Transition intensity matrix

Extract the estimated transition intensity matrix, and the corresponding standard errors, from a fitted multi-state model at a given set of covariate values.

Usage

qmatrix.msm(
  x,
  covariates = "mean",
  sojourn = FALSE,
  ci = c("delta", "normal", "bootstrap", "none"),
  cl = 0.95,
  B = 1000,
  cores = NULL
)

Arguments

x

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

covariates

The covariate values at which to estimate the intensity matrix. 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 more clearly, a named list,

list (age = 60, sex = 1)

If some covariates are specified but not others, the missing ones default to zero.

With covariates="mean", for factor / categorical variables, the mean of the 0/1 dummy variable for each factor level is used, representing an average over all values in the data, rather than a specific factor level.

sojourn

Set to TRUE if the estimated sojourn times and their standard errors should also be returned.

ci

If "delta" (the default) then confidence intervals are calculated by the delta method, or by simple transformation of the Hessian in the very simplest cases. Normality on the log scale is assumed.

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 transforming.

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.

cl

Width of the symmetric confidence interval to present. Defaults to 0.95.

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.

Value

A list with components:

estimate

Estimated transition intensity matrix.

SE

Corresponding approximate standard errors.

L

Lower confidence limits

U

Upper confidence limits

Or if ci="none", then qmatrix.msm just returns the estimated transition intensity matrix.

If sojourn is TRUE, extra components called sojourn, sojournSE, sojournL and sojournU are included, containing the estimates, standard errors and confidence limits, respectively, of the mean sojourn times in each transient state.

The default print method for objects returned by qmatrix.msm presents estimates and confidence limits. To present estimates and standard errors, do something like

qmatrix.msm(x)[c("estimates","SE")]

Details

Transition intensities and covariate effects are estimated on the log scale by msm. A covariance matrix is estimated from the Hessian of the maximised log-likelihood.

A more practically meaningful parameterisation of a continuous-time Markov model with transition intensities \(q_{rs}\) is in terms of the mean sojourn times \(-1 / q_{rr}\) in each state \(r\) and the probabilities that the next move of the process when in state \(r\) is to state \(s\), \(-q_{rs} / q_{rr}\).

See also

pmatrix.msm, sojourn.msm, deltamethod, ematrix.msm

Author

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