Institut de Santé Publique,
d'Épidémiologie et de Développement
Centre Inserm U897
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.
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.
The program MKVPCI was written in standard FORTRAN-77
language and can be run on any computer with a FORTRAN-77 compiler. MKVPCI
may be considered as an extension of MARKOV, the program proposed by
Marshall, Guo and Jones (1995), and the way of specifying some control
parameters needed to run MKVPCI are drawn from MARKOV.
A complete user guide is available: HERE
Alioum A, Commenges D.
Marshall G., Guo W. and Jones R. H.
Kalbfleisch J. D. and Lawless J. F.
We are interested in feed-back but can not guarantee support.
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.