linear.approx              package:boot              R Documentation

Linear Approximation of Bootstrap Replicates

Description:

     This function takes a bootstrap object and for each bootstrap
     replicate it calculates the linear approximation to the statistic
     of interest for  that bootstrap sample.

Usage:

     linear.approx(boot.out, L=NULL, index=1, type=NULL, t0=NULL, 
                   t=NULL, ...)

Arguments:

boot.out: An object of class '"boot"' representing a nonparametric
          bootstrap.  It will usually be created by the function
          'boot'. 

       L: A vector containing the empirical influence values for the
          statistic of interest.  If it is not supplied then 'L' is
          calculated through a call  to 'empinf'. 

   index: The index of the variable of interest within the output of 
          'boot.out$statistic'.  

    type: This gives the type of empirical influence values to be
          calculated.  It is not used if 'L' is supplied.  The possible
          types of empirical influence values are described in the
          helpfile for 'empinf'. 

      t0: The observed value of the statistic of interest.  The input
          value is used only  if one of 't' or 'L' is also supplied. 
          The default value is  'boot.out$t0[index]'.  If 't0' is
          supplied but neither 't' nor 'L' are supplied then 't0' is
          set to 'boot.out$t0[index]' and a warning is generated. 

       t: A vector of bootstrap replicates of the statistic of
          interest.  If 't0' is  missing then 't' is not used,
          otherwise it is used to calculate the empirical  influence
          values (if they are not supplied in 'L').   

     ...: Any extra arguments required by 'boot.out$statistic'.  These
          are needed if 'L' is not supplied as they are used by
          'empinf' to calculate empirical  influence values. 

Details:

     The linear approximation to a bootstrap replicate with frequency
     vector 'f' is given by 't0 + sum(L * f)/n' in the one sample with
     an easy extension to the stratified case.  The frequencies are
     found by calling 'boot.array'.

Value:

     A vector of length 'boot.out$R' with the linear approximations to
     the statistic of interest for each of the bootstrap samples.

References:

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

See Also:

     'boot', 'empinf', 'control'

Examples:

     # Using the city data let us look at the linear approximation to the 
     # ratio statistic and its logarithm. We compare these with the 
     # corresponding plots for the bigcity data 

     ratio <- function(d, w)
          sum(d$x * w)/sum(d$u * w)
     city.boot <- boot(city, ratio, R=499, stype="w")
     bigcity.boot <- boot(bigcity, ratio, R=499, stype="w")
     par(pty="s")
     par(mfrow=c(2,2))

     # The first plot is for the city data ratio statistic.
     city.lin1 <- linear.approx(city.boot)
     lim <- range(c(city.boot$t,city.lin1))
     plot(city.boot$t, city.lin1, xlim=lim,ylim=lim, 
          main="Ratio; n=10", xlab="t*", ylab="tL*")
     abline(0,1)

     # Now for the log of the ratio statistic for the city data.
     city.lin2 <- linear.approx(city.boot,t0=log(city.boot$t0), 
                                t=log(city.boot$t))
     lim <- range(c(log(city.boot$t),city.lin2))
     plot(log(city.boot$t), city.lin2, xlim=lim, ylim=lim, 
          main="Log(Ratio); n=10", xlab="t*", ylab="tL*")
     abline(0,1)

     # The ratio statistic for the bigcity data.
     bigcity.lin1 <- linear.approx(bigcity.boot)
     lim <- range(c(bigcity.boot$t,bigcity.lin1))
     plot(bigcity.lin1,bigcity.boot$t, xlim=lim,ylim=lim,
          main="Ratio; n=49", xlab="t*", ylab="tL*")
     abline(0,1)

     # Finally the log of the ratio statistic for the bigcity data.
     bigcity.lin2 <- linear.approx(bigcity.boot,t0=log(bigcity.boot$t0), 
                                   t=log(bigcity.boot$t))
     lim <- range(c(log(bigcity.boot$t),bigcity.lin2))
     plot(bigcity.lin2,log(bigcity.boot$t), xlim=lim,ylim=lim,
          main="Log(Ratio); n=49", xlab="t*", ylab="tL*")
     abline(0,1)

     par(mfrow=c(1,1))

