R: Bootstrap Resampling

boot {boot} | R Documentation

## 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

`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).
`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` is `"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 a 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` is `"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.
`...` | Any other arguments for `statistic` which are passed unchanged each time it is called. Any such arguments to `statistic` must follow the arguments which `statistic` is required to have for the simulation.

### 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 :

`t0` | The observed value of `statistic` applied to `data`.
`t` | A matrix with `R` rows each of which is a bootstrap replicate of `statistic`.
`R` | The value of `R` as passed to `boot`.
`data` | The `data` as passed to `boot`.
`seed` | The value of `.Random.seed` when `boot` was called.
`statistic` | The function `statistic` as passed to `boot`.
`sim` | Simulation type used.
`stype` | Statistic type as passed to `boot`.
`call` | The original call to `boot`.
`strata` | 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"`.
`weights` | 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"`.
`pred.i` | 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"`.
`L` | 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"`.
`ran.gen` | The random generator function used if `sim` is `"parametric"`. This component is omitted for any other value of `sim`.
`mle` | The parameter estimates passed to `boot` when `sim` is `"parametric"`. It is omitted for all other values of `sim`.

### 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)

[Package boot version 1.2-30 Index]