Nonparametric ABC Confidence Limits

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

abcnon(x, tt, epsilon=0.001, 
       alpha=c(0.025, 0.05, 0.1, 0.16, 0.84, 0.9, 0.95, 0.975))

Arguments

Value

list with following components

References

Efron, B, and DiCiccio, T. (1992) More accurate confidence intervals in exponential families. Biometrika 79, pages 231-245.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# compute abc intervals for the mean
x <- rnorm(10)
theta <- function(p,x) {sum(p*x)/sum(p)}
results <- abcnon(x, theta)  
# compute abc intervals for the correlation
x <- matrix(rnorm(20),ncol=2)
theta <- function(p, x)
{
    x1m <- sum(p * x[, 1])/sum(p)
    x2m <- sum(p * x[, 2])/sum(p)
    num <- sum(p * (x[, 1] - x1m) * (x[, 2] - x2m))
    den <- sqrt(sum(p * (x[, 1] - x1m)^2) *
              sum(p * (x[, 2] - x2m)^2))
    return(num/den)
}
results <- abcnon(x, theta)   

Parametric ABC Confidence Limits

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

abcpar(y, tt, S, etahat, mu, n=rep(1,length(y)),lambda=0.001, 
       alpha=c(0.025, 0.05, 0.1, 0.16))

Arguments

Value

list with the following components

,

References

Efron, B, and DiCiccio, T. (1992) More accurate confidence intervals in exponential families. Bimometrika 79, pages 231-245.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# binomial
# x is a p-vector of successes, n is a p-vector of 
#  number of trials
## Not run: 
S <- matrix(0,nrow=p,ncol=p)
S[row(S)==col(S)] <- x*(1-x/n)
mu <- function(eta,n){n/(1+exp(eta))}
etahat <- log(x/(n-x))
#suppose p=2 and we are interested in mu2-mu1
tt <- function(mu){mu[2]-mu[1]}
x <- c(2,4); n <- c(12,12)
a <- abcpar(x, tt, S, etahat,n)

## End(Not run)

Nonparametric BCa Confidence Limits

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

bcanon(x, nboot, theta, ..., 
       alpha=c(0.025, 0.05, 0.1, 0.16, 0.84, 0.9, 0.95, 0.975))

Arguments

Value

list with the following components

References

Efron, B. and Tibshirani, R. (1986). The Bootstrap Method for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, Vol 1., No. 1, pp 1-35.

Efron, B. (1987). Better bootstrap confidence intervals (with discussion). J. Amer. Stat. Assoc. vol 82, pg 171

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

#  bca limits for the  mean 
#  (this is for illustration; 
#   since "mean" is a built in function,
#   bcanon(x,100,mean) would be simpler!)
   x <- rnorm(20)                
   theta <- function(x){mean(x)}
   results <- bcanon(x,100,theta)   
                              
# To obtain bca limits for functions of more 
# complex data structures, write theta
# so that its argument x is the set of observation
# numbers and simply pass as data to bcanon 
# the vector 1,2,..n. 
# For example, find bca limits for
# the correlation coefficient from a set of 15 data pairs:
   xdata <- matrix(rnorm(30),ncol=2)
   n <- 15
   theta <- function(x,xdata){ cor(xdata[x,1],xdata[x,2]) }
   results <- bcanon(1:n,100,theta,xdata)

Bootstrap Estimates of Prediction Error

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

   bootpred(x,y,nboot,theta.fit,theta.predict,err.meas,...)

Arguments

Value

list with the following components

References

Efron, B. (1983). Estimating the error rate of a prediction rule: improvements on cross-validation. J. Amer. Stat. Assoc, vol 78. pages 316-31.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# bootstrap prediction error estimation in least squares
#  regression
   x <- rnorm(85)  
   y <- 2*x +.5*rnorm(85)                      
   theta.fit <- function(x,y){lsfit(x,y)}
   theta.predict <- function(fit,x){
               cbind(1,x)%*%fit$coef         
               }    
   sq.err <- function(y,yhat) { (y-yhat)^2}                   
   results <- bootpred(x,y,20,theta.fit,theta.predict,
     err.meas=sq.err)  
                                      
# for a classification problem, a standard choice 
# for err.meas would simply count up the
#  classification errors:
    miss.clas <- function(y,yhat){ 1*(yhat!=y)}
# with this specification,  bootpred estimates 
#  misclassification rate

Non-Parametric Bootstrapping

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

    bootstrap(x,nboot,theta,..., func=NULL)

Arguments

Value

list with the following components:

References

Efron, B. and Tibshirani, R. (1986). The bootstrap method for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, Vol 1., No. 1, pp 1-35.

Efron, B. (1992) Jackknife-after-bootstrap standard errors and influence functions. J. Roy. Stat. Soc. B, vol 54, pages 83-127

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# 100 bootstraps of the sample mean 
# (this is for illustration;  since "mean" is  a 
# built in function, bootstrap(x,100,mean) would be simpler!)
    x <- rnorm(20)                
    theta <- function(x){mean(x)} 
                              
    results <- bootstrap(x,100,theta)     
                             
# as above, but also estimate the 95th percentile   
# of the bootstrap dist'n of the mean, and         
# its jackknife-after-bootstrap  standard error    
                              
    perc95 <- function(x){quantile(x, .95)}
                             
                             
    results <-  bootstrap(x,100,theta, func=perc95)                                   
                                   
# To bootstrap functions of more complex data structures, 
# write theta so that its argument x
#  is the set of observation numbers  
#  and simply  pass as data to bootstrap the vector 1,2,..n. 
# For example, to bootstrap
# the correlation coefficient from a set of 15 data pairs:
   xdata <- matrix(rnorm(30),ncol=2)
   n <- 15
   theta <- function(x,xdata){ cor(xdata[x,1],xdata[x,2]) }
   results <- bootstrap(1:n,20,theta,xdata)

Internal functions of package bootstrap

Description

Internal functions of package bootstrap.

Usage

ctsub(x, y, z)
xinter(x, y, z, increasing = TRUE)
yinter(x, y, z, increasing = TRUE)

Details

These are not to be called by the user.

Bootstrap-t Confidence Limits

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

  boott(x,theta, ..., sdfun=sdfunboot, nbootsd=25, nboott=200,
        VS=FALSE, v.nbootg=100, v.nbootsd=25, v.nboott=200,
        perc=c(.001,.01,.025,.05,.10,.50,.90,.95,.975,.99,.999))

Arguments

Value

list with the following components:

References

Tibshirani, R. (1988) "Variance stabilization and the bootstrap". Biometrika (1988) vol 75 no 3 pages 433-44.

Hall, P. (1988) Theoretical comparison of bootstrap confidence intervals. Ann. Statisi. 16, 1-50.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

#  estimated confidence points for the mean
   x <- rchisq(20,1)
   theta <- function(x){mean(x)}
   results <- boott(x,theta)
# estimated confidence points for the mean, 
#  using variance-stabilization bootstrap-T method
   results <-  boott(x,theta,VS=TRUE)
   results$confpoints          # gives confidence points
# plot the estimated var stabilizing transformation
   plot(results$theta,results$g) 
# use standard formula for stand dev of mean
# rather than an inner bootstrap loop
   sdmean <- function(x, ...) 
       {sqrt(var(x)/length(x))}
   results <-  boott(x,theta,sdfun=sdmean) 
                                     
# To bootstrap functions of more  complex data structures, 
# write theta so that its argument x
#  is the set of observation numbers  
#  and simply  pass as data to boot the vector 1,2,..n. 
# For example, to bootstrap
# the correlation coefficient from a set of 15 data pairs:                              
    xdata <- matrix(rnorm(30),ncol=2)
    n <- 15
    theta <- function(x, xdata){ cor(xdata[x,1],xdata[x,2]) }
    results <- boott(1:n,theta, xdata)

Cell Survival data

Description

Data on cell survival under different radiation doses.

Usage

data(cell)

Format

A data frame with 14 observations on the following 2 variables.

dose

a numeric vector, unit rads/100

log.surv

a numeric vector, (natural) logarithm of proportion

Details

There are regression situations where the covariates are more naturally considered fixed rather than random. This cell survival data are an example. A radiologist has run an experiment involving 14 bacterial plates. The plates where exposed to different doses of radiation, and the proportion of surviving cells measured. Greater doses lead to smaller survival proportions, as would be expected. The investigator expressed some doubt as to the validity of observation 13.

So there is some interest as to the influence of observation 13 on the conclusions.

Two different theoretical models as to radiation damage were available, one predicting a linear regresion,

\mu_i = \mbox{E}(y_i \vert z_i) = \beta_1 z_i

and the other predicting a quadratic regression,

\mu_i = \mbox{E}(y_i \vert z_i) = \beta_1 z_i+\beta_2 z_i^2

Hypothesis tests on \beta_2 is of interest.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

plot(cell[,2:1],pch=c(rep(1,12),17,1),
                col=c(rep("black",12),"red", "black"),
                cex=c(rep(1,12), 2, 1))

The Cholostyramine Data

Description

n=164 men took part in an experiment to see if the drug cholostyramine lowered blood cholesterol levels. The men were supposed to take six packets of cholostyramine per day, but many actually took much less.

Usage

data(cholost)

Format

A data frame with 164 observations on the following 2 variables.

z

Compliance, a numeric vector

y

Improvement, a numeric vector

Details

In the book, this is used as an example for curve fitting, with two methods, traditional least-squares fitting and modern loess. In the book is considered linear and polynomial models for the dependence of Improvement upon Compliance.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(cholost)
summary(cholost)
plot(y ~ z, data=cholost, xlab="Compliance", 
            ylab="Improvement")
abline(lm(y ~ z, data=cholost), col="red")

K-fold Cross-Validation

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

   crossval(x, y, theta.fit, theta.predict, ..., ngroup=n)

Arguments

Value

list with the following components

References

Stone, M. (1974). Cross-validation choice and assessment of statistical predictions. Journal of the Royal Statistical Society, B-36, 111–147.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# cross-validation of least squares regression
# note that crossval is not very efficient, and being a
#  general purpose function, it does not use the
# Sherman-Morrison identity for this special case
   x <- rnorm(85)  
   y <- 2*x +.5*rnorm(85)                      
   theta.fit <- function(x,y){lsfit(x,y)}
   theta.predict <- function(fit,x){
               cbind(1,x)%*%fit$coef         
               }                       
   results <- crossval(x,y,theta.fit,theta.predict,ngroup=6)  
                                      

Blood Measurements on 43 Diabetic Children

Description

Measurements on 43 diabetic children of log-Cpeptide (a blood measurement) and age (in years). Interest is predicting the blood measurement from age.

Usage

data(diabetes)

Format

A data frame with 43 observations on the following 3 variables.

obs

a numeric vector

age

a numeric vector

logCpeptide

a numeric vector

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

plot(logCpeptide ~ age, data=diabetes)

Hormone Data from page 107

Description

The hormone data. Amount in milligrams of anti-inflammatory hormone remaining in 27 devices, after a certain number of hours (hrs) of wear.

Usage

data(hormone)

Format

A data frame with 27 observations on the following 3 variables.

Lot

a character vector

hrs

a numeric vector

amount

a numeric vector

Details

The hormone data. Amount in milligrams of anti-inflammatory hormone remaining in 27 devices, after a certain number of hours (hrs) of wear. The devices were sampled from 3 different manufacturing lots, called A, B and C. Lot C looks like it had greater amounts of remaining hormone, but it also was worn the least number of hours.

The book uses this as an example for regression analysis.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(hormone) 
if(interactive())par(ask=TRUE)
with(hormone, stripchart(amount ~ Lot))
with(hormone, plot(amount ~ hrs, pch=Lot))
abline( lm(amount ~ hrs, data=hormone, col="red2"))

Jackknife Estimation

Description

See Efron and Tibshirani (1993) for details on this function.

Usage

   jackknife(x, theta, ...)

Arguments

Value

list with the following components

References

Efron, B. and Tibshirani, R. (1986). The Bootstrap Method for standard errors, confidence intervals, and other measures of statistical accuracy. Statistical Science, Vol 1., No. 1, pp 1-35.

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

# jackknife values for the sample mean 
# (this is for illustration;  # since "mean" is  a 
#  built in function,  jackknife(x,mean) would be simpler!)
   x <- rnorm(20)               
   theta <- function(x){mean(x)}
                             
   results <- jackknife(x,theta)        
                              
# To jackknife functions of more  complex data structures, 
# write theta so that its argument x
#  is the set of observation numbers  
#  and simply  pass as data to jackknife the vector 1,2,..n. 
# For example, to jackknife
# the correlation coefficient from a set of 15 data pairs:      
                        
   xdata <- matrix(rnorm(30),ncol=2)
   n <- 15
   theta <- function(x,xdata){ cor(xdata[x,1],xdata[x,2]) }
   results <- jackknife(1:n,theta,xdata)

Law school data from Efron and Tibshirani

Description

The law school data. A random sample of size n=15 from the universe of 82 USA law schools. Two measurements: LSAT (average score on a national law test) and GPA (average undergraduate grade-point average). law82 contains data for the whole universe of 82 law schools.

Usage

        data(law)

Format

A data frame with 15 observations on the following 2 variables.

LSAT

a numeric vector

GPA

a numeric vector

Details

In the book for which this package is support software, this example is used to bootstrap the correlation coefficient.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

See Also

law82.

Examples

str(law)
if(interactive())par(ask=TRUE)
plot(law)
theta <- function(ind) cor(law[ind,1], law[ind,2])
theta(1:15) # sample estimate
law.boot <- bootstrap(1:15, 2000, theta)
sd(law.boot$thetastar) # bootstrap standard error
hist(law.boot$thetastar)
# bootstrap t confidence limits for the correlation coefficient:
theta <- function(ind) cor(law[ind,1], law[ind,2])
boott(1:15, theta, VS=FALSE)$confpoints
boott(1:15, theta, VS=TRUE)$confpoints
# Observe the difference! See page 162 of the book. 
# abcnon(as.matrix(law), function(p,x) cov.wt(x, p, cor=TRUE)$cor[1,2]  )$limits
# The above cannot be used, as the resampling vector can take negative values! 

Data for Universe of USA Law Schools

Description

This is the universe of 82 USA law schools for which the data frame law provides a sample of size 15. See documentation for law for more details.

Usage

data(law82)

Format

A data frame with 82 observations on the following 3 variables.

School

a numeric vector

LSAT

a numeric vector

GPA

a numeric vector

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

plot(law82[,2:3])
cor(law82[,2:3])

Luteinizing Hormone

Description

Five sets of levels of luteinizing hormone for each of 48 time periods

Usage

data(lutenhorm)

Format

A data frame with 48 observations on the following 5 variables.

V1

a numeric vector

V2

a numeric vector

V3

a numeric vector

V4

a numeric vector

V5

a numeric vector

Details

Five sets of levels of luteinizing hormone for each of 48 time periods, taken from Diggle (1990). These are hormone levels measured on a healty woman in 10 minute intervals over a period of 8 hours. The luteinizing hormone is one of the hormones that orchestrate the menstrual cycle and hence it is important to understand its daily variation.

This is a time series. The book gives only one time series, which correspond to V4. I don't know what are the other four series, the book does'nt mention them. They could be block bootstrap replicates?

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(lutenhorm) 
matplot(lutenhorm)

Experiments with mouse

Description

A small randomized experiment were done with 16 mouse, 7 to treatment group and 9 to control group. Treatment was intended to prolong survival after a test surgery.

Usage

data(mouse.c)

Format

The format is: num [1:9] 52 104 146 10 50 31 40 27 46

Details

The treatment group is is dataset mouse.t. mouse.c is the control group. The book uses this example to illustrate bootstrapping a sample mean. Measurement unit is days of survival following surgery.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(mouse.c)
if(interactive())par(ask=TRUE)
stripchart(list(treatment=mouse.t, control=mouse.c))
cat("bootstrapping the difference of means, treatment - control:\n")
cat("bootstrapping is done independently for the two groups\n")
mouse.boot.c <- bootstrap(mouse.c, 2000, mean)
mouse.boot.t <- bootstrap(mouse.t, 2000, mean)
mouse.boot.diff <- mouse.boot.t$thetastar - mouse.boot.c$thetastar
hist(mouse.boot.diff)
abline(v=0, col="red2")
sd(mouse.boot.diff)

Experiment with mouse

Description

A small randomized experiment were done with 16 mouse, 7 to treatment group and 9 to control group. Treatment was intended to prolong survival after a test surgery.

Usage

data(mouse.t)

Format

The format is: num [1:7] 94 197 16 38 99 141 23

Details

The control group is dataset mouse.c. This dataset is the treatment group. The book uses this for exemplifying bootstrapping the sample mean. Measurement unit is days of survival following surgery.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(mouse.t)
stripchart(list(treatment=mouse.t, control=mouse.c))

The Patch Data

Description

Eight subjects wore medical patches designed to infuse a naturally-occuring hormone into the blood stream.

Usage

data(patch)

Format

A data frame with 8 observations on the following 6 variables.

subject

a numeric vector

placebo

a numeric vector

oldpatch

a numeric vector

newpatch

a numeric vector

z

a numeric vector, oldpatch - placebo

y

a numeric vector, newpatch - oldpatch

Details

Eight subjects wore medical patches designed to infuse a certain naturally-occuring hormone into the blood stream. Each subject had his blood levels of the hormone measured after wearing three different patches: a placebo patch, an "old" patch manufactured at an older plant, and a "new" patch manufactured at a newly opened plant.

The purpose of the study was to show bioequivalence. Patchs from the old plant was already approved for sale by the FDA (food and drug administration). Patches from the new facility would not need a full new approval, if they could be shown bioequivalent to the patches from the old plant.

Bioequivalence was defined as

\frac{|E(\mbox{new}) - E(\mbox{old})|}{ E(\mbox{old})-E(\mbox{placebo})} \le .20

The book uses this to investigate bias of ratio estimation.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(patch) 
 theta <- function(ind){
      Y <- patch[ind,"y"]
      Z <- patch[ind,"z"]
      mean(Y)/mean(Z) }
patch.boot <- bootstrap(1:8, 2000, theta)
names(patch.boot)          
hist(patch.boot$thetastar)
abline(v=c(-0.2, 0.2), col="red2")
theta(1:8) #sample plug-in estimator
abline(v=theta(1:8) , col="blue")
# The bootstrap bias estimate:
mean(patch.boot$thetastar) - theta(1:8)
sd(patch.boot$thetastar) # bootstrapped standard error

Rainfall Data

Description

raifall data. The yearly rainfall, in inches, in Nevada City, California, USA, 1873 through 1978. An example of time series data.

Usage

data(Rainfall)

Format

The format is: Time-Series [1:106] from 1873 to 1978: 80 40 65 46 68 32 58 60 61 60 ...

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(Rainfall)  
plot(Rainfall) 

Open/Closed Book Examination Data

Description

This is data form mardia, Kent and Bibby on 88 students who took examinations in 5 subjects. Some where with open book and other with closed book.

Usage

data(scor)

Format

A data frame with 88 observations on the following 5 variables.

mec

mechanics, closed book note

vec

vectors, closed book note

alg

algebra, open book note

ana

analysis, open book note

sta

statistics, open book note

Details

The book uses this for bootstrap in principal component analysis.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(scor) 
if(interactive())par(ask=TRUE)
plot(scor) 
# The parameter of interest (theta) is the fraction of variance explained 
# by the first principal component.
# For principal components analysis svd is better numerically than 
# eigen-decomposistion, but for bootstrapping the latter is _much_ faster.
theta <- function(ind) {
   vals <- eigen(var(scor[ind,]), symmetric=TRUE, only.values=TRUE)$values
   vals[1] / sum(vals) }
scor.boot <- bootstrap(1:88, 500, theta)
sd(scor.boot$thetastar) # bootstrap standard error
hist(scor.boot$thetastar)
abline(v=theta(1:88), col="red2")
abline(v=mean(scor.boot$thetastar), col="blue")

Spatial Test Data

Description

Twenty-six neurologically impaired children have each taken two tests of spatial perception, called "A" and "B".

Usage

data(spatial)

Format

A data frame with 26 observations on the following 2 variables.

A

a numeric vector

B

a numeric vector

Details

In the book this is used as a test data set for bootstrapping confidence intervals.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(spatial)
plot(spatial) 
abline(0,1, col="red2")

Data on Thickness of Stamps

Description

Thickness in millimeters of 485 postal stamps, printed in 1872. The stamp issue of that year was thought to be a "philatelic mixture", that is, printed on more than one type of paper. It is of historical interest to determine how many different types of paper were used.

Usage

data(stamp)

Format

A data frame with 485 observations on the following variable.

Thickness

Thickness in millimeters, a numeric vector

Details

In the book, this is used to exemplify determination of number of modes. It is also used for kernel density estimation.

Note

The main example in the book is on page 227. See also the CRAN package diptest for an alternative method.

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

summary(stamp)
with(stamp, {hist(Thickness);
             plot(density(Thickness), add=TRUE)})

Tooth Strength Data

Description

Thirteen accident victims have had the strength of their teeth measured, It is desired to predict teeth strength from measurements not requiring destructive testing. Four such bvariables have been obtained for each subject, (D1,D2) are difficult to obtain, (E1,E2) are easy to obtain.

Usage

data(tooth)

Format

A data frame with 13 observations on the following 6 variables.

patient

a numeric vector

D1

a numeric vector

D2

a numeric vector

E1

a numeric vector

E2

a numeric vector

strength

a numeric vector

Details

Do the easy to obtain variables give as good prediction as the difficult to obtain ones?

Source

Efron, B. and Tibshirani, R. (1993) An Introduction to the Bootstrap. Chapman and Hall, New York, London.

Examples

str(tooth)
mod.easy <-  lm(strength ~ E1+E2, data=tooth)
mod.diffi <- lm(strength ~ D1+D2, data=tooth)
summary(mod.easy)
summary(mod.diffi)
if(interactive())par(ask=TRUE)
theta <- function(ind) {
    easy <- lm(strength ~ E1+E2, data=tooth, subset=ind)
    diffi<- lm(strength ~ D1+D2, data=tooth, subset=ind)
    (sum(resid(easy)^2) - sum(resid(diffi)^2))/13   }
tooth.boot <- bootstrap(1:13, 2000, theta)
hist(tooth.boot$thetastar)
abline(v=0, col="red2") 
qqnorm(tooth.boot$thetastar)
qqline(tooth.boot$thetastar, col="red2")