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\name{pmatrix.piecewise.msm}
\alias{pmatrix.piecewise.msm}
\title{Transition probability matrix for processes with
piecewise-constant intensities}
\description{
Extract the estimated transition probability matrix from a fitted
non-time-homogeneous multi-state model for a given time interval.
This is a generalisation of \code{\link{pmatrix.msm}} to
models with time-dependent covariates. Note that
\code{\link{pmatrix.msm}} is sufficient to calculate transition
probabilities for time-inhomogeneous
models fitted using the \code{pci} argument to \code{\link{msm}}.
}
\usage{
pmatrix.piecewise.msm(x, t1, t2, times, covariates,
ci=c("none","normal","bootstrap"), cl=0.95, B=1000, ...)
}
\arguments{
\item{x}{A fitted multi-state model, as returned by
\code{\link{msm}}. This should be a non-homogeneous model, whose
transition intensity matrix depends on a time-dependent covariate. }
\item{t1}{The start of the time interval to estimate the transition probabilities for.}
\item{t2}{The end of the time interval to estimate the transition probabilities
for.}
\item{times}{Cut points at which the transition intensity matrix changes.}
\item{covariates}{
A list with number of components one greater than the length of
\code{times}. Each component of the list is specified in the same
way as the \code{covariates} argument to \code{\link{pmatrix.msm}}.
The components correspond to the covariate values in the intervals
\code{(t1, times[1]], (times[1], times[2]], ...,
(times[length(times)], t2]}
(assuming that all elements of \code{times} are in the interval
\code{(t1, t2)}).
}
\item{ci}{If \code{"normal"}, then calculate a confidence interval for
the transition probabilities by simulating \code{B} random vectors
from the asymptotic multivariate normal distribution implied by the
maximum likelihood estimates (and covariance matrix) of the log
transition intensities and covariate effects, then calculating the
resulting transition probability matrix for each replicate.
If \code{"bootstrap"} then calculate a confidence interval by
non-parametric bootstrap refitting. This is 1-2 orders of magnitude
slower than the \code{"normal"} method, but is expected to be more
accurate. See \code{\link{boot.msm}} for more details of
bootstrapping in \pkg{msm}.
If \code{"none"} (the default) then no confidence interval is
calculated.}
\item{cl}{Width of the symmetric confidence interval, relative to 1.}
\item{B}{Number of bootstrap replicates, or number of normal
simulations from the distribution of the MLEs}
\item{...}{Optional arguments to be passed to \code{\link{MatrixExp}} to
control the method of computing the matrix exponential.}
}
\details{
Suppose a multi-state model has been fitted, in which
the transition intensity matrix \eqn{Q(x(t))} is modelled in terms of
time-dependent covariates \eqn{x(t)}. The transition probability
matrix \eqn{P(t_1, t_n)}{P(t1, tn)} for the time interval \eqn{(t_1,
t_n)}{(t1, tn)} cannot be calculated from the estimated intensity matrix as
\eqn{\exp((t_n - t_1) Q)}{exp((tn - t1) Q)}, because \eqn{Q} varies within the interval
\eqn{t_1, t_n}{t1, tn}. However, if the covariates are piecewise-constant,
or can be approximated as piecewise-constant, then we can calculate
\eqn{P(t_1, t_n)}{P(t1, tn)} by multiplying together individual matrices
\eqn{P(t_i,
t_{i+1}) = \exp((t_{i+1} - t_i) Q)}{P(t_i,
t_{i+1}) = exp((t_{i+1} - t_i) Q)}, calculated over intervals where Q is
constant:
\deqn{P(t_1, t_n) = P(t_1, t_2) P(t_2, t_3)\ldots P(t_{n-1},
t_n)}{P(t1, tn) = P(t1, t2) P(t2, t3)\ldotsP(tn-1, tn)}
}
\value{
The matrix of estimated transition probabilities \eqn{P(t)} for the
time interval \code{[t1, tn]}. That is, the probabilities of
occupying state \eqn{s} at time \eqn{t_n}{tn}
conditionally on occupying state \eqn{r} at time \eqn{t_1}{t1}.
Rows correspond to "from-state" and columns to "to-state".
}
\seealso{
\code{\link{pmatrix.msm}} }
\examples{
\dontrun{
## In a clinical study, suppose patients are given a placebo in the
## first 5 weeks, then they begin treatment 1 at 5 weeks, and
## a combination of treatments 1 and 2 from 10 weeks.
## Suppose a multi-state model x has been fitted for the patients'
## progress, with treat1 and treat2 as time dependent covariates.
## Cut points for when treatment covariate changes
times <- c(0, 5, 10)
## Indicators for which treatments are active in the four intervals
## defined by the three cut points
covariates <- list( list (treat1=0, treat2=0), list (treat1=0, treat2=0), list(treat1=1, treat2=0),
list(treat1=1, treat2=1) )
## Calculate transition probabilities from the start of the study to 15 weeks
pmatrix.piecewise.msm(x, 0, 15, times, covariates)
}
}
\author{C. H. Jackson \email{chris.jackson@mrc-bsu.cam.ac.uk}}
\keyword{models}
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