Concept Architecture
Concept
Theoretically, Bootstrap Standard Error is a resampling-based estimate of the sampling variability of a statistical estimator obtained by repeatedly drawing samples with replacement from the observed dataset. It represents the empirical standard deviation of bootstrap estimates and is founded on non-parametric bootstrap theory. The concept exists to estimate standard errors when analytical variance formulae are unavailable, difficult to derive or unreliable.
Mathematically, the Bootstrap Standard Error is calculated as the sample standard deviation of parameter estimates obtained from a large number of bootstrap resamples. The method approximates the sampling distribution of the estimator without requiring assumptions about its theoretical distribution. As the number of bootstrap replications increases, the bootstrap estimate converges to the true sampling variability under appropriate regularity conditions.
In practice, Bootstrap Standard Error is estimated by generating hundreds or thousands of bootstrap samples and recalculating the statistic of interest for each sample. It is widely applied in regression modelling, cost-effectiveness analysis, survival analysis and health economic evaluation to estimate uncertainty for complex estimators and support confidence interval construction and probabilistic sensitivity analysis.
Purpose
Used to estimate sampling variability for complex estimators, quantify parameter uncertainty, construct confidence intervals and support statistical and health economic analyses when analytical standard errors are unavailable or unreliable.
Mathematical Formulae
Primary Formula
SE? = �[(1 / (B ? 1)) ? ??????? (??? ? ??)�]
where:
- SE? = bootstrap standard error
- B = number of bootstrap samples
- ??? = estimate from bootstrap sample i
- ?? = mean bootstrap estimate
Supporting Formulae
?? = (1 / B) ? ??????? ???
Approximate 95% CI = ?? � 1.96 ? SE?
Related Mathematical Methods
- Bootstrapping
- Bootstrap Confidence Interval
- Non-Parametric Bootstrap
- Bias-Corrected and Accelerated Bootstrap
- Monte Carlo Simulation
- Standard Error
- Resampling Methods
Example
A health economic evaluation estimates the incremental cost-effectiveness ratio using 5,000 bootstrap resamples.
The bootstrap estimates have:
- Mean estimate = �18,450 per QALY
- Bootstrap Standard Error = �1,250 per QALY
Approximate 95% confidence interval:
95% CI = �18,450 � 1.96 ? �1,250
95% CI = �18,450 � �2,450
95% CI = �16,000 to �20,900 per QALY
The Bootstrap Standard Error quantifies the sampling uncertainty surrounding the estimated cost-effectiveness ratio.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| STDEV.S | =STDEV.S(B2:B5001) | Calculates the Bootstrap Standard Error from bootstrap estimates. |
| AVERAGE | =AVERAGE(B2:B5001) | Calculates the mean bootstrap estimate. |
| COUNT | =COUNT(B2:B5001) | Determines the number of bootstrap replications. |
| CONFIDENCE.NORM | =CONFIDENCE.NORM(0.05,STDEV.S(B2:B5001),COUNT(B2:B5001)) | Estimates an approximate confidence interval half-width from bootstrap estimates. |
VBA (Optional)
A VBA macro can automatically generate bootstrap resamples, calculate Bootstrap Standard Errors and produce confidence intervals for multiple health economic model outputs.
Sources
- Efron B, Tibshirani RJ. An Introduction to the Bootstrap.
- Davison AC, Hinkley DV. Bootstrap Methods and Their Application.
- Briggs A, Wonderling D, Mooney C. Pulling cost-effectiveness analysis up by its bootstraps. Health Economics. 1997;6(4):327?340.
- Carpenter J, Bithell J. Bootstrap confidence intervals: when, which, what? Statistics in Medicine. 2000;19:1141?1164.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (3)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a bootstrap standard error?
An estimate of a parameter's statistical uncertainty, calculated as the standard deviation of its value across many resampled datasets via bootstrapping.
Source: Efron 1979
How does a bootstrap standard error gauge an estimate's uncertainty?
A bootstrap standard error gauges the uncertainty of a statistic by resampling the observed data many times, recomputing the statistic on each resample, and taking the standard deviation of those values. It estimates how much the statistic would vary from sample to sample without relying on a formula, which is why it is valued for complex quantities whose uncertainty has no simple closed-form expression. The spread of the resampled estimates stands in for the sampling variability. Measuring variability by resampling is its approach. Efron and Tibshirani (1993) describe this.
Source: Efron & Tibshirani 1993
How is a bootstrap standard error calculated?
A bootstrap standard error is calculated by drawing many resamples, with replacement, from the original data, each the same size as the original; computing the statistic of interest on each resample; and taking the standard deviation of the statistic across all the resamples. This standard deviation is the bootstrap standard error. So a bootstrap standard error is calculated from the spread of the statistic over repeated bootstrap resamples, which empirically approximates its sampling distribution, providing an estimate of uncertainty that does not require an analytical formula and can be applied to complex statistics for which such formulas are difficult or unavailable.
Source: Efron 1979
When is a bootstrap standard error useful?
A bootstrap standard error is useful when an analytical formula for the standard error is unavailable, complex, or based on assumptions that may not hold, since it estimates uncertainty directly from the data by resampling. It is valuable for complicated statistics, such as medians, ratios, or the outputs of multi-step procedures. So a bootstrap standard error is useful for quantifying the uncertainty of estimates where standard formulas are hard to derive or their assumptions are doubtful, providing a flexible, data-driven measure of variability, which is why the bootstrap is widely used for standard errors and confidence intervals of statistics that do not have simple closed-form expressions.
Source: Efron 1979
What are the limitations of a bootstrap standard error?
The limitations of a bootstrap standard error include that it can be computationally intensive, requiring many resamples; that it may perform poorly for some statistics or with very small samples, where resampling cannot capture the true variability well; and that it assumes the original sample is representative of the population. So a bootstrap standard error is used with awareness that its accuracy depends on the sample size and the statistic, and that it is an approximation, since resampling from an unrepresentative or very small sample may not reflect the true sampling variability, which means the bootstrap is not a universal remedy and its results are interpreted in light of these constraints.
Source: Efron 1979
How does a bootstrap standard error relate to bootstrapping?
A bootstrap standard error is one product of bootstrapping, the general resampling technique of repeatedly drawing samples with replacement from the data to approximate a statistic's sampling distribution. From that approximated distribution, the standard deviation gives the bootstrap standard error, while other summaries give confidence intervals. So a bootstrap standard error relates to bootstrapping as a specific output of the method, derived from the variability of the statistic across the bootstrap resamples, and it is one of the main uses of the bootstrap, alongside confidence interval estimation, for quantifying the uncertainty of estimates without relying on analytical formulas.
Source: Efron 1979
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 11 Dec 2025
Content version: 1.0.0
Canonical Identity
- Term code
- HE-ES-SA-016
Stable URI · Machine-readable · Resolvable · CC BY 4.0