pmatrix.msm: 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

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,

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

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 c

Width of the symmetric confidence interval, relative to 1.

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

...

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 (in press).