R/kendall.ci.R

In NSM3: Functions and Datasets to Accompany Hollander, Wolfe, and Chicken - Nonparametric Statistical Methods, Third Edition

kendall.ci<-function (x=NULL, y=NULL, alpha=0.05, type="t", bootstrap=F, B=1000, example=F) 
{
	# This will produce a 1 - alpha CI for
	# Kendall's tau.  Based on sections 8.3 and 8.4 of:
	#
	#   Nonparametric Statistical Methods, 3e
	#   Hollander, Wolfe & Chicken 
	#
	# bootstrap = F will find the asymptotic CI as in section 8.3.
	# bootstrap = T will find a bootstrap CI as in section 8.4
	# type can be "t" (two-sided), "u" (upper) or "l" (lower).
	# B is the number of bootstrap replicates.
	#
	# Inefficiently programmed by Eric Chicken, October 2012.

	# Example 8.1 from HW&C
	if(example)
	{
		x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
		y <- c(2.6, 3.1, 2.5, 5, 3.6, 4, 5.2, 2.8, 3.8)
	}

	continue <- T

	if(is.null(x) | is.null(y)) 
	{	
		cat("\n")
		cat("You must supply an x sample and a y sample!", "\n")
		cat("\n")
		continue <- F
	}

	if(continue & (length(x) != length(y))) 
	{	
		cat("\n")
		cat("Samples must be of the same length!", "\n")
		cat("\n")
		continue <- F
	}

	if(continue & (length(x) <= 1)) 
	{	
		cat("\n")
		cat("Sample size n must be at least two!", "\n")
		cat("\n")
		continue <- F
	}

	if(continue & (type!="t" & type!="l" & type!="u")) 
	{	
		cat("\n")
		cat("Argument \"type\" must be one of \"s\" (symmetric), \"l\" (lower) or \"u\" (upper)!", "\n")
		cat("\n")
		continue <- F
	}

	# Q* from (8.17)
	Q <- function(i, j)
	{
		Q.ij <- 0
		ij <- (j[2] - i[2]) * (j[1] - i[1])
		if(ij > 0) Q.ij <- 1
		if(ij < 0) Q.ij <- -1
		Q.ij
	}

	# C.i from (8.37)
	C.i <- function(x, y, i)
	{
		C.i <- 0
		for(k in 1:length(x))
			if(k != i) 
				C.i <- C.i + Q(c(x[i], y[i]), c(x[k], y[k]))
		C.i		
	}

	if(continue & !bootstrap)
	{
		c.i <- numeric(0)
		n <- length(x)
		for(i in 1:n) c.i <- c(c.i, C.i(x, y, i))

		# Get the estimate of tau from the existing function cor.test
		# (8.34)
		# Temporarily disable warnings about p-values and ties
		options("warn" = -1)
		tau.hat <- cor.test(x, y, method="k")$estimate
		options("warn" = 0)
	
		# (8.38)
		sigma.hat.2 <- 2 * (n - 2) * var(c.i) / n / (n-1)
		sigma.hat.2 <- sigma.hat.2 + 1 - (tau.hat)^2
		sigma.hat.2 <- sigma.hat.2 * 2 / n / (n - 1)

		if(type=="t") z <- qnorm(alpha / 2, lower.tail = F)
		if(type!="t") z <- qnorm(alpha, lower.tail = F)
		# (8.39), (8.43), (8.45)
		tau.L <- tau.hat - z * sqrt(sigma.hat.2)
		tau.U <- tau.hat + z * sqrt(sigma.hat.2)
		if(type=="l") tau.U <- 1
		if(type=="u") tau.L <- -1
	}

	if(continue & bootstrap)
	{
		tau <- numeric(0)
		for(b in 1:B)
		{
			b.sample <- sample(1:length(x), length(x), replace=T)
			# Temporarily disable warnings about p-values and ties
			options("warn" = -1)
			tau.sample <- cor.test(x[b.sample], y[b.sample], method="k")
			options("warn" = 0)
			tau.sample <- tau.sample$estimate
			tau <- c(tau, tau.sample)
		}
		tau.hat <- sort(tau)
		hist(tau.hat)
		if(type=="t") k <- floor((B + 1) * alpha / 2)
		if(type!="t") k <- floor((B + 1) * alpha)
		tau.L <- tau.hat[k]
		tau.U <- tau.hat[(B + 1 - k)]
		if(type=="l") tau.U <- 1
		if(type=="u") tau.L <- -1

	}

	tau.L <- round(tau.L, 3)
	tau.U <- round(tau.U, 3)

	if(type=="t") print.type <- " two-sided CI for tau:"
	if(type=="l") print.type <- " lower bound for tau:"
	if(type=="u") print.type <- " upper bound for tau:"
		
	cat("\n")
	cat(paste("1 - alpha = ", 1 - alpha, print.type, sep=""))
	cat("\n")
	cat(paste(tau.L, ", ", tau.U, sep=""), "\n")
	cat("\n")

}