Statistical Research Report
Preprint 2, 2001
Author(s):
O. O. Aalen,
Ø. Borgan,
H. Fekjœr
Title:
Covariate adjustment of event
histories estimated from Markov chains: The additive approach.
Abstract:
Markov chain models are much used for studying
event histories that include transitions between several states.
An empirical transition matrix for non-homogeneous Markov chains
has previously been developed, including a detailed statistical
theory based on counting processes and martingales. In this paper
we show how to estimate transition probabilities dependent on
covariates. This technique may, for instance, be used for making
estimates of individual prognosis in epidemiological or clinical
studies. The covariates are included through non-parametric
additive models on the transition intensities of the Markov chain.
The additive model allows for estimation of covariate dependent
transition intensities, and again a detailed theory exists based
on counting processes. The martingale setting now allows for a
very natural combination of the empirical transition matrix and
the additive model, resulting in estimates that can be expressed
as stochastic integrals, and hence their properties are easily
evaluated. Two medical examples will be given. In the first
example we study how the lung cancer mortality of uranium miners
depend on smoking and radon exposure. In the second example we
study how the probability of being in response depends on patient
group and prophylactic treatment for leukemia patients who have
had a bone marrow transplantation. A program in R and S-PLUS that
can carry out the analyses described here is being developed and
will be freely available on the Internet.
A postscript version of the entire preprint.