What is stratified bootstrap?

What is Stratified Bootstrap?

Stratified bootstrap is a resampling technique used in statistics to estimate the sampling distribution of a statistic. It is a variation of the standard bootstrap method, but it incorporates stratification to improve the accuracy of the estimates, especially when dealing with data that is not uniformly distributed.

Understanding Stratification

What is Stratification?

Stratification is a sampling technique that involves dividing a population into subgroups, called strata, based on shared characteristics. This ensures that each subgroup is adequately represented in the sample. Stratification is often used when there is significant variation in the population and we want to ensure that the sample accurately reflects the distribution of the population.

Why Stratify in Bootstrapping?

When dealing with non-uniformly distributed data, using a simple bootstrap might not provide accurate estimates. The standard bootstrap might oversample certain groups and undersample others, leading to biased estimates. Stratified bootstrap addresses this by ensuring that each stratum is represented proportionally in the bootstrap samples.

How Stratified Bootstrap Works

The stratified bootstrap process involves the following steps:

1. Divide the population into strata: Identify the characteristic that you want to stratify by, and divide the population into subgroups based on that characteristic.
2. Sample within strata: From each stratum, draw a random sample with replacement. The size of the sample from each stratum should be proportional to its size in the original population.
3. Combine samples: Combine the samples from each stratum to form a bootstrap sample.
4. Repeat: Repeat steps 2 and 3 multiple times to create many bootstrap samples.
5. Estimate the statistic: Calculate the statistic of interest for each bootstrap sample.
6. Generate the distribution: The distribution of the statistic across the bootstrap samples provides an estimate of the sampling distribution of the statistic.

Example: Estimating Mean Income

Let’s say we want to estimate the mean income of a population using stratified bootstrap. Assume the population is divided into two strata: high-income and low-income. We would follow these steps:

1. Stratification: Divide the population into high-income and low-income strata.
2. Sampling: From each stratum, draw a random sample with replacement, ensuring that the sample sizes from each stratum are proportional to their population sizes.
3. Combination: Combine the samples from each stratum to form a bootstrap sample.
4. Repeat: Repeat steps 2 and 3 multiple times to create many bootstrap samples.
5. Calculation: Calculate the mean income for each bootstrap sample.
6. Distribution: The distribution of the mean incomes across the bootstrap samples provides an estimate of the sampling distribution of the mean income.

Code Example: R

library(boot)

# Example Data:  Income for two groups
income <- c(10000, 12000, 15000, 18000, 20000, 25000, 
           5000, 6000, 7000, 8000, 9000, 10000)
group <- c(rep("High", 6), rep("Low", 6))

# Function to calculate mean income for each bootstrap sample
mean_income <- function(data, indices){
  mean(data[indices])
}

# Stratified bootstrap
results <- boot(data = income, statistic = mean_income, 
                strata = group, R = 1000)

# Print bootstrap results
print(results)

Advantages of Stratified Bootstrap

• Improved Accuracy: By taking into account the population distribution, stratified bootstrap provides more accurate estimates, especially when dealing with non-uniform data.
• Reduced Variance: Stratified bootstrap can reduce the variance of the estimates, making them more reliable.
• Flexibility: The method can be applied to various statistics and different types of data.

Conclusion

Stratified bootstrap is a powerful resampling technique that offers significant advantages in situations where the data is not uniformly distributed. By stratifying the population and sampling within strata, it ensures that the bootstrap samples accurately reflect the population structure, leading to more reliable estimates.