: Results of a simulation study Booktitle society for prevention research annual meeting. Nylund, K. L. and Asparouhov, T. and Muthén, B. O. (2007). Deciding on the number of classes in latent class analysis and growth mixture modeling: a Monte Carlo simulation study Journal Structural Equation Modeling, 14, 535--569 Publisher Taylor \& Francis. Orchard, T. and Woodbury, M. A. (1972). A missing information principle: theory and applications In: Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, 697-715. Pennoni, F. (2014). Issues on the estimation of latent variable and latent class models Publisher Scholars' Press (2014). Saarb\"{u}cken. Pennoni, F. and Vittadini, G. (2015). Hidden Markov and mixture panel data models for ordinal variables derived from original continuous responses Proceedings of the 3rd International Conference on Mathematical, Computational and Statistical Sciences, 98-106. Pennoni, F. and Vittadini, G. (2013). Two competing models for ordinal longitudinal data with time-varying latent effects: an application to evaluate hospital efficiency Quaderni di Statistica, Journal of methodological and applied statistics, 15, 53-68. Ramaswamy, V. and DeSarbo, W. S. and Reibstein, D. J. and Robinson, W T. (1993). An empirical pooling approach for estimating marketing mix elasticities with PIMS data Journal Marketing Science, 12, 103--124. Rao, C. R. (1958). Some statistical methods for comparison of growth curves Journal Biometrics, 14, 1--17. Raudenbush, S. W. (2001). Comparing personal trajectories and drawing causal inferences from longitudinal data Annual review of psychology, 2, 501--525. Rusakov, D. and Geiger, D. (2002). Asymptotic model selection for naive Bayesian networks. Proceedings of the eighteenth conference on uncertainty in artificial intelligence Kaufmann Publishers Inc., 438--455. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6, 461-464. Tanner, M. A. (1996). Tools for statistical inference Springer, New York Tucker, L. R. (1958). Determination of parameters of a functional relation by factor analysis Journal Psychometrika, 23, 19--23 Twisk, J. and Hoekstra, T. (2012). Classifying developmental trajectories over time should be done with great caution: a comparison between methods Journal of clinical epidemiology, 65, 1078--1087. Vermunt, J K. (2010). Longitudinal research using mixture models In: Longitudinal research with latent variables Editors: Montfort, V. K. and Oud, J. and Satorra, A. 119--152, Springer, Heidelberg. Vermunt, J. K. and Van Dijk, L. (2001). A nonparametric random-coefficients approach: the latent class regression model. Journal Multilevel Modelling Newsletter, 13, 6--13. Viterbi, A. J. (1967). Error bounds for convolutional codes and an asymptotically optimum decoding algorithm. Journal IEEE Transactions on Information Theory, 13, 260-269. Wang, M. and Bodner, T. E. (2007). Growth mixture modeling identifying and predicting unobserved subpopulations with longitudinal data Journal Organizational Research Methods, 10, 635--656. Welch, L. R. (2003). Hidden Markov models and the Baum-Welch algorithm Journal IEEE Information Theory Society Newsletter, 53, 10--13. Wiggins, L. M. (1973). Panel Analysis: Latent probability models for attitude and behaviour processes Publisher Elsevier, Amsterdam. Celeux, G. and Soromenho, G. (1996). An entropy criterion for assessing the number of clusters in a mixture model Journal Journal of classification, 13 195--212. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/72939 |