Bootstrapping

Basic Nonparametric Bootstrap

As an example, we will consider the problem of estimating the standard error for the mean, x. This statistic can be calculated in a straightforward manner using the equation:

$$S{E}_x=S/\text{Square}\text{root}\left(N\right),$$

where S is the standard deviation of the sample and N is the sample size.

Equation 1 is based upon an assumption that the population distribution underlying the variable x is normal. However, if this is not the case, then Equation 1 no longer yields the appropriate standard error estimate of x. The bootstrap offers an alternative approach for calculating the standard error. The basic nonparametric bootstrap operates using the following steps:

• 1.
  Calculate sample statistic of interest (e.g., x) for the original sample.
• 2.
  Randomly sample n individuals from the original sample of size n, with replacement; individuals can appear multiple times in the bootstrap sample, while others may not appear at all.
• 3.
  Calculate the mean, xB*, for the bootstrap sample.
• 4.
  Repeat Steps 2 and 3 many times (e.g., B = 10,000) to create a sampling distribution for the statistic of interest.
• 5.
  Calculate the bootstrap standard error: SB = 1 BxB* − x* 2B−1.

To illustrate the bootstrap, consider the following simple example involving a sample of five individuals with scores 8, 3, 6, 1, and 5. The mean of these values is 4.6, and the standard deviation is 2.7. Based on Equation 1, the standard error is Sx = 2.75 = 1.2. Now, let’s draw five bootstrap samples (which would be far too few in practice but helps to illustrate how the bootstrap works) and calculate the mean for each. The samples appear below.

The standard deviation of the means for the five bootstrap samples is 0.68. Thus, based on these five samples, we would report that the bootstrap standard error of the mean is 0.68. In actual practice, we would use many more than five bootstrap samples, perhaps as many as 10,000. To finish this illustration of the basic bootstrap, a total of 1,000 bootstrap samples of the five data points were drawn using the software package SPSS, yielding a bootstrap standard error estimate of 1.06.

Bootstrap Confidence Intervals

Standard errors are frequently used to construct a confidence interval for the statistic of interest, in this case the mean. Confidence intervals reflect the neighborhood of values within which the population parameter is likely to reside. Using normal theory methods, the confidence interval for the mean is calculated

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