Total length of stay, or expected number of visits

Estimate the expected total length of stay, or the expected number of visits, in each state, for an individual in a given period of evolution of a multi-state model.

Usage

totlos.msm(
  x,
  start = 1,
  end = NULL,
  fromt = 0,
  tot = Inf,
  covariates = "mean",
  piecewise.times = NULL,
  piecewise.covariates = NULL,
  num.integ = FALSE,
  discount = 0,
  env = FALSE,
  ci = c("none", "normal", "bootstrap"),
  cl = 0.95,
  B = 1000,
  cores = NULL,
  ...
)

envisits.msm(
  x = NULL,
  start = 1,
  end = NULL,
  fromt = 0,
  tot = Inf,
  covariates = "mean",
  piecewise.times = NULL,
  piecewise.covariates = NULL,
  num.integ = FALSE,
  discount = 0,
  ci = c("none", "normal", "bootstrap"),
  cl = 0.95,
  B = 1000,
  cores = NULL,
  ...
)

Arguments

x

A fitted multi-state model, as returned by msm.

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 estimate the total length of stay (or number of visits) in. Defaults to all states. This is deprecated, since with the analytic solution (see "Details") it doesn't save any computation to only estimate for a subset of states.

fromt

Time from which to estimate. Defaults to 0, the beginning of the process.

tot

Time up to which the estimate is made. Defaults to infinity, giving the expected time spent in or number of visits to the state until absorption. However, the calculation will be much more efficient if a finite (potentially large) time is specified: see the "Details" section. For models without an absorbing state, t must be specified.

covariates

The covariate values to estimate for. 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)

piecewise.times

Times at which piecewise-constant intensities change. See pmatrix.piecewise.msm for how to specify this. This is only required for time-inhomogeneous models specified using explicit time-dependent covariates, and should not be used for models specified using "pci".

piecewise.covariates

Covariates on which the piecewise-constant intensities depend. See pmatrix.piecewise.msm for how to specify this.

num.integ

Use numerical integration instead of analytic solution (see below).

discount

Discount rate in continuous time.

env

Supplied to totlos.msm. If TRUE, return the expected number of visits to each state. If FALSE, return the total length of stay in each state. envisits.msm simply calls totlos.msm with env=TRUE.

ci

If "normal", then calculate a confidence interval 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 total length of stay 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

cores

Number of cores to use for bootstrapping using parallel processing. See boot.msm for more details.

...

Further arguments to be passed to the integrate function to control the numerical integration.

Value

A vector of expected total lengths of stay (totlos.msm), or expected number of visits (envisits.msm), for each transient state.

Details

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 $P(t) = Exp(tQ)$, where $Q$ is the transition intensity matrix.

The corresponding expected number of visits to state $j$ (excluding the stay in the current state at time 0) is $\sum_{i!=j} T_i Q_{i,j}$, where $T_i$ is the expected amount of time spent in state $i$.

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 $\mathbf{x}$ giving the expected lengths of stay in each state. The corresponding integral has the following solution (van Loan 1978; van Rosmalen et al. 2013)

$$\mathbf{x} = \left[ \begin{array}{ll} 1 & \mathbf{0}_K \end{array} \right] Exp(t Q') \left[ \begin{array}{l} \mathbf{0}_K\\I_K \end{array} \right] $$

where $$Q' = \left[ \begin{array}{ll} 0 & \mathbf{\pi}_0\\ \mathbf{0}_K & Q - rI_K \end{array} \right] $$

$\pi_0$ is the row vector of initial state probabilities supplied in start, $\mathbf{0}_K$ 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 is required where tot = Inf, since the package author is not aware of any analytic expression for the limit of the above formula as $t$ goes to infinity.

With the argument num.integ=TRUE, numerical integration is used even where the analytic solution is available. 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. This function automatically handles models fitted using the pci option to msm. For any other inhomogeneous models, the user must specify piecewise.times and piecewise.covariates arguments to totlos.msm.

References

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.

Author

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