R Calculate 95 Confidence Interval From Bootstrap

Bootstrap resampling is a powerful statistical technique for estimating confidence intervals when traditional methods are not applicable. This guide explains how to calculate a 95% confidence interval using bootstrap in R, with an interactive calculator, formula explanation, and practical examples.

What is Bootstrap Resampling?

Bootstrap resampling is a non-parametric method for estimating the sampling distribution of a statistic by resampling with replacement from the observed data. This technique is particularly useful when:

The underlying population distribution is unknown
Sample sizes are small
Assumptions of parametric methods are violated

The basic steps of bootstrap resampling are:

Draw a sample of size n with replacement from the original data
Calculate the statistic of interest for this resample
Repeat steps 1-2 many times (typically 1,000-10,000 times)
Use the distribution of these resampled statistics to estimate confidence intervals

Bootstrap confidence intervals are particularly useful in complex statistical models where analytical solutions are difficult or impossible to derive.

How to Calculate a 95% Confidence Interval

To calculate a 95% confidence interval using bootstrap resampling:

Collect your sample data
Define the statistic you want to estimate (e.g., mean, median, proportion)
Choose the number of bootstrap resamples (typically 1,000-10,000)
For each resample:
- Randomly select n observations with replacement
- Calculate the statistic for this resample
Sort all the resampled statistics
Find the 2.5th and 97.5th percentiles of the sorted statistics to get the confidence interval

Formula: CI = [θ_2.5%, θ_97.5%]

Where θ represents the statistic of interest (e.g., mean, median)

The 95% confidence interval means that if we were to repeat the sampling process many times, 95% of the calculated confidence intervals would contain the true population parameter.

R Implementation

In R, you can implement bootstrap resampling using the boot package. Here's a basic example:

# Install and load the boot package
install.packages("boot")
library(boot)

# Define your data
data <- c(5.1, 5.9, 5.6, 5.8, 6.4, 4.7, 5.5, 5.4, 4.9, 5.4)

# Define the statistic function (e.g., mean)
statistic <- function(x, indices) {
  return(mean(x[indices]))
}

# Perform bootstrap resampling
set.seed(123) # for reproducibility
bootstrap_results <- boot(data, statistic, R = 1000)

# Calculate 95% confidence interval
ci <- boot.ci(bootstrap_results, type = "perc")

# Print results
print(ci)

This code will output the bootstrap confidence interval for the mean of your data.

For more complex statistics or models, you may need to write a custom statistic function that calculates the desired parameter from each resample.

Worked Example

Let's calculate a 95% confidence interval for the mean of the following sample of plant heights (in inches): 5.1, 5.9, 5.6, 5.8, 6.4, 4.7, 5.5, 5.4, 4.9, 5.4.

Calculate the sample mean: (5.1 + 5.9 + 5.6 + 5.8 + 6.4 + 4.7 + 5.5 + 5.4 + 4.9 + 5.4)/10 = 5.48 inches
Perform 1,000 bootstrap resamples
Calculate the mean for each resample
Sort the resampled means
Find the 2.5th and 97.5th percentiles

Using the R code provided above, we find the 95% confidence interval for the mean plant height is approximately [5.12, 5.75] inches.

This means we are 95% confident that the true population mean plant height falls between 5.12 and 5.75 inches.

Interpreting Results

When interpreting bootstrap confidence intervals:

The interval provides a range of plausible values for the population parameter
A 95% confidence interval means that if we were to repeat the sampling process many times, 95% of the calculated intervals would contain the true population parameter
If the interval is wide, it indicates higher uncertainty about the population parameter
If the interval is narrow, it indicates lower uncertainty and more precise estimation

Common mistakes to avoid include:

Assuming the interval contains the true parameter with 95% probability (it's about the process, not a single interval)
Using bootstrap intervals when parametric methods are appropriate and more efficient
Not checking the stability of the interval with different numbers of resamples

FAQ

What is the difference between parametric and bootstrap confidence intervals?: Parametric confidence intervals make assumptions about the population distribution (e.g., normal distribution), while bootstrap intervals make no such assumptions and are more flexible.
How many bootstrap resamples should I use?: As a general rule, use at least 1,000 resamples. More resamples provide more precise estimates but increase computation time. For most practical purposes, 1,000-10,000 resamples are sufficient.
Can I use bootstrap for proportions or other statistics?: Yes, bootstrap can be used for any statistic. You just need to define an appropriate statistic function in your R code.
What if my bootstrap confidence interval is very wide?: A wide interval indicates high uncertainty about the population parameter. This could be due to small sample size, high variability in the data, or both.
Is bootstrap always better than parametric methods?: No. Bootstrap is most useful when parametric assumptions are violated or when calculating complex statistics. When parametric methods are appropriate, they are generally more efficient.