Predictions in an illness-death model

Multi-state models allow subjects to move among a finite number of states during a follow-up period. Most often, the objects of study are the transition intensities. The impact of covariates on them can also be studied by specifying regression models. Thus, estimation in multi-state models is usually focused on the transition intensities (or the cumulative transition intensities) and on the regression parameters. However, from a clinical or epidemiological point of view, other quantities could provide additional information and may be more relevant to answer practical questions. For example, given a set of covariates for a subject, it may be of interest to estimate the probability to experience a future event or the expected time without any event. To address these kinds of issues, we need to estimate quantities such as transition probabilities, cumulative probabilities and life expectancies. The purpose of this paper is to review a large number of these quantities in an illness-death model which is perhaps the most common multi-state model in the medical literature, and to propose a way to estimate them in addition to the transition intensities and the regression parameters. An illustration is given using interval-censored data from a large cohort study on cognitive ageing.

[1]  D Commenges,et al.  Incidence and mortality of Alzheimer's disease or dementia using an illness‐death model , 2004, Statistics in medicine.

[2]  N E Day,et al.  A Markov model for HIV disease progression including the effect of HIV diagnosis and treatment: application to AIDS prediction in England and Wales. , 1997, Statistics in medicine.

[3]  Maja Pohar Perme,et al.  Inference for outcome probabilities in multi-state models , 2008, Lifetime data analysis.

[4]  Andrew C Titman,et al.  Semi‐Markov Models with Phase‐Type Sojourn Distributions , 2010, Biometrics.

[5]  P. Grambsch.  Survival and Event History Analysis: A Process Point of View by AALEN, O. O., BORGAN, O., and GJESSING, H. K. , 2009 .

[6]  Ørnulf Borgan,et al.  Counting process models for life history data: a review , 1984 .

[7]  Thomas H. Scheike,et al.  A generalized additive regression model for survival times , 2001 .

[8]  J. Griffiths.  The Theory of Stochastic Processes , 1967 .

[9]  Carmen Cadarso-Suárez,et al.  Multi-state models for the analysis of time-to-event data , 2009, Statistical methods in medical research.

[10]  H Putter,et al.  Tutorial in biostatistics: competing risks and multi‐state models , 2007, Statistics in medicine.

[11]  Benoit Liquet,et al.  Choice between Semi‐parametric Estimators of Markov and Non‐Markov Multi‐state Models from Coarsened Observations , 2007 .

[12]  Jaroslaw Harezlak,et al.  An illness-death stochastic model in the analysis of longitudinal dementia data. , 2003, Statistics in medicine.

[13]  S Esbjerg,et al.  Multi-state models for bleeding episodes and mortality in liver cirrhosis. , 2000, Statistics in medicine.

[14]  B. Winblad,et al.  Very Old Women at Highest Risk of Dementia and Alzheimer's Disease , 1997, Neurology.

[15]  L. Ferrucci,et al.  Active Life Expectancy from Annual Follow–Up Data with Missing Responses , 2000, Biometrics.

[16]  D Commenges,et al.  Are sex and educational level independent predictors of dementia and Alzheimer’s disease? Incidence data from the PAQUID project , 1999, Journal of neurology, neurosurgery, and psychiatry.

[17]  O Borgan,et al.  Covariate Adjustment of Event Histories Estimated from Markov Chains: The Additive Approach , 2001, Biometrics.

[18]  Ren Johansen.  An Empirical Transition Matrix for Non-homogeneous Markov Chains Based on Censored Observations , 1978 .

[19]  J. Klein,et al.  Statistical Models Based On Counting Process , 1994 .

[20]  Halina Frydman,et al.  Nonparametric estimation of a Markov ‘illness-death’ process from interval-censored observations, with application to diabetes survival data , 1995 .

[21]  Christopher H. Jackson,et al.  Multi-State Models for Panel Data: The msm Package for R , 2011 .

[22]  Douglas A. Wolf,et al.  New Methods for Analyzing Active Life Expectancy , 1998 .

[23]  A Alioum,et al.  Modelling age-specific risk: application to dementia. , 1998, Statistics in medicine.

[24]  P Hougaard,et al.  Multi-state Models: A Review , 1999, Lifetime data analysis.

[25]  Ardo van den Hout,et al.  Estimating stroke-free and total life expectancy in the presence of non-ignorable missing values , 2010, Journal of the Royal Statistical Society. Series A,.

[26]  R Kay,et al.  A Markov model for analysing cancer markers and disease states in survival studies. , 1986, Biometrics.

[27]  Ardo van den Hout,et al.  Multi‐state analysis of cognitive ability data: A piecewise‐constant model and a Weibull model , 2008, Statistics in medicine.

[28]  Ardo van den Hout,et al.  Estimating dementia-free life expectancy for Parkinson's patients using Bayesian inference and microsimulation , 2009, Biostatistics.

[29]  H. Ramlau-Hansen.  Smoothing Counting Process Intensities by Means of Kernel Functions , 1983 .

[30]  Andrew C Titman,et al.  Flexible Nonhomogeneous Markov Models for Panel Observed Data , 2011, Biometrics.

[31]  N. Keiding,et al.  Multi-state models for event history analysis , 2002, Statistical methods in medical research.

[32]  Halina Frydman,et al.  Estimation of overall survival in an 'illness-death' model with application to the vertical transmission of HIV-1. , 2010, Statistics in medicine.

[33]  F. O’Sullivan.  Fast Computation of Fully Automated Log-Density and Log-Hazard Estimators , 1988 .

[34]  Halina Frydman,et al.  Nonparametric Estimation in a Markov “Illness–Death” Process from Interval Censored Observations with Missing Intermediate Transition Status , 2009, Biometrics.

[35]  J. Kalbfleisch,et al.  The Analysis of Panel Data under a Markov Assumption , 1985 .

[36]  Nicolas Brouard,et al.  THE ESTIMATION OF HEALTH EXPECTANCIES FROM CROSS-LONGITUDINAL SURVEYS , 2003 .

[37]  P. Andersen,et al.  Multistate models in survival analysis: a study of nephropathy and mortality in diabetes. , 1988, Statistics in medicine.

[38]  Daniel Commenges,et al.  A penalized likelihood approach for an illness-death model with interval-censored data: application to age-specific incidence of dementia. , 2002, Biostatistics.

[39]  O. Aalen.  Nonparametric Inference for a Family of Counting Processes , 1978 .

[40]  Micha Mandel,et al.  Simulation-Based Confidence Intervals for Functions With Complicated Derivatives , 2013 .

[41]  O. Aalen.  A linear regression model for the analysis of life times. , 1989, Statistics in medicine.

[42]  Stephen W Lagakos,et al.  Statistical methods for panel data from a semi-Markov process, with application to HPV. , 2007, Biostatistics.

[43]  M H Branson,et al.  Active life expectancy. , 1983, The New England journal of medicine.