Estimating survival curves via structural transformation model

trSurvfit estimates survival curves under dependent truncation and independent censoring via a structural transformation model.

trSurvfit(trun, obs, delta = NULL, tFun = "linear", plots = FALSE,
  control = trSurv.control(), ...)

Arguments

trun	left truncation time satisfying `trun` <= `obs`.
obs	observed failure time, must be the same length as `trun`, might be right-censored.
delta	an optional 0-1 vector of censoring indicator (0 = censored, 1 = event) for `obs`. If this vector is not specified, `cKendall` assumes no censoring and all observed failure time denote events.
tFun	a character string specifying the transformation function or a user specified function indicating the relationship between $X$, $T$, and $a$. When `tFun` is a character, the following are permitted: linear linear transformation structure; log log-linear transformation structure; exp exponential transformation structure.
plots	an optional logical value; if TRUE, a series of diagnostic plots as well as the survival curve for the observed failure time will be plotted.
control	controls the lower and upper bounds when `trans` is an user specified function.
...	for future methods.

Value

The output contains the following components:

surv

is a data.frame contains the survival probabilities estimates.

byTau

a list contains the estimator of transformation parameter:

par: is the best set of transformation parameter found;
obj: is the value of the inverse probability weighted Kendall's tau corresponding to 'par'.

byP

a list contains the estimator of transformation parameter:

par: is the best set of transformation parameter found;
obj: is the value of the inverse probability weighted Kendall's tau corresponding to 'par'.

qind

a data frame consists of two quasi-independent variables:

trun: is the transformed truncation time;
obs: is the corresponding uncensored failure time.

Details

A structural transformation model assumes there is a latent, quasi-independent truncation time that is associated with the observed dependent truncation time, the event time, and an unknown dependence parameter through a specified funciton. The dependence parameter is chosen to either minimize the absolute value of the restricted inverse probability weighted Kendall's tau or maximize the corresponding $p$-value. The marginal distribution for the truncation time and the event time are completely left unspecified.

The structure of the transformation model is of the form: $$h(U) = (1 + a)^{-1} \times (h(T) + ah(X)),$$ where $T$ is the truncation time, $X$ is the observed failure time, $U$ is the transformed truncation time that is quasi-independent from $X$ and $h(\cdot)$ is a monotonic transformation function. The condition, $T < X$, is assumed to be satisfied. The quasi-independent truncation time, $U$, is obtained by inverting the test for quasi-independence by either minimizing the absolute value of the restricted inverse probability weighted Kendall's tau or maximize the corresponding $p$-value.

At the current version, three transformation structures can be specified. trans = "linear" corresponds to $$h(X) = 1;$$ trans = "log" corresponds to $$h(X) = log(X);$$ trans = "exp" corresponds to $$h(X) = exp(X).$$

References

Martin E. and Betensky R. A. (2005), Testing quasi-independence of failure and truncation times via conditional Kendall's tau, Journal of the American Statistical Association, 100 (470): 484-492.

Austin, M. D. and Betensky R. A. (2014), Eliminating bias due to censoring in Kendall's tau estimators for quasi-independence of truncation and failure, Computational Statistics & Data Analysis, 73: 16-26.

Chiou, S., Austin, M., Qian, J. and Betensky R. A. (2018), Transformation model estimation of survival under dependent truncation and independent censoring, Statistical Methods in Medical Research, 28 (12): 3785-3798.

Examples

data(channing, package = "boot")
chan <- subset(channing, sex == "Male" & entry < exit)

## No display
(fit <- with(chan, trSurvfit(entry, exit, cens)))
#> 
#>  Fitting structural transformation model 
#> 
#>  Call: trSurvfit(trun = entry, obs = exit, delta = cens)
#> 
#>  Conditional Kendall's tau = 0.1967 , p-value = 0.0401
#>  Restricted inverse probability weighted Kendall's tau = 0.425 , p-value = 0.0146
#>  Transformation parameter by minimizing absolute value of Kendall's tau: -0.9231
#>  Transformation parameter by maximizing p-value of the test: -0.9231 
#> 

## With diagnostic plots and the survival estimate
with(chan, trSurvfit(entry, exit, cens, plots = TRUE))
#> 
#>  Fitting structural transformation model 
#> 
#>  Call: trSurvfit(trun = entry, obs = exit, delta = cens, plots = TRUE)
#> 
#>  Conditional Kendall's tau = 0.1967 , p-value = 0.0401
#>  Restricted inverse probability weighted Kendall's tau = 0.425 , p-value = 0.0146
#>  Transformation parameter by minimizing absolute value of Kendall's tau: -0.9212
#>  Transformation parameter by maximizing p-value of the test: -0.9212 
#> 

## Plots survival estimate

plot(fit)