Pairwise Likelihood Inference for Nested Hidden Markov Chain Models for Multilevel Longitudinal Data

In the context of multilevel longitudinal data, where sample units are collected in clusters, an important aspect that should be accounted for is the unobserved heterogeneity between sample units and between clusters. For this aim, we propose an approach based on nested hidden (latent) Markov chains, which are associated with every sample unit and with every cluster. The approach allows us to account for the previously mentioned forms of unobserved heterogeneity in a dynamic fashion; it also allows us to account for the correlation that may arise between the responses provided by the units belonging to the same cluster. Under the assumed model, computing the manifest distribution of these response variables is infeasible even with a few units per cluster. Therefore, we make inference on this model through a composite likelihood function based on all the possible pairs of subjects within each cluster. Properties of the composite likelihood estimator are assessed by simulation. The proposed approach is illustrated through an application to a dataset concerning a sample of Italian workers in which a binary response variable for the worker receiving an illness benefit was repeatedly observed. Supplementary materials for this article are available online.

[1]  Bengt Muthén,et al.  Multilevel regression mixture analysis , 2009 .

[2]  Florian Heiss.  Sequential numerical integration in nonlinear state space models for microeconometric panel data , 2008 .

[3]  C. Varin,et al.  A mixed autoregressive probit model for ordinal longitudinal data. , 2010, Biostatistics.

[4]  Jan Bulla,et al.  Computational issues in parameter estimation for stationary hidden Markov models , 2008, Comput. Stat..

[5]  C. Varin,et al.  A note on composite likelihood inference and model selection , 2005 .

[6]  N. Reid,et al.  AN OVERVIEW OF COMPOSITE LIKELIHOOD METHODS , 2011 .

[7]  Xin Gao,et al.  Composite Likelihood EM Algorithm with Applications to Multivariate Hidden Markov Model , 2009 .

[8]  Francesco Bartolucci,et al.  Longitudinal analysis of self‐reported health status by mixture latent auto‐regressive models , 2014 .

[9]  P. Diggle.  Analysis of Longitudinal Data , 1995 .

[10]  Francesco Bartolucci,et al.  Assessment of School Performance Through a Multilevel Latent Markov Rasch Model , 2009, 0909.4961.

[11]  Jeroen K. Vermunt,et al.  7. Multilevel Latent Class Models , 2003 .

[12]  Geert Molenberghs,et al.  A pairwise likelihood approach to estimation in multilevel probit models , 2004, Comput. Stat. Data Anal..

[13]  Antonello Maruotti,et al.  Multilevel multivariate modelling of legislative count data, with a hidden Markov chain , 2015 .

[14]  B. Muthén,et al.  Multilevel Mixture Models , 2006 .

[15]  B. Lindsay.  Conditional score functions: Some optimality results , 1982 .

[16]  Francesco Bartolucci,et al.  An extended class of marginal link functions for modelling contingency tables by equality and inequality constraints , 2007 .

[17]  Harvey Goldstein,et al.  Multilevel Statistical Models: Goldstein/Multilevel Statistical Models , 2010 .

[18]  John J. Hanfelt,et al.  Composite conditional likelihood for sparse clustered data , 2004 .

[19]  Cheng Hsiao,et al.  Analysis of Panel Data , 1987 .

[20]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[21]  Nils Lid Hjort,et al.  ML, PL, QL in Markov Chain Models , 2008 .

[22]  Harry Joe,et al.  Composite Likelihood Methods , 2012 .

[23]  Sophia Rabe-Hesketh,et al.  Handling initial conditions and endogenous covariates in dynamic/transition models for binary data with unobserved heterogeneity , 2014 .

[24]  J. H. Schuenemeyer,et al.  Generalized Linear Models (2nd ed.) , 1992 .

[25]  David J. Nott,et al.  A pairwise likelihood approach to analyzing correlated binary data , 2000 .

[26]  P. Song,et al.  Composite Likelihood Bayesian Information Criteria for Model Selection in High-Dimensional Data , 2010 .

[27]  L. R. Rabiner,et al.  An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition , 1983, The Bell System Technical Journal.

[28]  Edward W. Frees,et al.  Longitudinal and Panel Data , 2004 .

[29]  T. Rothenberg.  Identification in Parametric Models , 1971 .

[30]  D. Cox,et al.  A note on pseudolikelihood constructed from marginal densities , 2004 .

[31]  Mordecai Avriel,et al.  Nonlinear programming , 1976 .

[32]  J. Vermunt,et al.  Latent Gold 4.0 User's Guide , 2005 .

[33]  F. Bartolucci.  Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities , 2006 .

[34]  Christina M. Mastrangelo,et al.  Multilevel Statistical Models, 4th edition , 2011 .

[35]  A. Maruotti.  Mixed Hidden Markov Models for Longitudinal Data: An Overview , 2011 .

[36]  H. Akaike.  INFORMATION THEORY AS AN EXTENSION OF THE MAXIMUM LIKELIHOOD , 1973 .

[37]  F. V. D. Pol,et al.  MIXED MARKOV LATENT CLASS MODELS , 1990 .

[38]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[39]  R. McHugh.  Efficient estimation and local identification in latent class analysis , 1956 .

[40]  Scott L. Thomas,et al.  Multilevel Mixture Models , 2015 .

[41]  C. Varin.  On composite marginal likelihoods , 2008 .

[42]  W. Zucchini,et al.  Hidden Markov Models for Time Series: An Introduction Using R , 2009 .

[43]  Francesco Bartolucci,et al.  Latent Markov model for longitudinal binary data: An application to the performance evaluation of nursing homes , 2009, 0908.2300.

[44]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[45]  Robert Sabourin,et al.  On the memory complexity of the forward-backward algorithm , 2010, Pattern Recognit. Lett..

[46]  Geert Verbeke,et al.  MEANINGFUL STATISTICAL MODEL FORMULATIONS FOR REPEATED MEASURES , 2004 .

[47]  G. Molenberghs,et al.  Longitudinal data analysis , 2008 .

[48]  L. Baum,et al.  A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains , 1970 .

[49]  Francesco Bartolucci,et al.  Latent Markov Models for Longitudinal Data , 2012 .

[50]  Biing-Hwang Juang,et al.  Hidden Markov Models for Speech Recognition , 1991 .

[51]  A. Farcomeni,et al.  A Multivariate Extension of the Dynamic Logit Model for Longitudinal Data Based on a Latent Markov Heterogeneity Structure , 2009 .

[52]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[53]  P. McCullagh,et al.  Generalized Linear Models , 1984 .

[54]  Andrew J. Viterbi,et al.  Error bounds for convolutional codes and an asymptotically optimum decoding algorithm , 1967, IEEE Trans. Inf. Theory.