Statistics is the science of learning from data. Statistical knowledge aids in the proper methods for collecting data, analyzing data and effectively presenting the results derived from data. These methods are crucial to making decisions and predictions, whether it be predicting the consumer demand for a product, using text mining to filter spam emails or making real-time decisions in self-driving cars.
Bootstrapping Statistics Defined
Most of the time when you’re conducting research, it’s impractical to collect data from the entire population. This can be due to budget and/or time constraints, among other factors. Instead, a subset of the population is taken and insight is gathered from that subset to learn more about the population.
This means that suitably accurate information can be obtained quickly and relatively inexpensively from an appropriately drawn sample. However, many things can affect how well a sample reflects the population, and therefore, the validity and reliability of the conclusions. Because of this, let us talk about bootstrapping statistics.
What Is Bootstrapping Statistics?
“Bootstrapping is a statistical procedure that resamples a single data set to create many simulated samples. This process allows for the calculation of standard errors, confidence intervals, and hypothesis testing,” according to a post on bootstrapping statistics from statistician Jim Forst.
A bootstrapping approach is an extremely useful alternative to the traditional method of hypothesis testing, as it’s fairly simple and it mitigates some of the pitfalls encountered within the traditional approach.
Statistical inference generally relies on the sampling distribution and the standard error of the feature of interest. The traditional approach, or large sample approach, draws one sample of size n from the population, and that sample is used to calculate population estimates to then make inferences on. In reality, only one sample has been observed.
“Bootstrapping is a statistical procedure that resamples a single data set to create many simulated samples.”
However, a sampling distribution is a theoretical set of all possible estimates if the population were to be resampled. The theory states that, under certain conditions such as large sample sizes, the sampling distribution will be approximately normal, and the standard deviation of the distribution will be equal to the standard error. But what happens if the sample size is not sufficiently large enough? Then, it can’t necessarily be assumed that the theoretical sampling distribution is normal. This makes it difficult to determine the standard error of the estimate and harder to draw reasonable conclusions from the data.
How Bootstrapping Statistics Works
In the bootstrapping approach, a sample of size n is drawn from the population. Let’s call this sample S. Then, rather than using theory to determine all possible estimates, the sampling distribution is created by resampling observations with replacement from S m times, with each resampled set having n observations. Now, if sampled appropriately, S should be representative of the population. Therefore, by resampling S m times with replacement, it would be as if m samples were drawn from the original population, and the estimates derived would be representative of the theoretical distribution under the traditional approach.
Increasing the number of resamples, m, will not increase the amount of information in the data. That is, resampling the original set 100,000 times is not more useful than resampling it 1,000 times. The amount of information within the set is dependent on the sample size, n, which will remain constant throughout each resample. The benefit of more resamples, then, is to derive a better estimate of the sampling distribution.
Advantages of Bootstrapping Statistics
Now that we understand the bootstrapping approach, it must be noted that the results derived are basically identical to those of the traditional approach. Additionally, the bootstrapping approach will always work because it doesn’t assume any underlying distribution of the data.
This contrasts with the traditional approach which theoretically assumes that the data are normally distributed. Knowing how the bootstrapping approach works, you might wonder, does the bootstrapping approach rely too much on the observed data? This is a good question, given that the resamples are derived from the initial sample. And because of this, it’s logical to assume that an outlier will skew the estimates from the resamples.
“The advantages of bootstrapping are that it is a straightforward way to derive the estimates of standard errors and confidence intervals, and it is convenient since it avoids the cost of repeating the experiment to get other groups of sampled data.”
While this is true, if the traditional approach is considered, an outlier within the data set will also skew the mean and inflate the standard error of the estimate. While it might be tempting to think that an outlier can show up multiple times within the resampled data and skew the results and thus, making the traditional approach better, the bootstrapping approach relies as much on the data as the traditional approach.
“The advantages of bootstrapping are that it is a straightforward way to derive the estimates of standard errors and confidence intervals, and it is convenient since it avoids the cost of repeating the experiment to get other groups of sampled data. Although it is impossible to know the true confidence interval for most problems, bootstrapping is asymptotically consistent and more accurate than using the standard intervals obtained using sample variance and the assumption of normality,” according to author Graysen Cline in their book, Nonparametric Statistical Methods Using R.
Both approaches require the use of appropriately drawn samples to make inferences about populations. However, the biggest difference between these two methods is the mechanics behind estimating the sampling distribution. The traditional procedure requires one to have a test statistic that satisfies particular assumptions in order to achieve valid results, and this is largely dependent on the experimental design. The traditional approach also uses theory to tell what the sampling distribution should look like, but the results fall apart if the assumptions of the theory are not met.
The bootstrapping method, on the other hand, takes the original sample data and then resamples it to create many [simulated] samples. This approach doesn’t rely on theory since the sampling distribution can be observed, and you don’t have to worry about any assumptions. This technique allows for accurate estimates of statistics, which is crucial when using data to make decisions.