Bootstrap Resampling

Description

Generate R bootstrap replicates of a statistic applied to data. Both parametric and nonparametric resampling are possible. For the nonparametric bootstrap, possible resampling methods are the ordinary bootstrap, the balanced bootstrap, antithetic resampling, and permutation. For nonparametric multi-sample problems stratified resampling is used. This is specified by including a vector of strata in the call to boot. Importance resampling weights may be specified.

Usage

boot(data, statistic, R, sim="ordinary", stype="i", 
     strata=rep(1,n), L=NULL, m=0, weights=NULL, 
     ran.gen=function(d, p) d, mle=NULL, ...)

Arguments

Details

The statistic to be bootstrapped can be as simple or complicated as desired as long as its arguments correspond to the dataset and (for a nonparametric bootstrap) a vector of indices, frequencies or weights. statistic is treated as a black box by the boot function and is not checked to ensure that these conditions are met.

The first order balanced bootstrap is described in Davison, Hinkley and Schechtman (1986). The antithetic bootstrap is described by Hall (1989) and is experimental, particularly when used with strata. The other non-parametric simulation types are the ordinary bootstrap (possibly with unequal probabilities), and permutation which returns random permutations of cases. All of these methods work independently within strata if that argument is supplied.

For the parametric bootstrap it is necessary for the user to specify how the resampling is to be conducted. The best way of accomplishing this is to specify the function ran.gen which will return a simulated data set from the observed data set and a set of parameter estimates specified in mle.

Value

The returned value is an object of class "boot", containing the following components :

References

There are many references explaining the bootstrap and its variations. Among them are :

Booth, J.G., Hall, P. and Wood, A.T.A. (1993) Balanced importance resampling for the bootstrap. Annals of Statistics, 21, 286–298.

Davison, A.C. and Hinkley, D.V. (1997) Bootstrap Methods and Their Application. Cambridge University Press.

Davison, A.C., Hinkley, D.V. and Schechtman, E. (1986) Efficient bootstrap simulation. Biometrika, 73, 555–566.

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

Gleason, J.R. (1988) Algorithms for balanced bootstrap simulations. American Statistician, 42, 263–266.

Hall, P. (1989) Antithetic resampling for the bootstrap. Biometrika, 73, 713–724.

Hinkley, D.V. (1988) Bootstrap methods (with Discussion). Journal of the Royal Statistical Society, B, 50, 312–337, 355–370.

Hinkley, D.V. and Shi, S. (1989) Importance sampling and the nested bootstrap. Biometrika, 76, 435–446.

Johns M.V. (1988) Importance sampling for bootstrap confidence intervals. Journal of the American Statistical Association, 83, 709–714.

Noreen, E.W. (1989) Computer Intensive Methods for Testing Hypotheses. John Wiley & Sons.

See Also

boot.array, boot.ci, censboot, empinf, jack.after.boot, tilt.boot, tsboot

Examples

# usual bootstrap of the ratio of means using the city data
ratio <- function(d, w)
     sum(d$x * w)/sum(d$u * w)
boot(city, ratio, R=999, stype="w")

# Stratified resampling for the difference of means.  In this
# example we will look at the difference of means between the final
# two series in the gravity data.
diff.means <- function(d, f)
{    n <- nrow(d)
     gp1 <- 1:table(as.numeric(d$series))[1]
     m1 <- sum(d[gp1,1] * f[gp1])/sum(f[gp1])
     m2 <- sum(d[-gp1,1] * f[-gp1])/sum(f[-gp1])
     ss1 <- sum(d[gp1,1]^2 * f[gp1]) - 
            (m1 *  m1 * sum(f[gp1]))
     ss2 <- sum(d[-gp1,1]^2 * f[-gp1]) - 
            (m2 *  m2 * sum(f[-gp1]))
     c(m1-m2, (ss1+ss2)/(sum(f)-2))
}
grav1 <- gravity[as.numeric(gravity[,2])>=7,]
boot(grav1, diff.means, R=999, stype="f", strata=grav1[,2])

#  In this example we show the use of boot in a prediction from 
#  regression based on the nuclear data.  This example is taken 
#  from Example 6.8 of Davison and Hinkley (1997).  Notice also 
#  that two extra arguments to statistic are passed through boot.
nuke <- nuclear[,c(1,2,5,7,8,10,11)]
nuke.lm <- glm(log(cost)~date+log(cap)+ne+ ct+log(cum.n)+pt, data=nuke)
nuke.diag <- glm.diag(nuke.lm)
nuke.res <- nuke.diag$res*nuke.diag$sd
nuke.res <- nuke.res-mean(nuke.res)

#  We set up a new data frame with the data, the standardized 
#  residuals and the fitted values for use in the bootstrap.
nuke.data <- data.frame(nuke,resid=nuke.res,fit=fitted(nuke.lm))

#  Now we want a prediction of plant number 32 but at date 73.00
new.data <- data.frame(cost=1, date=73.00, cap=886, ne=0,
                       ct=0, cum.n=11, pt=1)
new.fit <- predict(nuke.lm, new.data)

nuke.fun <- function(dat, inds, i.pred, fit.pred, x.pred)
{
     assign(".inds", inds, envir=.GlobalEnv)
     lm.b <- glm(fit+resid[.inds] ~date+log(cap)+ne+ct+
          log(cum.n)+pt, data=dat)
     pred.b <- predict(lm.b,x.pred)
     remove(".inds", envir=.GlobalEnv)
     c(coef(lm.b), pred.b-(fit.pred+dat$resid[i.pred]))
}

nuke.boot <- boot(nuke.data, nuke.fun, R=999, m=1, 
     fit.pred=new.fit, x.pred=new.data)
#  The bootstrap prediction error would then be found by
mean(nuke.boot$t[,8]^2)
#  Basic bootstrap prediction limits would be
new.fit-sort(nuke.boot$t[,8])[c(975,25)]



#  Finally a parametric bootstrap.  For this example we shall look 
#  at the air-conditioning data.  In this example our aim is to test 
#  the hypothesis that the true value of the index is 1 (i.e. that 
#  the data come from an exponential distribution) against the 
#  alternative that the data come from a gamma distribution with
#  index not equal to 1.
air.fun <- function(data)
{    ybar <- mean(data$hours)
     para <- c(log(ybar),mean(log(data$hours)))
     ll <- function(k) {
          if (k <= 0) out <- 1e200 # not NA
          else out <- lgamma(k)-k*(log(k)-1-para[1]+para[2])
         out
     }
     khat <- nlm(ll,ybar^2/var(data$hours))$estimate
     c(ybar, khat)
}

air.rg <- function(data, mle)
#  Function to generate random exponential variates.  mle will contain 
#  the mean of the original data
{    out <- data
     out$hours <- rexp(nrow(out), 1/mle)
     out
}

air.boot <- boot(aircondit, air.fun, R=999, sim="parametric",
     ran.gen=air.rg, mle=mean(aircondit$hours))

# The bootstrap p-value can then be approximated by
sum(abs(air.boot$t[,2]-1) > abs(air.boot$t0[2]-1))/(1+air.boot$R)