jack.se {Monte.Carlo.se}

Jackknife Standard Error

Description

jack.se — gives jackknife standard error (SE=estimated standard deviation)

Usage

jack.se(x, theta, ...)

Arguments

x

vector of data

theta

— function (statistic) applied to the data (e.g., mean, median, var)

...

Additional arguments to be passed

Details

The code was modified from code associated with the appendix of Efron and Tibshirani (1993)

Value

Returns the jackknife standard error of theta(x)

Author(s)

Dennis Boos, Kevin Matthew, Jason Osborne

References

Efron and Tibshirani (1993), *An Introduction to the Bootstrap*.

Boos, D. D., and Osborne, J. A. (2015), "Assessing Variability of Complex Descriptive Statistics in Monte Carlo Studies using Resampling Methods," International Statistical Review, 25, 775-792.

See Also

boot.se — mc.se.vector — mc.se.matrix

Examples

# simple example, data from Boos and Osborne (2015, Table 3)
# using theta = coefficient of variation = mean/sd

x=c(1,2,79,5,17,11,2,15,85)
cv=function(x){sd(x)/mean(x)}
cv(x)
# [1] 1.383577

jack.se(x,theta=cv)
# [1] 0.3435321

# More complex example using two samples, se for ratio of means
# data from Higgins (2003, problem 4.4, p. 142), LDH readings on 7 patients

before=c(89,90,87,98,120,85,97)
after=c(76,101,84,86,105,84,93)

# requires function using row index as "data,"
# real data is extra parameter xdata

ratio.means <- function(index,xdata){
 mean(xdata[index,1])/mean(xdata[index,2])}

# ratio of means for before-after data

ratio.means(index=1:7,xdata=data.frame(before,after))
# [1] 1.058824

# jackknife SE for ratio of means

jack.se(x=1:7,theta=ratio.means,xdata=data.frame(before,after))
# [1] 0.03913484

# To illustrate use with Monte Carlo output, first create some sample data
# 10,000 samples of size 15 from the Laplace (double exp) distribution

N<-10000
set.seed(450)
z1 <- matrix(rexp(N*15),nrow=N)
z2 <- matrix(rexp(N*15),nrow=N)
z<-(z1-z2)/sqrt(2)              # subtract standard exponentials
out.m.15   <- apply(z,1,mean)
out.t20.15 <- apply(z,1,mean,trim=0.20)
out.med.15 <- apply(z,1,median)

# The three datasets (out.m.15,out.t20.15,out.med.15) each contain 10,000 values.
# If we want use the variance of each column in a table, then to get
# the Monte Carlo standard error of those 3 variances,

jack.se(out.m.15,theta = var)
# [1] 0.0009612314

jack.se(out.t20.15,theta = var)
# [1] 0.0007008859

jack.se(out.med.15,theta = var)
# [1] 0.0008130531

# Function Code

jack.se=function(x, theta, ...){
 call <- match.call()
 n <- length(x)
 u <- rep(0, n)
 for(i in 1:n) {u[i] <- theta(x[ - i], ...)}
 jack.se <- sqrt(((n - 1)/n) * sum((u - mean(u))^2))
 return(jack.se)}