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Bootstrapping in R Programming

  • Last Updated : 26 May, 2020

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

  • Select number of bootstrap samples.
  • Select 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 a sample data in statistics. It produces a range of values or an interval where true value lies in 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:




# 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



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