Fitting regression model via structural transformation model

trReg fits transformation model under dependent truncation and independent censoring via a structural transformation model.

trReg(formula, data, subset, tFun = "linear", method = c("kendall",
  "adjust"), B = 0, control = list())

Arguments

formula	a formula expression, of the form response ~ predictors. The response is assumed to be a survival::Surv object with both left truncation and right censoring. When there is no covariates, e.g., when the right hand side of the formula is ~ 1, the trReg() function returns a trSurvfit object. See ?survival::Surv for more details.
data	an optional data frame in which to interpret the variables occurring in the formula.
subset	an optional vector specifying a subset of observations to be used in the fitting process.
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.
method	a character string specifying how the transformation parameter is estimated. The available options are "kendall" and "adjust". See Details.
B	a numerical value specifies the bootstrap size for estimating the standard error. When B = 0 (default), the bootstrap standard errors will not be computed.
control	a list of control parameters. The following arguments are allowed: lower The lower bound to search for the transformation parameter; default at -1. upper The upper bound to search for the transformation parameter; default at 20. tol The tolerance used in the search for the transformation parameter; default at 0.01. G The number of grids used in the search for the transformation parameter; default is 50. A smaller G could results in faster search, but might be inaccurate. Q The number of cutpoints for the truncation time used when method = "adjust". The default is 0. P The number of breakpoints to divide the event times into equally spaced segmenets. When P > 1, the latent truncation time, \(T'(a)\) will be computed in each subset. The transformation model is then applied to the aggregated data. a The transformation parameter. When this is specified, the transformation model is applied based on the specified a. When this is not specified, an optimized a will be determined by optimization one of the quasi-independence measure. See Details. parallel an logical value indicating whether parallel computation will be applied when B > 0. parCl an integer value specifying the number of CPU cores to be used when parallel = TRUE. The default value is half the CPU cores on the current host.

Details

The main assumption on the structural transformation model is that it 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 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 one of the following methods:

method = "kendall"

by minimizing the absolute value of the restricted inverse probability weighted Kendall's tau or maximize the corresponding \(p\)-value. This is the same procedure used in the trSUrvfit() function.

method = "adjust"

includes a function of latent truncation time, \(U\), as a covariate. A piece-wise function is constructed based on (\(Q + 1\)) indicator functions on whether \(U\) falls in the \(Q\)th and the (\(Q+1\))th percentile, where \(Q\) is the number of cutpoints used. See control for details. The transformation parameter, \(a\), is then chosen to minimize the significance of the coefficient parameter.

See also

trSurvfit

Examples

data(channing, package = "boot")
chan <- subset(channing, entry < exit)
trReg(Surv(entry, exit, cens) ~ sex, data = chan)
#> 
#>  Call:trReg(formula = Surv(entry, exit, cens) ~ sex, data = chan)
#> 
#>  Sample size = 457
#>  Number of events =  175
#> 
#>  Transformation parameter is -0.8033718 
#> 
#>  Standard errors obtained from 0 bootstrap samples.
#>         coef se(coef)  z Pr(>|z|)
#> sexMale 0.51       NA NA       NA
#> 
trReg(Surv(entry, exit, cens) ~ sex, data = chan, method = "adjust", control = list(G = 10))
#> 
#>  Call:trReg(formula = Surv(entry, exit, cens) ~ sex, data = chan, method = "adjust", 
#>     control = list(G = 10))
#> 
#>  Sample size = 457
#>  Number of events =  175
#> 
#>  Transformation parameter is -0.8354068 
#> 
#>  Standard errors obtained from 0 bootstrap samples.
#>          coef se(coef)  z Pr(>|z|)
#> sexMale 0.503       NA NA       NA
#> 
#> 
#>  Coefficient estimates for transformed truncation times used in the adjusted model:
#>       coef se(coef)      z Pr(>|z|)
#> T'(a)    0        0 -0.001        1
#>