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.

```
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)
```

data

The data as a vector, matrix or data frame. If it is a matrix or data frame then each row is considered as one multivariate observation.

statistic

A function which when applied to data returns a vector containing
the statistic(s) of interest. When `sim = "parametric"`

, the
first argument to `statistic`

must be the data. For each
replicate a simulated dataset returned by `ran.gen`

will be
passed. In all other cases `statistic`

must take at least two
arguments. The first argument passed will always be the original
data. The second will be a vector of indices, frequencies or weights
which define the bootstrap sample. Further, if predictions are
required, then a third argument is required which would be a vector
of the random indices used to generate the bootstrap predictions.
Any further arguments can be passed to `statistic`

through the
`…`

argument.

R

The number of bootstrap replicates. Usually this will be a single
positive integer. For importance resampling, some resamples may use
one set of weights and others use a different set of weights. In
this case `R`

would be a vector of integers where each
component gives the number of resamples from each of the rows of
weights.

sim

A character string indicating the type of simulation required.
Possible values are `"ordinary"`

(the default),
`"parametric"`

, `"balanced"`

, `"permutation"`

, or
`"antithetic"`

. Importance resampling is specified by
including importance weights; the type of importance resampling must
still be specified but may only be `"ordinary"`

or
`"balanced"`

in this case.

stype

A character string indicating what the second argument of `statistic`

represents. Possible values of stype are `"i"`

(indices - the
default), `"f"`

(frequencies), or `"w"`

(weights). Not
used for `sim = "parametric"`

.

strata

An integer vector or factor specifying the strata for multi-sample
problems. This may be specified for any simulation, but is ignored
when `sim = "parametric"`

. When `strata`

is
supplied for a nonparametric bootstrap, the simulations are done
within the specified strata.

L

Vector of influence values evaluated at the observations. This is
used only when `sim`

is `"antithetic"`

. If not supplied,
they are calculated through a call to `empinf`

. This will use
the infinitesimal jackknife provided that `stype`

is
`"w"`

, otherwise the usual jackknife is used.

m

The number of predictions which are to be made at each bootstrap
replicate. This is most useful for (generalized) linear models.
This can only be used when `sim`

is `"ordinary"`

.
`m`

will usually be a single integer but, if there are strata,
it may be a vector with length equal to the number of strata,
specifying how many of the errors for prediction should come from
each strata. The actual predictions should be returned as the final
part of the output of `statistic`

, which should also take an
argument giving the vector of indices of the errors to be used for
the predictions.

weights

Vector or matrix of importance weights. If a vector then it should
have as many elements as there are observations in `data`

.
When simulation from more than one set of weights is required,
`weights`

should be a matrix where each row of the matrix is
one set of importance weights. If `weights`

is a matrix then
`R`

must be a vector of length `nrow(weights)`

. This
parameter is ignored if `sim`

is not `"ordinary"`

or
`"balanced"`

.

ran.gen

This function is used only when `sim = "parametric"`

when it describes how random values are to be generated. It should
be a function of two arguments. The first argument should be the
observed data and the second argument consists of any other
information needed (e.g. parameter estimates). The second argument
may be a list, allowing any number of items to be passed to
`ran.gen`

. The returned value should be a simulated data set
of the same form as the observed data which will be passed to
`statistic`

to get a bootstrap replicate. It is important that the
returned value be of the same shape and type as the original
dataset. If `ran.gen`

is not specified, the default is a
function which returns the original `data`

in which case all
simulation should be included as part of `statistic`

. Use of
`sim = "parametric"`

with a suitable `ran.gen`

allows the
user to implement any types of nonparametric resampling which are
not supported directly.

mle

The second argument to be passed to `ran.gen`

. Typically these
will be maximum likelihood estimates of the parameters. For
efficiency `mle`

is often a list containing all of the objects
needed by `ran.gen`

which can be calculated using the original
data set only.

simple

logical, only allowed to be `TRUE`

for
`sim = "ordinary", stype = "i", n = 0`

(otherwise ignored with a
warning). By default a `n`

by `R`

index array is created:
this can be large and if `simple = TRUE`

this is avoided by
sampling separately for each replication, which is slower but uses
less memory.

…

Other named arguments for `statistic`

which are passed
unchanged each time it is called. Any such arguments to
`statistic`

should follow the arguments which `statistic`

is
required to have for the simulation. Beware of partial matching to
arguments of `boot`

listed above, and that arguments named
`X`

and `FUN`

cause conflicts in some versions of
boot (but not this one).

parallel

The type of parallel operation to be used (if any). If missing, the
default is taken from the option `"boot.parallel"`

(and if that
is not set, `"no"`

).

ncpus

integer: number of processes to be used in parallel operation: typically one would chose this to the number of available CPUs.

cl

An optional parallel or snow cluster for use if
`parallel = "snow"`

. If not supplied, a cluster on the
local machine is created for the duration of the `boot`

call.

The returned value is an object of class `"boot"`

, containing the
following components:

The observed value of `statistic`

applied to `data`

.

A matrix with `sum(R)`

rows each of which is a bootstrap replicate
of the result of calling `statistic`

.

The value of `R`

as passed to `boot`

.

The `data`

as passed to `boot`

.

The value of `.Random.seed`

when `boot`

started work.

The function `statistic`

as passed to `boot`

.

Simulation type used.

Statistic type as passed to `boot`

.

The original call to `boot`

.

The strata used. This is the vector passed to `boot`

, if it
was supplied or a vector of ones if there were no strata. It is not
returned if `sim`

is `"parametric"`

.

The importance sampling weights as passed to `boot`

or the empirical
distribution function weights if no importance sampling weights were
specified. It is omitted if `sim`

is not one of
`"ordinary"`

or `"balanced"`

.

If predictions are required (`m > 0`

) this is the matrix of
indices at which predictions were calculated as they were passed to
statistic. Omitted if `m`

is `0`

or `sim`

is not
`"ordinary"`

.

The influence values used when `sim`

is `"antithetic"`

.
If no such values were specified and `stype`

is not `"w"`

then `L`

is returned as consecutive integers corresponding to
the assumption that data is ordered by influence values. This
component is omitted when `sim`

is not `"antithetic"`

.

The random generator function used if `sim`

is
`"parametric"`

. This component is omitted for any other value
of `sim`

.

The parameter estimates passed to `boot`

when `sim`

is
`"parametric"`

. It is omitted for all other values of
`sim`

.

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.

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`

.

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.

`boot.array`

, `boot.ci`

,
`censboot`

, `empinf`

,
`jack.after.boot`

, `tilt.boot`

,
`tsboot`

.

# NOT RUN { # 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) # }