# pmatrix.msm: Transition probability matrix In msm: Multi-State Markov and Hidden Markov Models in Continuous Time

## 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

 ```1 2 3 4``` ```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.

## 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.

## 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.

## Author(s)

C. H. Jackson [email protected].

## References

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`