totlos.msm {msm} | R Documentation |
Estimate the expected total length of stay in each set of states, for a given period of evolution of a multi-state model.
totlos.msm(x, start=1, end=NULL, fromt=0, tot=Inf, covariates="mean", num.integ=FALSE, discount=0, ci=c("none","normal","bootstrap"), cl=0.95, B=1000, ...)
x |
A fitted multi-state model, as returned by
|
start |
Either a single number giving the state at the beginning of the period, or a vector of probabilities of being in each state at this time. |
end |
States to forecast the total length of stay in. Defaults to all states. |
fromt |
Time from which to estimate total length of stay. Defaults to 0, the beginning of the process. |
tot |
Time up to which total length of stay is estimated. Defaults
to infinity, giving the expected time spent in the state until
absorption. For models without an absorbing state, |
covariates |
The covariate values to estimate for. 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,
|
num.integ |
Use numerical integration instead of analytic solution (see below). |
discount |
Discount rate in continuous time. |
ci |
If If If |
cl |
Width of the symmetric confidence interval, relative to 1 |
B |
Number of bootstrap replicates |
... |
Further arguments to be passed to the
|
The expected total length of stay in state j between times t_1 and t_2, from the point of view of an individual in state i at time 0, is defined by the integral from t_1 to t_2 of the i,j entry of the transition probability matrix P(t) = Exp(tQ).
More generally, suppose that pi_0
is the vector of probabilities of being in each state at time 0,
supplied in start
, and we want the vector x giving the expected
lengths of stay in each state. The corresponding integral has the
following solution (van Loan 1978; van Rosmalen et al. 2013)
x = [1, 0_K] Exp(t Q') [0_K, I_K]'
where
Q' = rbind(c(0, pi_0), cbind(0_K, Q - r I_K)),
pi_0 is the row vector of initial state probabilities
supplied in start
, 0_K is
the row vector of K zeros, r is the discount rate, I_K is the K x K identity matrix, and
Exp is the matrix exponential.
Alternatively, the integrals can be calculated numerically, using the
integrate
function. This may take a long time for
models with many states where P(t) is expensive to calculate.
This facility is just provided for checking results against versions 1.2.4 and
earlier, and will be removed eventually. Please let the package maintainer
know if any results are different.
For a model where the individual has only one place to go from each state, and each state is visited only once, for example a progressive disease model with no recovery or death, these are equal to the mean sojourn time in each state. However, consider a three-state health-disease-death model with transitions from health to disease, health to death, and disease to death, where everybody starts healthy. In this case the mean sojourn time in the disease state will be greater than the expected length of stay in the disease state. This is because the mean sojourn time in a state is conditional on entering the state, whereas the expected total time diseased is a forecast for a healthy individual, who may die before getting the disease.
In the above formulae, Q is assumed to be constant over time,
but the results generalise easily to piecewise-constant intensities.
Though the only time-inhomogeneous models currently handled by this function are those
specified using the pci
option to msm
.
For any other inhomogeneous models, the function assumes that
covariates are constant on the forecasted interval.
A vector of expected total lengths of stay for each transient state.
C. H. Jackson chris.jackson@mrc-bsu.cam.ac.uk
C. van Loan (1978). Computing integrals involving the matrix exponential. IEEE Transactions on Automatic Control 23(3)395-404.
J. van Rosmalen, M. Toy and J.F. O'Mahony (2013). A mathematical approach for evaluating Markov models in continuous time without discrete-event simulation. Medical Decision Making 33:767-779.
sojourn.msm
, pmatrix.msm
, integrate
, boot.msm
.