The Annals of Applied Statistics

Latent Markov model for longitudinal binary data: An application to the performance evaluation of nursing homes

Francesco Bartolucci, Monia Lupparelli, and Giorgio E. Montanari

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Abstract

Performance evaluation of nursing homes is usually accomplished by the repeated administration of questionnaires aimed at measuring the health status of the patients during their period of residence in the nursing home. We illustrate how a latent Markov model with covariates may effectively be used for the analysis of data collected in this way. This model relies on a not directly observable Markov process, whose states represent different levels of the health status. For the maximum likelihood estimation of the model we apply an EM algorithm implemented by means of certain recursions taken from the literature on hidden Markov chains. Of particular interest is the estimation of the effect of each nursing home on the probability of transition between the latent states. We show how the estimates of these effects may be used to construct a set of scores which allows us to rank these facilities in terms of their efficacy in taking care of the health conditions of their patients. The method is used within an application based on data concerning a set of nursing homes located in the Region of Umbria, Italy, which were followed for the period 2003–2005.

Article information

Source
Ann. Appl. Stat., Volume 3, Number 2 (2009), 611-636.

Dates
First available in Project Euclid: 22 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1245676188

Digital Object Identifier
doi:10.1214/08-AOAS230

Mathematical Reviews number (MathSciNet)
MR2750675

Zentralblatt MATH identifier
1166.62330

Keywords
EM algorithm hidden Markov chains item response theory latent variable models

Citation

Bartolucci, Francesco; Lupparelli, Monia; Montanari, Giorgio E. Latent Markov model for longitudinal binary data: An application to the performance evaluation of nursing homes. Ann. Appl. Stat. 3 (2009), no. 2, 611--636. doi:10.1214/08-AOAS230. https://projecteuclid.org/euclid.aoas/1245676188


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References

  • Bartolucci, F. (2006). Likelihood inference for a class of latent Markov models under linear hypotheses on the transition probabilities. J. Roy. Statist. Soc. Ser. B 68 155–178.
  • Bartolucci, F. and Forcina, A. (2005). Likelihood inference on the underlying structure of IRT models. Psychometrika 70 31–43.
  • Bartolucci, F., Pennoni, F. and Francis, B. (2007). A latent Markov model for detecting patterns of criminal activity. J. Roy. Statist. Soc. Ser. A 170 115–132.
  • Bartolucci, F., Lupparelli, M. and Montanari, G. E. (2009). Supplement to “Latent Markov model for longitudinal binary data: An application to the performance evaluation of nursing homes.” DOI: 10.1214/08-AOAS230SUPP.
  • Bartolucci, F., Pennoni, F. and Lupparelli, M. (2008). Likelihood inference for the latent Markov Rasch model. In Mathematical Methods for Survival Analysis, Reliability and Quality of Life (C. Huber, N. Limnios, M. Mesbah and M. Nikulin, eds.) 239–254. Wiley, London.
  • Baum, L. E., Petrie, T., Soules, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41 164–171.
  • Biernacki, C., Celeux, G. and Govaert, G. (1999). An improvement of the NEC criterion for assessing the number of clusters in a mixture model. Pattern Recognition Letters 20 267–272.
  • Biernacki, C., Celeux, G. and Govaert, G. (2003). Choosing starting values for the EM algorithm for getting the highest likelihood in multivariate Gaussian mixture models. Comput. Statist. Data Anal. 41 561–575.
  • Boucheron, S. and Gassiat, E. (2007). An information-theoretic perspective on order estimation. In Inference in Hidden Markov Models (O. Cappé, E. Moulines and T. Rydén, eds.) 565–602. Springer, New York.
  • Celeux, G. and Durand, J.-B. (2008). Selecting hidden Markov chain states number with cross-validation likelihood. Comput. Statist. Data Anal. 23 541–564.
  • Celeux, G. and Soromenho, G. (1996). An entropy criterion for assessing the number of clusters in a mixture model. J. Classification 13 195–212.
  • Cox, D. R. and Snell, E. G. (1989). The Analysis of Binary Data, 2nd ed. Chapman and Hall, London.
  • Dardanoni, V. and Forcina, A. (1998). A unified approach to likelihood inference on stochastic orderings in a nonparametric context. J. Amer. Statist. Assoc. 93 1112–1123.
  • Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). J. Roy. Statist. Soc. Ser. B 39 1–38.
  • Forcina, A. and Bartolucci, F. (2004). Modelling quality of life variables with non-parametric mixtures. Environmetrics 15 519–528.
  • Hambleton, R. K. and Swaminathan, H. (1985). Item Response Theory: Principles and Applications. Kluwer, Boston.
  • Harrington, C., Collier, E., O’Meara, J., Kitchener, M., Simon, L. P. and Schnelle, J. F. (2003). Federal and state nursing facility websites: Just what the consumer needs? Amer. J. Med. Quality 18 21–37.
  • Hirdes, J. P., Zimmerman, D., Hallman, K. G. and Soucie, P. S. (1998). Use of the MDS quality indicators to assess quality of care in institutional settings. Canadian Journal for Quality in Health Care 14 5–11.
  • Jacobs, R., Jordan, M. I., Nowlan. S. J. and Hinton, G. E. (1991). Adaptive mixtures of local experts. Neural Computation 3 79–87.
  • Kane, R. L. (1998). Assuring quality in nursing homes care. Journal of the American Geriatrics Society 46 232–237.
  • Keribin, C. (2000). Consistent estimation of the order of mixture models. Sankhyā Ser. A 62 49–66.
  • Langeheine, R. and van de Pol, F. (2002). Latent Markov chains. In Applied Latent Class Analysis (J. A. Hagenaars and A. L. McCutcheon, eds.) 304–341. Cambridge Univ. Press, Cambridge.
  • Leroux, B. G. (1992). Consistent estimation of a mixing distribution. Ann. Statist. 20 1350–1360.
  • Levinson, S. E., Rabiner, L. R. and Sondhi, M. M. (1983). An introduction to the application of the theory of probabilistic functions of a Markov process to automatic speech recognition. Bell System Technical Journal 62 1035–1074.
  • MacDonald, I. L. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman & Hall, London.
  • McCullagh, P. (1980). Regression models for oridinal data (with discussion). J. Roy. Statist. Soc. Ser. B 42 109–142.
  • McLachlan, G. J. and Peel, D. (2000). Finite Mixture Models. Wiley, New York.
  • Mesbah, M., Cole, B. F. and Lee, M.-L. T. (2002). Statistical Methods for Quality of Life Studies: Design, Measurements and Analysis. Kluwer, Boston.
  • Mor, V., Berg, K., Angelelli, J., Gifford, D., Morris, J. and Moore, T. (2003). The quality of quality measurement in U.S. nursing homes. Gerontologist 43 37–46.
  • Normand, S. L. T., Glickman, M. E. and Gatsonis, C. A. (1997). Statistical methods for profiling providers of medical care: Issues and applications. J. Amer. Statist. Assoc. 92 803–814.
  • Normand, S. L. T. and Shahian, D. M. (2007). Statistical and clinical aspects of hospital outcomes profiling. Statist. Sci. 22 206–226.
  • Ohlssen, D. I., Sharples, L. D. and Spiegelhalter, D. J. (2007a). Flexible random-effects models using Bayesian semi-parametric models: Applications to institutional comparisons. Statist. Med. 26 2088–2112.
  • Ohlssen, D. I., Sharples, L. D. and Spiegelhalter, D. J. (2007b). A hierarchical modelling framework for identifying unusual performance in health care providers. J. Roy. Statist. Soc. Ser. A 26 865–890.
  • Phillips, C. D., Hawes, C., Lieberman, T. and Koren, M. J. (2007). Where should Momma go? Current nursing home performance measurement strategies and a less ambitious approach. BMC Health Services Research 7.
  • Pongsapukdee, V. and Sukgumphaphan, S. (2007). Goodness of fit of cumulative logit models for ordinal response categories and nominal explanatory variables with two-factor interaction. Silpakorn University Science and Technology Journal 1 29–38.
  • Spiegelhalter, D. J. (2003). Ranking institutions. Journal of Thoracic and Cardiovascular Surgery 125 1171–1173.
  • Spiegelhalter, D. J. (2005). Funnel plots for comparing institutional performance. Statist. Med. 24 1185–1202.
  • Vermunt, J. K., Langeheine, R. and Böckenholt, U. (1999). Discrete-time discrete-state latent Markov models with time-constant and time-varying covariates. Journal of Educational and Behavioral Statistics 24 179–207.
  • Wiggins, L. M. (1973). Panel Analysis: Latent Probability Models for Attitude and Behavior Processes. Elsevier, Amsterdam.

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