Extract the estimated transition probability matrix from a fitted continuous-time multi-state model for a given time interval, at a given set of covariate values.
pmatrix.msm(x=NULL, t=1, t1=0, covariates="mean",
ci=c("none","normal","bootstrap"), cl=0.95, B=1000,
cores=NULL, qmatrix=NULL,
...)The matrix of estimated transition probabilities \(P(t)\) in the given time. Rows correspond to "from-state" and columns to "to-state".
Or if ci="normal" or ci="bootstrap", pmatrix.msm
returns a list with
components estimates and ci, where estimates is
the matrix of estimated transition probabilities, and ci is a
list of two matrices containing the upper and lower confidence
limits.
A fitted multi-state model, as returned by msm.
The time interval to estimate the transition probabilities for, by default one unit.
The starting time of
the interval. Used for models x with piecewise-constant intensities fitted
using the pci option to msm. The probabilities will be computed on the interval [t1, t1+t].
The covariate values at which to estimate the transition
probabilities. This can either be:
the string "mean", denoting the means of the covariates in
the data (this is the default),
the number 0, indicating that all the covariates should be
set to zero,
or a list of values, with optional names. For example
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,
list (age = 60, sex = 1)
If some covariates are specified but not others, the missing ones default to zero.
For time-inhomogeneous models fitted using the pci option to
msm, "covariates" here include only those specified using the
covariates argument to 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 covariates argument to
msm, the function pmatrix.piecewise.msm
should be used to to calculate transition probabilities.
If "normal", then calculate a confidence interval for
the transition probabilities by simulating 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. See,
e.g. Mandel (2013) for a discussion of this approach.
If "bootstrap" then calculate a confidence interval by
non-parametric bootstrap refitting. This is 1-2 orders of magnitude
slower than the "normal" method, but is expected to be more
accurate. See boot.msm for more details of
bootstrapping in msm.
If "none" (the default) then no confidence interval is
calculated.
Width of the symmetric confidence interval, relative to 1.
Number of bootstrap replicates, or number of normal simulations from the distribution of the MLEs
Number of cores to use for bootstrapping using parallel
processing. See boot.msm for more details.
A transition intensity matrix. Either this or
a fitted model x must be supplied. No confidence intervals
are available if qmatrix is supplied.
Optional arguments to be passed to MatrixExp to
control the method of computing the matrix exponential.
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk.
For a continuous-time homogeneous Markov process with transition intensity matrix \(Q\), the probability of occupying state \(s\) at time \(u + t\) conditionally on occupying state \(r\) at time \(u\) is given by the \((r,s)\) entry of the matrix \(P(t) = \exp(tQ)\), where \(\exp()\) is the matrix exponential.
For non-homogeneous processes, where covariates and hence the
transition intensity matrix \(Q\) are piecewise-constant in time,
the transition probability matrix is calculated as
a product of matrices over a series of intervals, as explained in
pmatrix.piecewise.msm.
The pmatrix.piecewise.msm
function is only necessary for models fitted using a
time-dependent covariate in the covariates argument to
msm. For time-inhomogeneous models fitted using "pci",
pmatrix.msm can be used, with arguments t and t1,
to calculate transition probabilities over any time period.
Mandel, M. (2013). "Simulation based confidence intervals for functions with complicated derivatives." The American Statistician 67(2):76-81
qmatrix.msm, pmatrix.piecewise.msm, boot.msm