Total length of stay in particular states for a fully-parametric, time-inhomogeneous Markov multi-state model

The matrix whose $r,s$ entry is the expected amount of time spent in state $s$ for a time-inhomogeneous, continuous-time Markov multi-state process that starts in state $r$, up to a maximum time $t$. This is defined as the integral of the corresponding transition probability up to that time.

Usage

totlos.fs(
  x,
  trans = NULL,
  t = 1,
  newdata = NULL,
  ci = FALSE,
  tvar = "trans",
  sing.inf = 1e+10,
  B = 1000,
  cl = 0.95,
  ...
)

Arguments

x

A model fitted with flexsurvreg. See msfit.flexsurvreg for the required form of the model and the data. Additionally, this must be a Markov / clock-forward model, but can be time-inhomogeneous. See the package vignette for further explanation.

x can also be a list of models, with one component for each permitted transition, as illustrated in msfit.flexsurvreg.

trans

Matrix indicating allowed transitions. See msfit.flexsurvreg.

t

Time or vector of times to predict up to. Must be finite.

newdata

A data frame specifying the values of covariates in the fitted model, other than the transition number. See msfit.flexsurvreg.

ci

Return a confidence interval calculated by simulating from the asymptotic normal distribution of the maximum likelihood estimates. Turned off by default, since this is computationally intensive. If turned on, users should increase B until the results reach the desired precision.

tvar

Variable in the data representing the transition type. Not required if x is a list of models.

sing.inf

If there is a singularity in the observed hazard, for example a Weibull distribution with shape < 1 has infinite hazard at t=0, then as a workaround, the hazard is assumed to be a large finite number, sing.inf, at this time. The results should not be sensitive to the exact value assumed, but users should make sure by adjusting this parameter in these cases.

B

Number of simulations from the normal asymptotic distribution used to calculate variances. Decrease for greater speed at the expense of accuracy.

cl

Width of symmetric confidence intervals, relative to 1.

...

Arguments passed to ode in deSolve.

Value

The matrix of lengths of stay $T(t)$, if t is of length 1, or a list of matrices if t is longer.

If ci=TRUE, each element has attributes "lower" and "upper" giving matrices of the corresponding confidence limits. These are formatted for printing but may be extracted using attr().

The result also has an attribute P giving the transition probability matrices, since these are unavoidably computed as a side effect. These are suppressed for printing, but can be extracted with attr(...,"P").

Details

This is computed by solving a second order extension of the Kolmogorov forward differential equation numerically, using the methods in the deSolve package. The equation is expressed as a linear system

$$\frac{dT(t)}{dt} = P(t)$$ $$\frac{dP(t)}{dt} = P(t) Q(t)$$

and solved for $T(t)$ and $P(t)$ simultaneously, where $T(t)$ is the matrix of total lengths of stay, $P(t)$ is the transition probability matrix for time $t$, and $Q(t)$ is the transition hazard or intensity as a function of $t$. The initial conditions are $T(0) = 0$ and $P(0) = I$.

Note that the package msm has a similar method totlos.msm. totlos.fs should give the same results as totlos.msm when both of these conditions hold:

the time-to-event distribution is exponential for all transitions, thus the flexsurvreg model was fitted with dist="exp", and is time-homogeneous.
the msm model was fitted with exacttimes=TRUE, thus all the event times are known, and there are no time-dependent covariates.

msm only allows exponential or piecewise-exponential time-to-event distributions, while flexsurvreg allows more flexible models. msm however was designed in particular for panel data, where the process is observed only at arbitrary times, thus the times of transition are unknown, which makes flexible models difficult.

This function is only valid for Markov ("clock-forward") multi-state models, though no warning or error is currently given if the model is not Markov. See totlos.simfs for the equivalent for semi-Markov ("clock-reset") models.

Author

Christopher Jackson chris.jackson@mrc-bsu.cam.ac.uk.

Examples


# BOS example in vignette, and in msfit.flexsurvreg
bexp <- flexsurvreg(Surv(Tstart, Tstop, status) ~ trans,
                    data=bosms3, dist="exp")
tmat <- rbind(c(NA,1,2),c(NA,NA,3),c(NA,NA,NA))

# predict 4 years spent without BOS, 3 years with BOS, before death
# As t increases, this should converge

totlos.fs(bexp, t=10, trans=tmat)
#>          [,1]     [,2]      [,3]
#> [1,] 3.749105 2.126570  4.124326
#> [2,] 0.000000 3.518329  6.481671
#> [3,] 0.000000 0.000000 10.000000
totlos.fs(bexp, t=1000, trans=tmat)
#>          [,1]     [,2]      [,3]
#> [1,] 4.109742 2.956493  992.9338
#> [2,] 0.000000 3.788904  996.2111
#> [3,] 0.000000 0.000000 1000.0000
totlos.fs(bexp, t=c(5,10), trans=tmat)
#> $`5`
#>          [,1]     [,2]     [,3]
#> [1,] 2.892316 1.068225 1.039459
#> [2,] 0.000000 2.776392 2.223608
#> [3,] 0.000000 0.000000 5.000000
#> 
#> $`10`
#>          [,1]     [,2]      [,3]
#> [1,] 3.749105 2.126570  4.124326
#> [2,] 0.000000 3.518329  6.481671
#> [3,] 0.000000 0.000000 10.000000
#> 

# Answers should match results in help(totlos.simfs) up to Monte Carlo
# error there / ODE solving precision here, since with an exponential
# distribution, the "semi-Markov" model there is the same as the Markov
# model here