# Bootstrapping

## Nonparametric Bootstrapping

The **boot **package provides extensive facilities for bootstrapping and related resampling methods. You can bootstrap a single statistic (e.g. a median), or a vector (e.g., regression weights). This section will get you started with basic nonparametric bootstrapping.

The main bootstrapping function is **boot( ) **and has the following format:

** bootobject <- boot(data= , statistic= , R=, ...) **where

parameter |
description |

data |
A vector, matrix, or dataframe |

statistic |
A function that produces the k statistics to be bootstrapped (k=1 if bootstrapping a single statistic). The function should include an indices parameter that the boot() function can use to select cases for each replication (see examples below). |

R |
Number of bootstrap replicates |

... |
Additional parameters to be passed to the function that produces the statistic of interest |

**boot( )** calls the statistic function *R* times. Each time, it generates a set of random indices, with replacement, from the integers 1:nrow(*data*). These indices are used within the statistic function to select a sample. The statistics are calculated on the sample and the results are accumulated in the *bootobject*. The *bootobject* structure includes

element |
description |

t0 |
The observed values of k statistics applied to the orginal data. |

t |
An R x k matrix where each row is a bootstrap replicate of the k statistics. |

You can access these as *bootobject*$t0 and *bootobject*$t.

Once you generate the bootstrap samples, **print(***bootobject***)** and **plot(***bootobject***)** can be used to examine the results. If the results look reasonable, you can use** boot.ci(**** )** function to obtain confidence intervals for the statistic(s).

The format is

**boot.ci(***bootobject, ***conf=, type=** ** )** where

parameter |
description |

bootobject |
The object returned by the boot function |

conf |
The desired confidence interval (default: conf=0.95) |

type |
The type of confidence interval returned. Possible values are "norm", "basic", "stud", "perc", "bca" and "all" (default: type="all") |

### Bootstrapping a Single Statistic (k=1)

The following example generates the bootstrapped 95% confidence interval for R-squared in the linear regression of miles per gallon (mpg) on car weight (wt) and displacement (disp). The data source is mtcars. The bootstrapped confidence interval is based on 1000 replications.

`# Bootstrap 95% CI for R-Squared`

library(boot)

# function to obtain R-Squared from the data

rsq <- function(formula, data, indices)
{

d <- data[indices,] # allows boot to select sample

fit <- lm(formula, data=d)

return(summary(fit)$r.square)

}

# bootstrapping with 1000 replications

results <- boot(data=mtcars, statistic=rsq,

R=1000, formula=mpg~wt+disp)

# view results

results

plot(results)

# get 95% confidence interval

boot.ci(results, type="bca")

### Bootstrapping several Statistics (k>1)

In example above, the function rsq returned a number and boot.ci returned a single confidence interval. The statistics function you provide can also return a vector. In the next example we get the 95% CI for the three model regression coefficients (intercept, car weight, displacement). In this case we add an index parameter to **plot( ) and boot.ci( )** to indicate which column in *bootobject*$t is to analyzed.

`# Bootstrap 95% CI for regression coefficients `

library(boot)

# function to obtain regression weights

bs <- function(formula, data, indices)
{

d <- data[indices,] # allows boot to select sample

fit <- lm(formula, data=d)

return(coef(fit))

}

# bootstrapping with 1000 replications

results <- boot(data=mtcars, statistic=bs,

R=1000, formula=mpg~wt+disp)

# view results

results

plot(results, index=1) # intercept

plot(results, index=2) # wt

plot(results, index=3) # disp

# get 95% confidence intervals

boot.ci(results, type="bca", index=1)
# intercept

boot.ci(results, type="bca", index=2)
# wt

boot.ci(results, type="bca", index=3)
# disp

## Going Further

The **boot( )** function can generate both nonparametric and parametric resampling. For the nonparametric bootstrap, resampling methods include ordinary, balanced, antithetic and permutation. For the nonparametric bootstrap, stratified resampling is supported. Importance resampling weights can also be specified.

The **boot.ci( ) **function takes a bootobject and generates 5 different types of two-sided nonparametric confidence intervals. These include the first order normal approximation, the basic bootstrap interval, the studentized bootstrap interval, the bootstrap percentile interval, and the adjusted bootstrap percentile (BCa) interval.

Look at **help(boot)**, **help(boot.ci)**, and **help(plot.boot) **for more details.

## Learning More

Good sources of information include **Resampling Methods in R: The boot Package** by Angelo Canty, **Getting started with the boot package** by Ajay Shah, **Bootstrapping Regression Models **by John Fox, and **Bootstrap Methods and Their Applications** by Davison and Hinkley.