boot.ci                 package:boot                 R Documentation

Nonparametric Bootstrap Confidence Intervals

Description:

     This function generates 5 different types of equi-tailed two-sided
     nonparametric confidence intervals.  These are the first order
     normal  approximation, the basic bootstrap interval, the
     studentized bootstrap  interval, the bootstrap percentile
     interval, and the adjusted bootstrap  percentile (BCa) interval. 
     All or a subset of these intervals can be  generated.

Usage:

     boot.ci(boot.out, conf = 0.95, type = "all", 
             index = 1:min(2,length(boot.out$t0)), var.t0 = NULL, 
             var.t = NULL, t0 = NULL, t = NULL, L = NULL, h = function(t) t,
             hdot = function(t) rep(1,length(t)), hinv = function(t) t, ...)

Arguments:

boot.out: An object of class '"boot"' containing the output of a
          bootstrap calculation.   

    conf: A scalar or vector containing the confidence level(s) of the
          required interval(s). 

    type: A vector of character strings representing the type of
          intervals required. The value should be any subset of the
          values 'c("norm","basic", "stud", "perc", "bca")' or simply
          '"all"' which will compute all five types of intervals. 

   index: This should be a vector of length 1 or 2.  The first element
          of 'index' indicates the position of the variable of interest
          in 'boot.out$t0' and the relevant column in 'boot.out$t'. 
          The second element indicates the position of the variance of
          the variable of interest.  If both 'var.t0' and 'var.t' are
          supplied then the second element of 'index' (if present) is
          ignored.  The default is that the variable of interest is in
          position 1 and its variance is in position 2 (as long as
          there are 2 positions in 'boot.out$t0'). 

  var.t0: If supplied, a value to be used as an estimate of the
          variance of the statistic for the normal approximation and
          studentized intervals. If it is not supplied and
          'length(index)' is 2 then 'var.t0' defaults to
          'boot.out$t0[index[2]]' otherwise 'var.t0' is undefined.  For
          studentized intervals 'var.t0' must be defined. For the
          normal approximation, if 'var.t0' is undefined it defaults to
          'var(t)'.  If a transformation is supplied through the
          argument 'h' then 'var.t0' should be the variance of the
          untransformed statistic. 

   var.t: This is a vector (of length 'boot.out$R') of variances of the
          bootstrap replicates of the variable of interest.  It is used
          only for studentized intervals.  If it is not supplied and
          'length(index)' is 2 then 'var.t' defaults to
          'boot.out$t[,index[2]]', otherwise its value is undefined
          which will cause an error for studentized intervals.  If a
          transformation is supplied through the argument 'h' then
          'var.t' should be the variance of the untransformed bootstrap
          statistics. 

      t0: The observed value of the statistic of interest.  The default
          value is 'boot.out$t0[index[1]]'.  Specification of 't0' and
          't' allows the user to get intervals for a transformed
          statistic which may not be in the bootstrap output object. 
          See the second example below.  An alternative way of
          achieving this would be to supply the functions 'h', 'hdot',
          and 'hinv' below. 

       t: The bootstrap replicates of the statistic of interest.  It
          must be a vector of length 'boot.out$R'.  It is an error to
          supply one of 't0' or 't' but not the other.  Also if
          studentized intervals are required and 't0' and 't' are
          supplied then so should be 'var.t0' and 'var.t'.  The default
          value is 'boot.out$t[,index]'. 

       L: The empirical influence values of the statistic of interest
          for the observed data.  These are used only for BCa
          intervals.  If a transformation is supplied through the
          parameter 'h' then 'L' should be the influence values for
          't'; the values for 'h(t)' are derived from these and 'hdot'
          within the function. If 'L' is not supplied then the values
          are calculated using 'empinf' if they are needed. 

       h: A function defining a transformation.  The intervals are
          calculated on the scale of 'h(t)' and the inverse function
          'hinv' applied to the resulting intervals.  It must be a
          function of one variable only and for a vector argument, it
          must return a vector of the same length, i.e.
          'h(c(t1,t2,t3))' should return 'c(h(t1),h(t2),h(t3))'. The
          default is the identity function. 

    hdot: A function of one argument returning the derivative of 'h'. 
          It is a required argument if 'h' is supplied and normal,
          studentized or BCa intervals are required.  The function is
          used for approximating the variances of 'h(t0)' and 'h(t)'
          using the delta method, and also for finding the empirical
          influence values for BCa intervals.  Like 'h' it should be
          able to take a vector argument and return a vector of the
          same length.  The default is the constant function 1. 

    hinv: A function, like 'h', which returns the inverse of 'h'. It is
          used to transform the intervals calculated on the scale of
          'h(t)' back to the original scale. The default is the
          identity function.  If 'h' is supplied but 'hinv' is not,
          then the intervals returned will be on the transformed scale. 

     ...: Any extra arguments that 'boot.out$statistic' is expecting.
          These arguments are needed only if BCa intervals are required
          and 'L' is not supplied since in that case 'L' is calculated
          through a call to 'empinf' which calls 'boot.out$statistic'. 

Details:

     The formulae on which the calculations are based can be found in
     Chapter 5 of Davison and Hinkley (1997).  Function 'boot' must be
     run prior to running this function to create the object to be
     passed as 'boot.out'.

     Variance estimates are required for studentized intervals.  The
     variance of the observed statistic is optional for normal theory
     intervals.  If it is not supplied then the bootstrap estimate of
     variance is used.  The normal intervals also use the bootstrap
     bias correction.

     Interpolation on the normal quantile scale is used when a
     non-integer order statistic is required.  If the order statistic
     used is the smallest or largest of the R values in boot.out a
     warning is generated and such intervals should not be considered
     reliable.

Value:

     An object of type '"bootci"' which contains the intervals. It has
     components 

       R: The number of bootstrap replicates on which the intervals
          were based. 

      t0: The observed value of the statistic on the same scale as the
          intervals. 

    call: The call to 'boot.ci' which generated the object.

          It will also contain one or more of the following components
          depending on the value of 'type' used in the call to
          'bootci'. 

  normal: A matrix of intervals calculated using the normal
          approximation.  It will have 3 columns, the first being the
          level and the other two being the upper and lower endpoints
          of the intervals. 

   basic: The intervals calculated using the basic bootstrap method. 

 student: The intervals calculated using the studentized bootstrap
          method. 

 percent: The intervals calculated using the bootstrap percentile
          method. 

     bca: The intervals calculated using the adjusted bootstrap
          percentile (BCa) method.

          These latter four components will be matrices with 5 columns,
           the first column containing the level, the next two
          containing the indices of the order statistics used in the
          calculations and the final two the calculated endpoints
          themselves.  

References:

     Davison, A.C. and Hinkley, D.V. (1997)  _Bootstrap Methods and
     Their Application_, Chapter 5. Cambridge University Press.

     DiCiccio, T.J. and Efron  B. (1996) Bootstrap confidence intervals
     (with  Discussion). _Statistical Science_, *11*, 189-228.

     Efron, B. (1987) Better bootstrap confidence intervals (with
     Discussion). _Journal of the American Statistical Association_,
     *82*, 171-200.

See Also:

     'abc.ci', 'boot', 'empinf', 'norm.ci'

Examples:

     # confidence intervals for the city data
     ratio <- function(d, w) sum(d$x * w)/sum(d$u * w)
     city.boot <- boot(city, ratio, R = 999, stype = "w",sim = "ordinary")
     boot.ci(city.boot, conf = c(0.90,0.95),
             type = c("norm","basic","perc","bca"))

     # studentized confidence interval for the two sample 
     # difference of means problem using the final two series
     # of 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,]
     grav1.boot <- boot(grav1, diff.means, R=999, stype="f", strata=grav1[,2])
     boot.ci(grav1.boot, type=c("stud","norm"))

     # Nonparametric confidence intervals for mean failure time 
     # of the air-conditioning data as in Example 5.4 of Davison
     # and Hinkley (1997)
     mean.fun <- function(d, i) 
     {    m <- mean(d$hours[i])
          n <- length(i)
          v <- (n-1)*var(d$hours[i])/n^2
          c(m, v)
     }
     air.boot <- boot(aircondit, mean.fun, R=999)
     boot.ci(air.boot, type = c("norm", "basic", "perc", "stud"))

     # Now using the log transformation
     # There are two ways of doing this and they both give the
     # same intervals.

     # Method 1
     boot.ci(air.boot, type = c("norm", "basic", "perc", "stud"), 
             h = log, hdot = function(x) 1/x)

     # Method 2
     vt0 <- air.boot$t0[2]/air.boot$t0[1]^2
     vt <- air.boot$t[,2]/air.boot$t[,1]^2
     boot.ci(air.boot, type = c("norm", "basic", "perc", "stud"), 
             t0 = log(air.boot$t0[1]), t = log(air.boot$t[,1]),
             var.t0 = vt0, var.t = vt)

