# Nonparametric Bootstrap Confidence Intervals 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. ```r 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. ## Returns 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 ```r # 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) ```

boot.ci function