Transition probability matrix

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.

Usage

pmatrix.msm(
  x = NULL,
  t = 1,
  t1 = 0,
  covariates = "mean",
  ci = c("none", "normal", "bootstrap"),
  cl = 0.95,
  B = 1000,
  cores = NULL,
  qmatrix = NULL,
  ...
)

Arguments

x

A fitted multi-state model, as returned by msm.

t

The time interval to estimate the transition probabilities for, by default one unit.

t1

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].

covariates

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.

ci

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.

cl

Width of the symmetric confidence interval, relative to 1.

B

Number of bootstrap replicates, or number of normal simulations from the distribution of the MLEs

cores

Number of cores to use for bootstrapping using parallel processing. See boot.msm for more details.

qmatrix

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.

Value

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.

Details

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.

References

Mandel, M. (2013). "Simulation based confidence intervals for functions with complicated derivatives." The American Statistician 67(2):76-81

Author

C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk.