# 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

1 2 3 4 |

### Arguments

`x` |
A fitted multi-state model, as returned by |

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

`t1` |
The starting time of
the interval. Used for models |

`covariates` |
The covariate values at which to estimate the transition
probabilities. This can either be: the string the number or a list of values, with optional names. For example
where the order of the list follows the order of the covariates originally given in the model formula, or a named list,
For time-inhomogeneous models fitted using the For time-inhomogeneous models fitted "by hand" by using a
time-dependent covariate in the |

`ci` |
If If If |

`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 |

`qmatrix` |
A transition intensity matrix. Either this or
a fitted model |

`...` |
Optional arguments to be passed to |

### 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 chris.jackson@mrc-bsu.cam.ac.uk.

### References

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

### See Also

`qmatrix.msm`

, `pmatrix.piecewise.msm`

, `boot.msm`