Bootstrapping in R Programming (original) (raw)

Last Updated : 16 Dec, 2021

Bootstrapping is a technique used in inferential statistics that work on building random samples of single datasets again and again. Bootstrapping allows calculating measures such as mean, median, mode, confidence intervals, etc. of the sampling.

R - Bootstrapping

Following is the process of bootstrapping in R Programming Language:

• Select the number of bootstrap samples.
• Select the size of each sample.
• For each sample, if the size of the sample is less than the chosen sample, then select a random observation from the dataset and add it to the sample.
• Measure the statistic on the sample.
• Measure the mean of all calculated sample values.

Methods of Bootstrapping

There are 2 methods of bootstrapping:

• Residual Resampling: This method is also called as model-based resampling. This method assumes that model is correct and errors are independent and distributed identically. After each resampling, variables are redefined and new variables are used to measure the new dependent variables.
• Bootstrap Pairs: In this method, dependent and independent variables are used together as pairs for sampling.

Types of Confidence Intervals in Bootstrapping

Confidence Interval (CI) is a type of computational value calculated on sample data in statistics. It produces a range of values or an interval where the true value lies for sure. There are 5 types of confidence intervals in bootstrapping as follows:

• Basic: It is also known as Reverse Percentile Interval and is generated using quantiles of bootstrap data distribution. Mathematically,

\left(2 \widehat{\theta}-\theta_{(1-\alpha / 2)}^{*}, 2 \widehat{\theta}-\theta_{(\alpha / 2)}^{*}\right)

where,
\alpha represents confidence interval, mostly \alpha = 0.95
\theta^{*} represents bootstrapped coefficients
\theta_{(1-\alpha / 2)}^{*} represents 1-\alpha / 2 percentile of bootstrapped coefficients

• Normal: Normal CI is mathematically given as,

\begin{array}{c} t_{0}-b \pm Z_{\alpha} \cdot \mathrm{se}^{*} \\ 2 t_{0}-t^{*} \pm Z_{\alpha} \cdot \mathrm{se}^{*} \end{array}
where,

t_{0} represents a value from dataset t
b is the bias of bootstrap estimate i.e.,

\mathbf{b}=\mathbf{t}^{*}-\mathbf{t}_{\mathrm{o}}
Z_{\alpha} represents 1-\alpha / 2 quantile of bootstrap distribution
se^{*} represents standard error oft^{*}

• Stud: In studentized CI, data is normalized with center at 0 and standard deviation 1 correcting the skew of distribution.
• Perc - Percentile CI is similar to basic CI but with different formula,

\left(\theta_{(\alpha / 2)}^{*}, \theta_{(1-\alpha / 2)}^{*}\right)

• BCa: This method adjusts for both bias and skewness but can be unstable when outliers are extreme. Mathematically,

\left(\theta_{0}+\frac{\theta_{0}+\theta_{\alpha}}{1-a\left(\theta_{0}-\theta_{\alpha}\right)}, \theta_{0}+\frac{\theta_{0}+\theta_{(1-\alpha)}}{1-a\left(\theta_{0}-\theta_{(1-\alpha)}\right)}\right)

The syntax to perform bootstrapping in R programming is as follows:

Syntax: boot(data, statistic, R)

Parameters:

• data represents dataset
• statistic represents statistic functions to be performed on dataset
• R represents number of samples

To learn about more optional arguments of boot() function, use below command:

help("boot")

Example:

R `

Library required for boot() function

install.packages("boot")

Load the library

library(boot)

Creating a function to pass into boot() function

bootFunc <- function(data, i){ df <- data[i, ] c(cor(df[, 2], df[, 3]), median(df[, 2]), mean(df[, 1]) ) }

b <- boot(mtcars, bootFunc, R = 100)

print(b)

Show all CI values

boot.ci(b, index = 1)

`

Output:

ORDINARY NONPARAMETRIC BOOTSTRAP Call: boot(data = mtcars, statistic = bootFunc, R = 100)

Bootstrap Statistics : original bias std. error t1* 0.9020329 -0.002195625 0.02104139 t2* 6.0000000 0.340000000 0.85540468 t3* 20.0906250 -0.110812500 0.96052824

BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS Based on 100 bootstrap replicates

CALL : boot.ci(boot.out = b, index = 1)

Intervals : Level Normal Basic
95% ( 0.8592, 0.9375 ) ( 0.8612, 0.9507 )

Level Percentile BCa
95% ( 0.8534, 0.9429 ) ( 0.8279, 0.9280 )
Calculations and Intervals on Original Scale Some basic intervals may be unstable Some percentile intervals may be unstable Warning : BCa Intervals used Extreme Quantiles Some BCa intervals may be unstable Warning messages: 1: In boot.ci(b, index = 1) : bootstrap variances needed for studentized intervals 2: In norm.inter(t, adj.alpha) : extreme order statistics used as endpoints