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 = c("i", "f", "w"), 
     strata = rep(1,n), L = NULL, m = 0, weights = NULL, 
     ran.gen = function(d, p) d, mle = NULL, simple = FALSE, ...,
     parallel = c("no", "multicore", "snow"),
     ncpus = getOption("boot.ncpus", 1L), cl = 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:

There are c, plot and print methods for this class.

Parallel operation

When parallel = "multicore" is used (not available on Windows), each worker process inherits the environment of the current session, including the workspace and the loaded namespaces and attached packages (but not the random number seed: see below).

More work is needed when parallel = "snow" is used: the worker processes are newly created R processes, and statistic needs to arrange to set up the environment it needs: often a good way to do that is to make use of lexical scoping since when statistic is sent to the worker processes its enclosing environment is also sent. (E.g. see the example for jack.after.boot where ancillary functions are nested inside the statistic function.) parallel = "snow" is primarily intended to be used on multi-core Windows machine where parallel = "multicore" is not available.

For most of the boot methods the resampling is done in the master process, but not if simple = TRUE nor sim = "parametric". In those cases (or where statistic itself uses random numbers), more care is needed if the results need to be reproducible. Resampling is done in the worker processes by censboot(sim = "wierd") and by most of the schemes in tsboot (the exceptions being sim == "fixed" and sim == "geom" with the default ran.gen).

Where random-number generation is done in the worker processes, the default behaviour is that each worker chooses a separate seed, non-reproducibly. However, with parallel = "multicore" or parallel = "snow" using the default cluster, a second approach is used if RNGkind("L'Ecuyer-CMRG") has been selected. In that approach each worker gets a different subsequence of the RNG stream based on the seed at the time the worker is spawned and so the results will be reproducible if ncpus is unchanged, and for parallel = "multicore" if parallel::mc.reset.stream() is called: see the examples for mclapply.

Note that loading the parallel namespace may change the random seed, so for maximum reproducibility this should be done before calling this function.

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)
{
     lm.b <- glm(fit+resid[inds] ~ date+log(cap)+ne+ct+log(cum.n)+pt,
                 data = dat)
     pred.b <- predict(lm.b, x.pred)
     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 squared 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) 1e200 else lgamma(k)-k*(log(k)-1-para[1]+para[2])
     }
     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)