Two-Sample Inference with Tau

The R code in this repository is used to implement the inference procedures based on Kendall’s tau (τ_b) between a binary group indicator and a continuous variable which may be subject to right-censoring. The methods are proposed by Yi-Cheng Tai, Weijing Wang and Martin T. Wells and will be submitted for publication.
(Tai, Y.C., Wang, W., & Wells, M.. (2022). Kendall's Tau for Two-Sample Inference Problems. https://arxiv.org/abs/2207.14445)

tau.bar_func()

When the observed failure times do not subject to censoring, testing results for H₀: S₀ = S₁
and H₀: τ_b = 0 and confidence intervals of τ_b are given.

Arguments

X: a non-empty numeric vector of group indicators, encoded as 0 or 1
observed.time: a non-empty numeric vector of data

Value

A list containing the following components
tau.bar: the estimated value of τ_b
var.fixed: the variance of the estimator of τ_b when the group indicators are fixed
var.random: the variance of the estimator of τ_b when the group indicators are random
var.null.0: the variance of the estimator of τ_b under H₀: S₀ = S₁
var.null.tau: the variance of the estimator of τ_b under H₀: τ_b = 0
ci.fixed: the 95% confidence interval of τ_b when the group indicators are fixed
ci.random: the 95% confidence interval of τ_b when the group indicators are random
z.val.0: the z-score under H₀: S₀ = S₁
z.val.tau: the z-score under H₀: τ_b = 0
p.value.0: p-value under H₀: S₀ = S₁
p.value.tau: p-value under H₀: τ_b = 0

tau.hat_func()

An IPCW estimator of τ_b is given when the observations are subject to right-censoring. The testing results for H₀: S₀ = S₁
and H₀: τ_b = 0 and confidence intervals of τ_b are given.

Arguments

X: a non-empty numeric vector of group indicators, encoded as 0 or 1
observed.time: a non-empty numeric vector of follow-up time
delta: the status indicator. Typically, 0: censored, 1: died

Value

A list containing the following components
tau.hat: the estimated value of τ_b
U: the sum of scores assigned. var.fixed: the variance of the estimator of τ_b when the group indicators are fixed
var.random: the variance of the estimator of τ_b when the group indicators are random
var.null.tau: the variance of the estimator of τ_b under H₀: τ_b = 0
ci.fixed: the 95% confidence interval of τ_b when the group indicators are fixed
ci.random: the 95% confidence interval of τ_b when the group indicators are random
z.val.tau: the z-score under H₀: τ_b = 0
p.value.tau: p-value under H₀: τ_b = 0

imputed.tau.hat_func()

When the upper bound of the support of censoring variable is shorter than the variable of interest, the estimate of τ_b with imputed association pattern in unidentifiable region is provided. Several parametric distributions are used.

Arguments

X: a non-empty numeric vector of group indicators, encoded as 0 or 1
observed.time: a non-empty numeric vector of follow-up time
delta: the status indicator. Typically, 0: censored, 1: died
t.star: a pre-specified value sets the identifiable region

Value

A list containing the following components
weibull: the estimated value of τ_b with imputed weibull tail
exp: the estimated value of τ_b with imputed exponential tail
lnorm: the estimated value of τ_b with imputed log-normal tail
logis: the estimated value of τ_b with imputed logistic tail

Example

Complete Data

The soil water contents (% water by volume) collected from two experimental fields growing bell peppers are under comparison (Gumpertz et al., 1997).

tau.bar_func(X = group, observed.time = obs)
$tau.bar
[1] 0.1901042

$var.fixed
[1] 0.0087689

$var.random
[1] 0.0087689

$var.null.0
[1] 0.008796296

$var.null.tau
[1] 0.009722584

$ci.fixed
[1] 0.006568424 0.373639910

$ci.random
[1] 0.006568424 0.373639910

$z.val.0
[1] 2.026944

$z.val.tau
[1] 1.927972

$p.val.0
[1] 0.04266816

$p.val.tau
[1] 0.05385857

Censored Data

The dataset is obtained from the book "Survival Analysis Techniques for Censored and Truncated Data" (Klein, John P., Moeschberger, Melvin L., 2003). It recoreded the time to infection for patients receiving Kidney Dialysis.

tau.hat_func(X = KD$treatment, observed.time = KD$time, delta = KD$delta)
$tau.hat
[1] -0.4437778

$U
[1] -1450.266

$var.fixed
[1] 0.1051801

$var.random
[1] 0.1237697

$var.null.tau
[1] 0.130941

$ci.fixed
[1] -1.0794231  0.1918676

$ci.random
[1] -1.1333112  0.2457556

$z.val.tau
[1] -1.226388

$p.val.tau
[1] 0.2200529

imputed.tau.hat_func(X = KD$treatment, observed.time = KD$time, delta = KD$delta, t.star = max(KD$time))
$weibull
[1] -0.6192046

$exp
[1] -0.5082488

$lnorm
[1] -0.6497025

$logis
[1] -0.483848

Remark

The dependency packages include survival, parmsurvfit.