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\name{pmatrix.msm}
\alias{pmatrix.msm}
\title{Transition probability matrix}
\description{
Extract the estimated transition probability matrix from a fitted multi-state
model for a given time interval, at a given set of covariate values.
}
\usage{
pmatrix.msm(x, t=1, t1=0, covariates="mean",
ci=c("none","normal","bootstrap"), cl=0.95, B=1000, ...)
}
\arguments{
\item{x}{A fitted multi-state model, as returned by \code{\link{msm}}.}
\item{t}{The time interval to estimate the transition probabilities
for, by default one unit. }
\item{t1}{The starting time of
the interval. Used for models \code{x} with piecewise-constant intensities fitted
using the \code{pci} option to \code{\link{msm}}. The probabilities will be computed on the interval [t1, t1+t].}
\item{covariates}{
The covariate values at which to estimate the transition
probabilities. This can either be:\cr
the string \code{"mean"}, denoting the means of the covariates in
the data (this is the default),\cr
the number \code{0}, indicating that all the covariates should be
set to zero,\cr
or a list of values, with optional names. For example
\code{list (60, 1)}
where the order of the list follows the order of the covariates
originally given in the model formula, or a named list,
\code{list (age = 60, sex = 1)}
For time-inhomogeneous models fitted using the \code{pci} option to
\code{\link{msm}}, "covariates" here include only those specified using the
\code{covariates} argument to \code{\link{msm}}, and exclude the
artificial covariates representing the time period.
For time-inhomogeneous models fitted "by hand" by using a
time-dependent covariate in the \code{covariates} argument to
\code{\link{msm}}, the function \code{\link{pmatrix.piecewise.msm}}
should be used to to calculate transition probabilities.
}
\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.}
}
\value{
The matrix of estimated transition probabilities \eqn{P(t)} in the given time.
Rows correspond to "from-state" and columns to "to-state".
Or if \code{ci="normal"} or \code{ci="bootstrap"}, \code{pmatrix.msm}
returns a list with
components \code{estimates} and \code{ci}, where \code{estimates} is
the matrix of estimated transition probabilities, and \code{ci} is a
list of two matrices containing the upper and lower confidence
limits.
}
\details{
For a continuous-time homogeneous Markov process with transition
intensity matrix
\eqn{Q}, the probability of occupying state \eqn{s} at time \eqn{u + t}
conditionally on occupying state \eqn{r} at time \eqn{u} is given by the
\eqn{(r,s)} entry of the matrix \eqn{P(t) = \exp(tQ)}{P(t) = exp(tQ)},
where \eqn{\exp()}{exp()} is the matrix exponential.
For non-homogeneous processes, where covariates and hence the
transition intensity matrix \eqn{Q} are piecewise-constant in time,
the transition probability matrix is calculated as
a product of matrices over a series of intervals, as explained in
\code{\link{pmatrix.piecewise.msm}}.
The \code{\link{pmatrix.piecewise.msm}}
function is only necessary for models fitted using a
time-dependent covariate in the \code{covariates} argument to
\code{\link{msm}}. For time-inhomogeneous models fitted using "pci",
\code{pmatrix.msm} can be used, with arguments \code{t} and \code{t1},
to calculate transition probabilities over any time period.
}
\seealso{
\code{\link{qmatrix.msm}}, \code{\link{pmatrix.piecewise.msm}}, \code{\link{boot.msm}}
}
\author{C. H. Jackson \email{chris.jackson@mrc-bsu.cam.ac.uk}.}
\keyword{models}
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