Abstract
Inference for semi-Markov models under panel data presents considerable computational difficulties. In general the likelihood is intractable, but a tractable likelihood with the form of a hidden Markov model can be obtained if the sojourn times in each of the states are assumed to have phase-type distributions. However, using phase-type distributions directly may be undesirable as they require estimation of parameters which may be poorly identified. In this article, an approach to fitting semi-Markov models with standard parametric sojourn distributions is developed. The method involves establishing a family of Coxian phase-type distribution approximations to the parametric distribution and merging approximations for different states to obtain an approximate semi-Markov process with a tractable likelihood. Approximations are developed for Weibull and Gamma distributions and demonstrated on data relating to post-lung-transplantation patients.
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References
Asmussen, S., Nerman, O., Olsson, M.: Fitting phase-type distribution via the EM algorithm. Scand. J. Stat. 23, 419–441 (1996)
Bladt, M., Sorensen, M.: Efficient estimation of transition rates between credit ratings from observations at discrete time points. Quant. Finance 9, 147–160 (2009)
Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes. Chapman and Hall, London (1965)
Crespi, C.M., Cumberland, W.G., Blower, S.: A queuing model for chronic recurrent conditions under panel observation. Biometrics 61, 193–198 (2005)
De Dominics, R., Manca, R.: An algorithmic approach to non homogeneous semi Markov processes. Commun. Stat., Simul. Comput. 13, 823–838 (1984)
Foucher, Y., Giral, M., Soulillou, J.-P., Daures, J.-P.: A flexible semi-Markov model for interval-censored data and goodness-of-fit testing. Stat. Methods Med. Res. 19, 127–145 (2010)
Gentleman, R.C., Lawless, J.F., Lindsey, J.C., Yan, P.: Multi-state Markov models for analysing incomplete disease history data with illustrations for HIV disease. Stat. Med. 13, 805–821 (1994)
Howard, R.A.: System analysis of semi-Markov processes. IEEE Trans. Mil. Electron. 8, 114–124 (1964)
Jackson, C.H., Sharples, L.D., Thompson, S.G., Duffy, S.W., Couto, E.: Multistate Markov models for disease progression with classification error. Statistician 52, 193–209 (2003)
Kalbfleisch, J.D., Lawless, J.F.: The analysis of panel data under a Markov assumption. J. Am. Stat. Assoc. 80, 863–871 (1985)
Kang, M., Lagakos, S.W.: Statistical methods for panel data from a semi-Markov process, with application to HPV. Biostatistics 8, 252–264 (2007)
Lancaster, T., Nickell, S.: The analysis of re-employment probabilities for the unemployed. J. R. Stat. Soc. A 143, 141–165 (1980)
Mandel, M.: Estimating disease progression using panel data. Biostatistics 11, 304–316 (2010)
Neuts, M.F.: Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach. John Hopkins University Press, Baltimore (1981)
Olsson, M.: The EMpht-Programme Manual. Gothenburg University, Gotenburg (1998)
Smith, D.R.: Variational Methods in Optimization. Prentice-Hall, Englewood Cliffs (1974)
Titman, A.C., Sharples, L.D.: Semi-Markov models with phase-type sojourn distributions. Biometrics 66, 742–752 (2010)
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Titman, A.C. Estimating parametric semi-Markov models from panel data using phase-type approximations. Stat Comput 24, 155–164 (2014). https://doi.org/10.1007/s11222-012-9360-6
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DOI: https://doi.org/10.1007/s11222-012-9360-6