 Institut de Santé Publique, d'Épidémiologie et de Développement |
Centre Inserm U897 Equipe Biostatistique |
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MKVPCI
1.0 Computer program for Markov models with piecewise constant intensities and covariates. Programme pour l’estimation des modèles de Markov avec intensités de transition constantes par morceaux et covariables. |
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Presentation The program MKVPCI is designed to fit a multi-state Markov model with piecewise constant transition intensities and covariates. A modified homogeneous Markov model is used with appropriate time-dependant covariates to fit a Markov model in which the transition intensities are piecewise constant. The Markov model consists of s states and the exact transition times are generally not observed. We consider a regression model expressed in terms of transition intensities from a state h to state j (h, j=1,2,…,s ; h ≠ j) as :
where αhj0 is the baseline constant transition intensity, β'hj is a vector of regression coefficients and Z(t) is a q-vector of observed covariates. The model can handle time-dependant observed covariates by assuming that the covariates remain constant between two consecutive observation times. The main idea of the proposed approach is to consider a partition of time [ τk -1 , τk ) , where k=1,2,…,r+1 and τr+1 = ∞, assuming constant intensity for each type of transition in each interval. Accordingly, we used a vector Z *(t) = ( Z *1(t), Z *2(t), …, Z *r(t) )' of artificial time-dependant covariates defined as
Z *k(t) = 0 if
τ0 ≤ t < τk for k=1,2,…r, and fitted the model with the following transition intensities:
In this model, the intensities vary with time t as a step-functions defined on the pre-specified intervals: [ τk -1 , τ k ) , where k=1,2,…,r+1; time is measured from the beginning of the process. The parameters of the model are the baselines intensities αhj0 which represent the transition intensities in the interval [ τ0 , τ1 ), the vector of regression coefficients β*hj and βhj associated with the artificial time-dependant covariates and the observed covariates, respectively. These parameters are estimated by maximizing a modified likelihood for time-homogeneous Markov model to handle the introduction of artificial covariates. Note that a model including only the vector of artificial covariates Z-(t) leads to a non-homogeneous Markov model in which the transition intensities are step-functions of time and are defined as follows:
If observed covariates are added in the model, then αhj1 represents the baseline transition intensity in the interval [ τk -1 , τk ) , k=1,2,…,r+1, and the regression coefficients associated with the observed covariates can be interpreted, as usual, in terms of relative risks of making the transition from h to j. A time-homogeneous Markov model is obtained if there is no artificial covariate, that is r=0. It is important to note that, in the present version of the program, the maximum number of artificial covariates is fixed to two (rmax=2); this implies a maximum of three intervals [ τk -1 , τk ) in which any transition intensity may take different values. |
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Alioum A, Commenges D. Marshall G., Guo W. and Jones R. H. Kalbfleisch J. D. and Lawless J. F. Author Ahmadou Alioum Daniel Commenges Contact E-mail: Alioum.Ahmadou@isped.u-bordeaux2.fr.We are interested in feed-back but can not guarantee support. Licence This program is free software; you can
redistribute it and/or modify it under the terms of the GNU General Public
License as published by the Free Software Foundation; either version 2 of
the License, or (at your option) any later version. |
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