In stc04003/tranSurv: Transformation Model Based Estimation of Survival and Regression Under Dependent Truncation and Independent Censoring

knitr::opts_chunk$set(comment = NA, prompt = TRUE)

In this vignette, we demonstrate how to test quasi-independence of Failure and Truncation Times with the tranSurv package.

Loading package

The tranSurv package can be installed from CRAN with ```{R, eval = FALSE} install.packages("tranSurv")

or from GitHub with
```{R, eval = FALSE}
devtools::install_github("stc04003/tranSurv")

Once installed, the tranSurv package can be loaded with

library(tranSurv)
packageVersion("tranSurv")

Unconditional Kendall's tau

Suppose we have the correlated data generated from the following codes with the copula package:

library(copula)
set.seed(1)
rho <- iTau(normalCopula(dim = 2), .5) ## convert kendall's tau to pearson's rho
dat <- rCopula(3000, normalCopula(rho, dim = 2)) 
dat <- data.frame(qweibull(dat, 2, 1))
colnames(dat) <- c("T", "X")
head(dat)

The above codes generate random sample of points $(T, X)$ from the bivariate Weibull distribution with the Kendall's tau ($\tau$) of 0.5. The pairs $(T, X)$ are stored in the rows of the data frame dat.

The consistent estimator of $\tau$ is the U-statistics $$ \widehat\tau = \frac{1}{\binom{n}{2}} \sum_{i = 1}^{n - 1}\sum_{j = i + 1}^n\mbox{sgn}{(X_i - X_j)(T_i - T_j)},$$ where $n$ is the sample size, and $\mbox{sgn}(u)$ is the sign of $u$, e.g., @schucany1985introduction. The estimator, $\widehat\tau$ can be computed with cor:

cor(dat, method = "kendall")

The tranSurv package has a faster implementation to compute the Kendall' tau: ```{R, cache = TRUE} kendall(dat) microbenchmark::microbenchmark(kendall(dat), cor(dat, method = "kendall"))

## Conditional Kendall's tau
Assume $X$ is (left) truncated by $T$ as in the scenario of left-truncation 
so that samples with $X < T$ are not observed. 
We further assume $X$ is subject to independent right-censoring time $C$, 
and $Y = \min(X, C)$ and $\Delta = I(X \le C)$ are also observed.
The observed data is then iid copies of $(Y, T, \Delta)|Y\ge T$.

```{R}
dat$C <- rexp(3000, .5)
dat$Y <- pmin(dat$C, dat$X)
dat$delta <- 1 * (dat$X <= dat$C)
dat <- subset(dat, Y >= T)
dim(dat)
head(dat)

The tranSurv package provides three implementations of conditional Kendall's tau ($\tau_c$) for left-truncated, right-censored data. These implementations are called with function cKendall, whose arguments are as follows:

args(cKendall)

• trun left truncation time satisfying trun <= obs.
• obs observed failure time. Must be the same length as trun, and 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 all observed failure times are events.
• method a character string specifying the different version of conditional Kendall's tau to be computed. The possible methods are:
  • MB conditional Kendall's tau proposed in @tsai1990testing and @martin2005testing
  • IPW1 inverse probability weighted conditional Kendall's tau proposed in @austin2014eliminating
  • IPW2 restricted inverse probability weighted conditional Kendall's tau proposed in @austin2014eliminating
• weights an optional vector of user-specified sampling weights
• a a numeric transformation parameter. The default value is 0, indicating no transformation. See @chiou2018transformation for more details.
• trans a character string specifying the transformation structure. See @chiou2018transformation for more details. The following are permitted:
  • linear linear transformation structure
  • log log-linear transformation structure
  • exp exponential transformation structure

@martin2005testing

@tsai1990testing and @martin2005testing consider the consistent estimator: $$\widetilde\tau_c = \frac{1}{M}\sum_{i = 1}^{n - 1}\sum_{j = i + 1}^n \mbox{sgn}{(Y_i - Y_j)(T_i - T_j)}I(\Lambda_{ij}),$$ where $\Lambda_{ij} = {\max(T_i, T_j)\le \min(Y_i, Y_j)}\cap {\Delta_{ij}^{(1)}=1}$, $\Delta_{ij}^{(1)} = 1$ if $\min{Y_i, Y_j}$ is not a censored event and 0 otherwise. The notation $\Lambda_{ij}$ ensures $Y_i$ and $Y_j$ are orderable and comparable. The corresponding quasi-independence test can be called with the function cKendall

cKendall(dat$T, dat$Y, dat$delta)

where the asymptotic variance is computed using the asymptotic U-statistic variance outlined in Section 6 of @martin2005testing. The conditional Kendall's tau value and the corresponding $p$-value can be called individually:

cKendall(dat$T, dat$Y, dat$delta)$PE
cKendall(dat$T, dat$Y, dat$delta)$p.value

@austin2014eliminating

@austin2014eliminating proposed an inverse probability weighted versions of the conditional Kendall's tau estimator. When method = "IPW1", cKendall compute the conditional Kendall's tau defined as $\widehat\tau_{c2}$ in Section 4 of @austin2014eliminating.

cKendall(dat$T, dat$Y, dat$delta, method = "IPW1")

@austin2014eliminating

When method = "IPW2", cKendall compute the conditional Kendall's tau defined as $\widehat\tau_{c3}$ in Section 4 of @austin2014eliminating.

cKendall(dat$T, dat$Y, dat$delta, method = "IPW2")

Channing House study

We illustrate the use of cKendall on the Channing House data set, which consists of 96 male retirees from their entry to the Channing House retirement community. The data can be prepared with the following codes.

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

The conditional Kendall's tau values and the associated $p$-value with the different methods are:

attach(chan)
cKendall(entry, exit, cens)
cKendall(entry, exit, cens, method = "IPW1")
cKendall(entry, exit, cens, method = "IPW2")
detach(chan)

The results are similar to these reported in @austin2014eliminating.

Reference

stc04003/tranSurv documentation built on July 21, 2021, 6:46 p.m.