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Non-homogeneous Markov process models with informative observations with an application to Alzheimer’s disease. (English) Zbl 1213.62132
Summary: Identifying risk factors for transition rates among normal cognition, mildly cognitive impairment, dementia and death in an Alzheimer’s disease study is very important. It is known that transition rates among these states are strongly time dependent. While Markov process models are often used to describe these disease progressions, the literature mainly focuses on time homogeneous processes, and limited tools are available for dealing with non-homogeneity. Further, patients may choose when they want to visit the clinics, which creates informative observations.
We develop methods to deal with non-homogeneous Markov processes through time scale transformations when observation times are pre-planned with some observations missing. Maximum likelihood estimation via the EM algorithm is derived for parameter estimation. Simulation studies demonstrate that the proposed method works well under a variety of situations. An application to the Alzheimer’s disease study identifies that there is a significant increase in transition rates as a function of time. Furthermore, our models reveal that the non-ignorable missing mechanism is perhaps reasonable.

MSC:
62M05 Markov processes: estimation; hidden Markov models
62P10 Applications of statistics to biology and medical sciences; meta analysis
65C60 Computational problems in statistics (MSC2010)
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References:
[1] Albert, A two-state Markov chain for heterogeneous transitional data: a quasi-likelihood approach, Statistics in Medicine 17 pp 1481– (1998)
[2] Andersen, Statistical Models Based on Counting Processes (1993) · Zbl 0769.62061
[3] Bartholomew, Some recent developments in social statistics, International Statistical Review 51 pp 1– (1983) · Zbl 0504.62110
[4] Beekly, The National Alzheimer’s Coordinating Center (NACC) database: the Uniform Data Set, Alzheimer Disease and Related Disorders 21 pp 249– (2007)
[5] Breto, Time series analysis via mechanistic models, Annals of Applied Statistics 3 pp 319– (2009) · Zbl 1160.62080
[6] Cappe, Inference in Hidden Markov Models (2005) · Zbl 1080.62065
[7] Chen, Regression modeling with recurrent events and time-dependent interval-censored marker data, Lifetime Data Analysis 9 pp 275– (2003) · Zbl 1116.62427
[8] Chen, Analysis of interval-censored disease progression data via multi-state models under a nonignorable inspection process, Statistics in Medicine 29 pp 1175– (2010)
[9] Cook, A generalized mover-stayer model for panel data, Biostatistics 3 pp 407– (2002) · Zbl 1135.62363
[10] Cook, A conditional Markov model for clustered progressive multistate processes under incomplete observation, Biometrics 60 pp 436– (2004) · Zbl 1274.62752
[11] Cook, A multistate model for bivariate interval-censored failure time data, Biometrics 64 pp 1100– (2008) · Zbl 1152.62078
[12] Copas, Local sensitivity approximations for selectivity bias, Journal of the Royal Statistical Society, Series B 63 pp 871– (2001) · Zbl 0988.62074
[13] Copas, Inference for non-random samples (with discussion), Journal of the Royal Statistical Society, Series B 59 pp 55– (1997) · Zbl 0896.62003
[14] Copas, Meta-analysis, funnel plots and sensitivity analysis, Biometrics 1 pp 247– (2000) · Zbl 0958.62102
[15] Cox, The Theory of Stochastic Processes (1977) · Zbl 0359.60004
[16] Dempster, Maximum likelihood from incomplete data via the EM algorithm (with discussion), Journal of the Royal Statistical Society, Series B 39 pp 1– (1977) · Zbl 0364.62022
[17] Gentleman, Multi-state Markov models for analysing incomplete disease history data with illustrations for HIV disease, Statistics in Medicine 13 pp 805– (1994)
[18] Goggins, Applying the Cox proportional hazards model when the change time of a binary time-varying covariate is interval-censored, Biometrics 55 pp 445– (1999) · Zbl 1059.62652
[19] Grüger, The validity of inferences based on incomplete observations in disease state models, Biometrics 47 pp 595– (1991)
[20] Harezlak, An illness-death stochastic model in the analysis of longitudinal dementia data, Statistics in Medicine 22 pp 1465– (2003)
[21] Hubbard, Modeling nonhomogeneous Markov processes via time transformation, Biometrics 64 pp 843– (2008) · Zbl 1146.62089
[22] Ibrahim, Missing responses in generalized linear mixed models when the missing data mechanism is nonignorable, Biometrika 88 pp 551– (2001) · Zbl 0984.62047
[23] Ibrahim, Missing-data methods for generalized linear models: a comparative review, Journal of the American Statistical Association 100 pp 332– (2005) · Zbl 1117.62360
[24] Jansen, A local influence approach to binary data from a psychiatric study, Biometrics 59 pp 410– (2003) · Zbl 1210.62169
[25] Kalbfleisch, The analysis of panel data under a Markov assumption, Journal of the American Statistical Association 80 pp 863– (1985) · Zbl 0586.62136
[26] Kalbfleish, Proceedings of the Statistics Canada Symposium on Analysis of Data in Time pp 185– (1989)
[27] Kenward, Selection models for repeated measurements with nonrandom dropout: an illustration of sensitivity, Statistics in Medicine 17 pp 2723– (1998)
[28] Laird, Missing data in longitudinal studies, Statistics in Medicine 7 pp 305– (1988)
[29] Little, Statistical Analysis with Missing Data (2002)
[30] Louis, Finding the observed information matrix when using the EM algorithm, Journal of the Royal Statistical Society, Series B 44 pp 226– (1982) · Zbl 0488.62018
[31] Molenberghs, Sensitivity analysis for incomplete contingency tables, Applied Statistics 50 pp 15– (2001) · Zbl 1021.62045
[32] Molenberghs, Models for Discrete Longitudinal Data (2005) · Zbl 1093.62002
[33] Ocana-Riola, Non-homogeneous Markov processes for biomedical data analysis, Biometrical Journal 47 pp 369– (2005)
[34] Perez-Ocon, Non-homogeneous Markov models in the analysis of survival after breast cancer, Journal of the Royal Statistical Society, Series C 50 pp 111– (2001) · Zbl 1021.62096
[35] Saint-Pierre, The analysis of asthma control under a Markov assumption with use of covariates, Statistics in Medicine 22 pp 3755– (2003)
[36] Slazar, Shared random effects analysis of multi-state Markov models: application to a longitudinal study of transitions to dementia, Statistics in Medicine 26 pp 568– (2007)
[37] Singer, The representation of social processes by Markov models, American Journal of Sociology 82 pp 1– (1976a)
[38] Singer, Some methodoloical issues in the analysis of longitudinal surveys, Annals of Economic and Sociological Measurement 5 pp 447– (1976b)
[39] Sweeting, Multi-state Markov models for disease progression in the presence of informative examination times: an application to hepatitis C, Statistics in Medicine 29 pp 1161– (2010)
[40] Van Steen, A local influence approach to sensitivity analysis of incomplete longitudinal ordinal data, Statistical Modelling: An International Journal 1 pp 125– (2001) · Zbl 1022.62062
[41] Verbeke, Linear Mixed Models for Longitudinal Data (2000)
[42] Verbeke, Sensitivity analysis for non-random dropout: a local influence approach, Biometrics 57 pp 43– (2001) · Zbl 1209.62170
[43] Wasserman, Analyzing social networks as stochastic processes, Journal of the American Statistical Association 75 pp 280– (1980) · Zbl 0439.92029
[44] Yesavage, Modeling the prevalence and incidence of Alzheimer’s disease and mild cognitive impairment, Journal of Psychiatric Research 36 pp 281– (2002)
[45] Zhu, Local inference for incomplete data models, Journal of the Royal Statistical Society, Series B 63 pp 111– (2001) · Zbl 0976.62071
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