Transition probability matrix for processes with piecewise-constant intensities

Extract the estimated transition probability matrix from a fitted non-time-homogeneous multi-state model for a given time interval. This is a generalisation of pmatrix.msm to models with time-dependent covariates. Note that pmatrix.msm is sufficient to calculate transition probabilities for time-inhomogeneous models fitted using the pci argument to msm.

Usage

pmatrix.piecewise.msm(
  x = NULL,
  t1,
  t2,
  times,
  covariates = NULL,
  ci = c("none", "normal", "bootstrap"),
  cl = 0.95,
  B = 1000,
  cores = NULL,
  qlist = NULL,
  ...
)

Arguments

x

A fitted multi-state model, as returned by msm. This should be a non-homogeneous model, whose transition intensity matrix depends on a time-dependent covariate.

t1

The start of the time interval to estimate the transition probabilities for.

t2

The end of the time interval to estimate the transition probabilities for.

times

Cut points at which the transition intensity matrix changes.

covariates

A list with number of components one greater than the length of times. Each component of the list is specified in the same way as the covariates argument to pmatrix.msm. The components correspond to the covariate values in the intervals

(t1, times[1]], (times[1], times[2]], ..., (times[length(times)], t2]

(assuming that all elements of times are in the interval (t1, t2)).

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.

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.

qlist

A list of transition intensity matrices, of length one greater than the length of times. Either this or a fitted model x must be supplied. No confidence intervals are available if (just) qlist 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 $P(t)$ for the time interval [t1, tn]. That is, the probabilities of occupying state $s$ at time $t_n$ conditionally on occupying state $r$ at time $t_1$. Rows correspond to "from-state" and columns to "to-state".

Details

Suppose a multi-state model has been fitted, in which the transition intensity matrix $Q(x(t))$ is modelled in terms of time-dependent covariates $x(t)$. The transition probability matrix $P(t_1, t_n)$ for the time interval $(t_1, $$ t_n)$ cannot be calculated from the estimated intensity matrix as $\exp((t_n - t_1) Q)$, because $Q$ varies within the interval $t_1, t_n$. However, if the covariates are piecewise-constant, or can be approximated as piecewise-constant, then we can calculate $P(t_1, t_n)$ by multiplying together individual matrices $P(t_i, $$ t_{i+1}) = \exp((t_{i+1} - t_i) Q)$, calculated over intervals where Q is constant:

$$P(t_1, t_n) = P(t_1, t_2) P(t_2, t_3)\ldots P(t_{n-1}, $$$$ t_n)$$

Author

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

Examples


if (FALSE) { # \dontrun{
## In a clinical study, suppose patients are given a placebo in the
## first 5 weeks, then they begin treatment 1 at 5 weeks, and
## a combination of treatments 1 and 2 from 10 weeks.
## Suppose a multi-state model x has been fitted for the patients'
## progress, with treat1 and treat2 as time dependent covariates.

## Cut points for when treatment covariate changes
times <- c(0, 5, 10)

## Indicators for which treatments are active in the four intervals
## defined by the three cut points
covariates <- list( list (treat1=0, treat2=0), list (treat1=0, treat2=0), list(treat1=1, treat2=0),
list(treat1=1, treat2=1) )

## Calculate transition probabilities from the start of the study to 15 weeks
pmatrix.piecewise.msm(x, 0, 15, times, covariates)
} # }