Mixed Hidden Markov Models for Clinical Research with Discrete Repeated Measurements
American Journal of Theoretical and Applied Statistics
Volume 6, Issue 6, November 2017, Pages: 290-296
Received: Oct. 4, 2017;
Accepted: Oct. 28, 2017;
Published: Dec. 7, 2017
Views 1728 Downloads 130
Yosuke Inaba, Department of Data Science, National Center for Global Health and Medicine, Tokyo, Japan; Department of Mathematical Science for Information Science, Tokyo University of Science, Tokyo, Japan
Asanao Shimokawa, Department of Mathematics, Tokyo University of Science, Tokyo, Japan
Etsuo Miyaoka, Department of Mathematics, Tokyo University of Science, Tokyo, Japan
A hidden Markov model (HMM) is a method for analyzing a sequence of transitions for a set of data by considering the outcomes Y to be output from latent state X, which has the Markov property. The HMM has been widely applied, with applications that include speech recognition, genomic analysis, and finance forecasting. The HMM was originally a method for dealing with single-process data. Thus, it is a natural extension to apply it to data with a repeated measure structure by incorporating random effects in it. This is called the mixed hidden Markov model (MHMM). With this extension, the MHMM was recently applied to clinical research data with repeated measurements, e.g. multiple sclerosis, alcohol consumption, and primary biliary cirrhosis. In relation to parameter inference, because regular HMM methods can be used in an MHMM framework, some legacy knowledge is applicable. The likelihood can be obtained by simply adding a random effect parameter to a single process HMM, and the conventional maximum-likelihood method can be used for parameter estimation. On the other hand, much work must still be performed. For instance, the mathematical property of the maximum likelihood estimator has not yet been thoroughly examined. In this study, the asymptotic normality and consistency of the maximum likelihood estimator of the MHMM concerned with time points are examined via simulation, and found that these properties were almost fine. These methods are applied to actual study data, and future perspectives are provided in the conclusion.
Mixed Hidden Markov Models for Clinical Research with Discrete Repeated Measurements, American Journal of Theoretical and Applied Statistics.
Vol. 6, No. 6,
2017, pp. 290-296.
Zucchini W, MacDonald IL. Hidden Markov models for time series: an introduction using R: CRC press Boca Raton; 2009.
Baum LE, Petrie T, Soules G, Weiss N. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The annals of mathematical statistics. 1970; 41 (1): 164-71.
Baum LE. An equality and associated maximization technique in statistical estimation for probabilistic functions of Markov processes. Inequalities. 1972; 3: 1-8.
Baum LE, Petrie T. Statistical inference for probabilistic functions of finite state Markov chains. The annals of mathematical statistics. 1966; 37 (6): 1554-63.
Baum LE. An Inequality with Applications to Statistical Estimation for Probabilistic Functions of a Markov Process and to a Model for Ecology. Bulletin of the American Mathematical Society. 1967; 73: 360-3.
Rabiner LR. A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE. 1989; 77 (2): 257-86.
Durbin R, Eddy SR, Krogh A, Mitchison G. Biological sequence analysis: probabilistic models of proteins and nucleic acids: Cambridge university press; 1998.
Rydén T, Teräsvirta T, Åsbrink S. Stylized facts of daily return series and the hidden Markov model. Journal of applied econometrics. 1998: 217-44.
Faundez-Zanuy M. On-line signature recognition based on VQ-DTW. Pattern Recognition. 2007; 40 (3): 981-92.
Lander ES, Green P. Construction of multilocus genetic linkage maps in humans. Proceedings of the National Academy of Sciences. 1987; 84 (8): 2363-7.
Churchill GA. Stochastic models for heterogeneous DNA sequences. Bulletin of mathematical biology. 1989; 51 (1): 79-94.
Krogh A, Brown M, Mian IS, Sjölander K, Haussler D. Hidden Markov models in computational biology: Applications to protein modeling. Journal of molecular biology. 1994; 235 (5): 1501-31.
Scharpf RB, Parmigiani G, Pevnser J, Ruczinski I. A hidden Markov model for joint estimation of genotype and copy number in high-throughput SNP chips. 2007.
Solomon J, Butman JA, Sood A. Segmentation of brain tumors in 4D MR images using the hidden Markov model. Computer methods and programs in biomedicine. 2006; 84 (2): 76-85.
Marshall AH, Donaghy R, editors. Intelligent Patient Management using Dynamic Models of Clinical Variables. Computer-Based Medical Systems, 2006 CBMS 2006 19th IEEE International Symposium on; 2006: IEEE.
Diggle P. Analysis of longitudinal data: Oxford University Press; 2002.
Crespi CM, Cumberland WG, Blower S. A queueing model for chronic recurrent conditions under panel observation. Biometrics. 2005; 61 (1): 193-8.
Altman RM. Mixed hidden Markov models: an extension of the hidden Markov model to the longitudinal data setting. Journal of the American Statistical Association. 2007; 102 (477): 201-10.
Shirley KE, Small DS, Lynch KG, Maisto SA, Oslin DW. Hidden Markov models for alcoholism treatment trial data. The Annals of Applied Statistics. 2010: 366-95.
Bartolucci F, Farcomeni A. A discrete time event‐history approach to informative drop‐out in mixed latent Markov models with covariates. Biometrics. 2015; 71 (1): 80-9.
Leroux BG. Maximum-likelihood estimation for hidden Markov models. Stochastic processes and their applications. 1992; 40 (1): 127-43.
Bickel PJ, Ritov Ya, Ryden T. Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. The Annals of Statistics. 1998; 26 (4): 1614-35.
Geert M, Geert V. Models for discrete longitudinal data. New York: Spriner. 2005.
Wang P, Puterman ML. Analysis of Longitudinal Data of Epileptic Seizure Counts–A Two‐State Hidden Markov Regression Approach. Biometrical journal. 2001; 43 (8): 941-62.
Harville DA. Maximum likelihood approaches to variance component estimation and to related problems. Journal of the American Statistical Association. 1977; 72 (358): 320-38.
Lee Y, Nelder JA, Pawitan Y. Generalized linear models with random effects: unified analysis via H-likelihood: CRC Press; 2006.
Faught E, Wilder B, Ramsay R, Reife R, Kramer L, Pledger G, et al. Topiramate placebo-controlled dose-ranging trial in refractory partial epilepsy using 200-, 400-, and 600-mg daily dosages. Neurology. 1996; 46 (6): 1684-90.
Marino, Maria Francesca, Nikos Tzavidis, and Marco Alfò. Mixed hidden Markov quantile regression models for longitudinal data with possibly incomplete sequences. Statistical methods in medical research 2016: 0962280216678433.
Marino, Maria Francesca, and Marco Alfó. Gaussian quadrature approximations in mixed hidden Markov models for longitudinal data: A simulation study. Computational Statistics & Data Analysis 94 2016: 193-209.
DeRuiter, Stacy L., et al. "A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure." The Annals of Applied Statistics 11.1 2017: 362-392.