etm {etm}R Documentation

Computation of the empirical transition matrix

Description

This function computes the empirical transition matrix, also called Aalen-Johansen estimator, of the transition probability matrix of any multistate model. The covariance matrix is also computed.

Usage

etm(data, state.names, tra, cens.name, s, t = "last",
    covariance = TRUE)

Arguments

data data.frame of the form data.frame(id,from,to,time) or (id,from,to,entry,exit)
id:
patient id
from:
the state from where the transition occurs
to:
the state to which a transition occurs
time:
time when a transition occurs
entry:
entry time in a state
exit:
exit time from a state
This data.frame is transition-oriented, i.e. it contains one row per transition, and possibly several rows per patient. Specifying an entry and exit time permits to take into account left-truncation.
state.names A vector of characters giving the states names.
tra A quadratic matrix of logical values describing the possible transitions within the multistate model.
cens.name A character giving the code for censored observations in the column 'to' of data. If there is no censored observations in your data, put 'NULL'.
s Starting value for computing the transition probabilities.
t Ending value. Default is "last", meaning that the transition probabilities are computed over (s, t], t being the last event time in the data set.
covariance Logical. Decide whether or not computing the covariance matrix. May be useful for, say, simulations, as the variance computation is a bit long. Default is TRUE.

Details

Data are considered to arise from a time-inhomogeneous Markovian multistate model with finite state space, and possibly subject to independent right-censoring and left-truncation.

The matrix of the transition probabilities is estimated by the Aalen-Johansen estimator / empirical transition matrix (Andersen et al., 1993), which is the product integral over the time period (s, t] of I + the matrix of the increments of the Nelson-Aalen estimates of the cumulative transition hazards. The (i, j)-th entry of the empirical transition matrix estimates the transition probability of being in state j at time t given that one has been in state j at time s.

The covariance matrix is computed using the recursion formula (4.4.19) in Anderson et al. (1993, p. 295). This estimator of the covariance matrix is an estimator of the Greenwood type.

If the multistate model is not Markov, but censorship is entirely random, the Aalen-Johansen estimator still consistently estimates the state occupation probabilities of being in state i at time t (Datta & Satten, 2001; Glidden, 2002)

Value

est Transition probability estimates. This is a 3 dimension array with the first dimension being the state from where transitions occur, the second the state to which transitions occur, and the last one being the event times.
cov Estimated covariance matrix. Each cell of the matrix gives the covariance between the transition probabilities given by the rownames and the colnames, respectively.
time Event times at which the transition probabilities are computed. That is all the observed event times between (s, t].
s Start of the time interval.
t End of the time interval.
trans A data.frame giving the possible transitions.
state.names A vector of character giving the state names.
cens.name How the censored observation are coded in the data set.
n.risk Matrix indicating the number of individuals at risk just before an event
n.event Array containing the number of transitions at each event times

Note

Transitions into a same state, mathematically superfluous, are not allowed. If transitions into the same state are detected in the data, the function will stop. Equally, diag(tra) must be set to FALSE, see the example below.

Author(s)

Arthur Allignol, arthur.allignol@fdm.uni-freiburg.de

References

Andersen, P.K., Borgan, O., Gill, R.D. and Keiding, N. (1993). Statistical models based on counting processes. Springer Series in Statistics. New York, NY: Springer.

Aalen, O. and Johansen, S. (1978). An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scandinavian Journal of Statistics, 5: 141-150.

Gill, R.D. and Johansen, S. (1990). A survey of product-integration with a view towards application in survival analysis. Annals of statistics, 18(4): 1501-1555.

Datta, S. and Satten G.A. (2001). Validity of the Aalen-Johansen estimators of stage occupation probabilities and Nelson-Aalen estimators of integrated transition hazards for non-Markov models. Statistics and Probability Letters, 55(4): 403-411.

Glidden, D. (2002). Robust inference for event probabilities with non-Markov data. Biometrics, 58: 361-368.

See Also

print.etm, summary.etm, sir.cont, xyplot.etm

Examples

data(sir.cont)

# Modification for patients entering and leaving a state
# at the same date
# Change on ventilation status is considered
# to happen before end of hospital stay
sir.cont <- sir.cont[order(sir.cont$id, sir.cont$time), ]
for (i in 2:nrow(sir.cont)) {
  if (sir.cont$id[i]==sir.cont$id[i-1]) {
    if (sir.cont$time[i]==sir.cont$time[i-1]) {
      sir.cont$time[i-1] <- sir.cont$time[i-1] - 0.5
    }
  }
}

### Computation of the transition probabilities
# Possible transitions.
tra <- matrix(ncol=3,nrow=3,FALSE)
tra[1, 2:3] <- TRUE
tra[2, c(1, 3)] <- TRUE

# etm
tr.prob <- etm(sir.cont, c("0", "1", "2"), tra, "cens", 1)

tr.prob
summary(tr.prob)

# plotting
xyplot(tr.prob, tr.choice=c("0 0", "1 1", "0 1", "0 2", "1 0", "1 2"),
       layout=c(2, 3), strip=strip.custom(bg="white",
         factor.levels=
           c("0 to 0", "1 to 1", "0 to 1", "0 to 2", "1 to 0", "1 to 2")))

[Package etm version 0.3-7 Index]