Introducing the bootstrap confidence interval

A bootstrap confidence interval is a type of confidence interval that is calculated using the bootstrapping technique. Bootstrapping involves resampling from a dataset with replacement to estimate the sampling distribution of a statistic of interest. The resulting sampling distribution can then be used to estimate the confidence interval for the population parameter of interest.

How to calculate bootstrap confidence interval

To calculate a bootstrap confidence interval, we start by creating multiple resamples of the original dataset. For each resample, we calculate the statistic of interest, such as the mean, median, or proportion. We then use these resampled statistics to estimate the sampling distribution of the statistic.

Once we have the sampling distribution, we can calculate a confidence interval around our estimate of the population parameter. For example, a 95% confidence interval would include the middle 95% of the sampling distribution. This interval can be used to estimate the range of values that the true population parameter is likely to fall within.

The benefit of using a bootstrap confidence interval is that it does not rely on any assumptions about the underlying population distribution. This makes it particularly useful when the population distribution is unknown or non-normal. Additionally, bootstrap confidence intervals can be used for a wide range of statistical tests and analyses, including hypothesis testing and regression analysis.

Overall, bootstrap confidence intervals provide a useful tool for estimating the uncertainty associated with a statistic of interest, allowing researchers to make inferences about the population with more accuracy and confidence.

Two methods for constructing a confidence interval

Once we have a bootstrap sampling distribution there are two methods for constructing a confidence interval:

1. The standard deviation of the bootstrap distribution is the standard error which we can use to construct a bootstrap confidence interval. Recall that for a 95% confidence interval, given that the sampling distribution is approximately normal, the 95% confidence interval will be

CI = point estimate ± (z-value * standard error)

2. For a 95% confidence interval we can find the middle 95% bootstrap statistics. This is known as the percentile method. This is the preferred method because it works regardless of the shape of the sampling distribution.

Example – bootstrap confidence interval

For example, if we want to calculate a 95% bootstrap confidence interval for the mean height of a population based on a sample of 100 individuals, and the point estimate is 68 inches, and the estimated standard error is 0.5 inches, the 95% bootstrap confidence interval would be:

CI = 68 ± (1.96 * 0.5) = (67.02, 68.98)

Therefore, we can say with 95% confidence that the true population mean height is likely to fall within the range of 67.02 to 68.98 inches.

Example – bootstrap confidence interval percentile method

The percentile method is a non-parametric approach to constructing bootstrap confidence intervals. It involves finding the upper and lower percentiles of the bootstrap sampling distribution that correspond to the desired confidence level, and using those values to define the confidence interval.

For example, to construct a 95% confidence interval using the percentile method, we would find the 2.5th percentile and the 97.5th percentile of the bootstrap sampling distribution. The lower bound of the confidence interval would be the value of the statistic at the 2.5th percentile, and the upper bound would be the value of the statistic at the 97.5th percentile.

The percentile method is preferred over other methods (such as the t-interval or normal approximation interval) because it does not require any assumptions about the shape of the sampling distribution. It works well for both skewed and symmetric distributions, and is generally more robust and accurate.

However, it’s important to keep in mind that the percentile method can be computationally intensive, especially for larger sample sizes or when using many resamples. In these cases, other methods such as the t-interval or normal approximation interval may be more efficient.