NADA2 (version 2.0.1)

cenken: Compute Kendall's Tau Correlation and ATS Line for Censored Data

Description

Computes Kendall's tau for singly (y only) or doubly (x and y) censored data. Also computes the Akritas-Theil-Sen (ATS) nonparametric regression line, with the Turnbull estimate of the intercept.

Usage

cenken(y, ycen = NULL, x = NULL, xcen = NULL)

Value

A list containing:

tau

Kendall's tau correlation coefficient.

slope

The estimated slope from the ATS line.

p.value

P-value for testing the null hypothesis of no association.

Arguments

y

A numeric vector of observations or a formula.

ycen

A logical vector indicating TRUE where an observation in y is censored (a less-than value) and FALSE otherwise. Can be omitted for the case where y is not censored.

x

A numeric vector of observations.

xcen

default is NULL for trend analysis purposes. If included, a logical vector indicating TRUE where an observation in x is censored (a less-than value) and FALSE otherwise.

Details

If using the formula interface, the ycen, x, and xcen arguments are not specified. All information is instead provided through a formula via the y argument. The formula must use a Cen object as the response (on the left-hand side of ~), and predictors (optional) on the right-hand side separated by +. See examples below.

Kendall's tau is a nonparametric correlation coefficient that measures monotonic association between y and x. For left-censored data, concordant and discordant pairs are evaluated wherever possible. For example, with increasing x values, a change in y from <1 to 10 is considered an increase (concordant), while a change from <1 to 0.5 or <5 is considered a tie.

The ATS line is defined as the slope resulting in Kendall's tau of 0 between residuals (y - slope * x) and x. The routine uses an iterative bisection search to find this slope. The intercept is the median residual, computed using the Turnbull estimate for interval-censored data as implemented in the Icens package.

References

Helsel, D. R. (2005). Nondetects and Data Analysis: Statistics for Censored Environmental Data. John Wiley & Sons.

Akritas, M. G., Murphy, S. A., & LaValley, M. P. (1995). The Theil-Sen Estimator with Doubly Censored Data and Applications to Astronomy. Journal of the American Statistical Association, 90, 170–177.

Examples

# Both y and x are censored
data(PbHeron)
with(PbHeron,cenken(Blood,BloodCen,Kidney,KidneyCen))

# x is not censored
data(TCEReg)
with(TCEReg, cenken(log(TCEConc), TCECen, PopDensity))

# Synthetic time-series with trend analysis
set.seed(123)

## Parameters
n <- 15  # 15 years of data
time <- 1:n

## Components
trend <- 0.235 * time
noise <- rnorm(n, mean = 5, sd = 1.5)

syn_dat <- data.frame(Yr = 1989 + time, value = trend + noise)
syn_dat$censored <- syn_dat$value < quantile(syn_dat$value, 0.2)

with(syn_dat,cenken(value,censored,Yr))
if (FALSE) {
plot(value~Yr,syn_dat,pch=21,bg=ifelse(syn_dat$censored==TRUE,"red","blue",cex=1.5))
abline(h=quantile(syn_dat$value, 0.2),lty=2,col="red")
}