Concept Architecture
Concept
Theoretically, Bootstrapping is a computational resampling method used to estimate the sampling distribution of a statistic by repeatedly drawing samples with replacement from the observed dataset. It represents a non-parametric approach to statistical inference that avoids reliance on analytical sampling distributions or restrictive distributional assumptions. The concept is founded on resampling theory and exists to estimate uncertainty for complex estimators when theoretical methods are unavailable or unreliable.
Mathematically, Bootstrapping generates a large number of bootstrap samples, each having the same size as the original dataset and obtained by random sampling with replacement. The statistic of interest is recalculated for every bootstrap sample, producing an empirical approximation to its sampling distribution. Standard errors, confidence intervals and measures of bias are subsequently derived from the distribution of bootstrap estimates.
In practice, Bootstrapping is implemented using hundreds or thousands of resamples within statistical software or simulation frameworks. It is widely applied in regression modelling, survival analysis, health economic evaluation and cost-effectiveness analysis to estimate parameter uncertainty, generate confidence intervals, quantify sampling variability and support probabilistic decision-making.
Purpose
Used to estimate sampling distributions, quantify statistical uncertainty, calculate standard errors and confidence intervals and support complex statistical and health economic analyses without requiring strong parametric assumptions.
Mathematical Formulae
Primary Formula
??? = Statistic(X?*)
where:
- ??? = estimate from bootstrap sample i
- X?* = bootstrap sample drawn with replacement from the original data
Supporting Formulae
?? = (1 / B) ? ??????? ???
SE? = �[(1 / (B ? 1)) ? ??????? (??? ? ??)�]
Bias = ?? ? ??
where:
- B = number of bootstrap samples
- ?? = estimate from the original sample
Related Mathematical Methods
- Bootstrap Standard Error
- Bootstrap Confidence Interval
- Bias-Corrected and Accelerated Bootstrap
- Non-Parametric Bootstrap
- Monte Carlo Simulation
- Jackknife
- Resampling Methods
Example
A health economic evaluation estimates the Incremental Cost-Effectiveness Ratio using patient-level trial data.
Five thousand bootstrap samples are generated by sampling patients with replacement.
The Incremental Cost-Effectiveness Ratio is recalculated for each bootstrap sample, producing an empirical distribution from which the standard error, confidence interval and cost-effectiveness acceptability curve are estimated.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| RANDARRAY | =RANDARRAY(1000,1) | Generates random values for bootstrap resampling. |
| INDEX | =INDEX(B$2:B$101,RANDBETWEEN(1,100)) | Samples observations with replacement from the original dataset. |
| AVERAGE | =AVERAGE(C2:C101) | Calculates the statistic of interest for each bootstrap sample. |
| STDEV.S | =STDEV.S(D2:D5001) | Calculates the Bootstrap Standard Error from repeated bootstrap estimates. |
VBA (Optional)
A VBA macro can automate bootstrap resampling, repeatedly estimate model parameters and generate empirical sampling distributions, confidence intervals and uncertainty summaries.
Sources
- Efron B. Bootstrap methods: another look at the jackknife. Annals of Statistics. 1979;7(1):1?26.
- 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.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (3)
Library
Publications
2
Economic Evaluation in Clinical Trials — Glick, Doshi, Sonnad & Polsky, 2nd Edition ed., 2015 (Oxford University Press)
Practical guidance on conducting cost-effectiveness analyses alongside controlled trials, covering trial design, measurement of costs and quality-adjusted life years, handling censored and missing data, and reporting stochastic uncertainty. Volume 4 in the Handbooks in Health Economic Evaluation series.
BookView source →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 →
Media
1
Using and Interpreting Cost-Effectiveness Acceptability Curves (AFFIRM Example) — Fenwick, Marshall, Levy & Nichol, Open access ed., 2006 (BMC Health Services Research (Open Access))
An open-access tutorial article with annotated diagrams walking through the incremental cost-effectiveness plane and the construction and interpretation of cost-effectiveness acceptability curves, using atrial fibrillation trial data.
Web (Open Access)View source →
Frequently Asked Questions (6)
What is bootstrapping?
A resampling technique estimating a statistic's sampling distribution by repeatedly drawing random samples, with replacement, from the original observed data.
Source: Efron 1979
How does bootstrapping approximate a statistic's sampling distribution?
Bootstrapping approximates the sampling distribution of a statistic by drawing many new samples from the observed data itself, each formed by sampling with replacement to the original size, and computing the statistic on each. Treating the sample as a stand-in for the population, it builds up the distribution of the statistic empirically rather than deriving it mathematically. This allows confidence intervals and standard errors for quantities too awkward for formulas, provided the original sample represents the population reasonably. Resampling the data to see how a statistic varies is its logic. Efron and Tibshirani (1993) describe this.
Source: Efron & Tibshirani 1993
How does bootstrapping work?
Bootstrapping works by treating the observed sample as a stand-in for the population and repeatedly drawing new samples, called bootstrap resamples, from it with replacement, each the same size as the original. The statistic of interest is computed on each resample, and the collection of these values approximates the statistic's sampling distribution, from which standard errors and confidence intervals are derived. So bootstrapping works by resampling the data many times with replacement and recomputing the statistic, building up an empirical distribution that stands in for the sampling distribution, which allows uncertainty to be quantified without formulas or strong distributional assumptions.
Source: Efron 1979
When is bootstrapping used?
Bootstrapping is used when the sampling distribution of a statistic is unknown, complex, or hard to derive analytically, or when the assumptions of formula-based methods are doubtful, since it estimates the distribution directly from the data. It is valuable for complicated statistics and for constructing confidence intervals without assuming normality. So bootstrapping is used to quantify uncertainty for statistics where standard analytical approaches are unavailable or unreliable, providing a general, flexible tool for inference, which is why it is widely applied to estimate standard errors and confidence intervals for a broad range of statistics and models.
Source: Efron 1979
What are the advantages of bootstrapping?
The advantages of bootstrapping include that it requires few distributional assumptions, estimating uncertainty directly from the data; that it applies to complex statistics for which analytical formulas are difficult or unavailable; and that it is conceptually straightforward and general. So bootstrapping is advantageous for its flexibility and minimal assumptions, allowing uncertainty to be assessed for many statistics and situations where traditional methods are hard to apply, which makes it a valuable and widely used tool, though it depends on the sample being representative and can be computationally demanding, so its advantages are realised where these conditions are acceptable.
Source: Efron 1979
What are the limitations of bootstrapping?
The limitations of bootstrapping include that it assumes the observed sample adequately represents the population, so it can mislead with very small or unrepresentative samples; that it can be computationally intensive, requiring many resamples; and that it may perform poorly for certain statistics, such as extreme quantiles. So bootstrapping is used with awareness that its validity rests on the sample being representative and sufficiently large, and that it is an approximation rather than a guarantee, since resampling cannot create information not present in the data, which means its results are interpreted with attention to sample size and the nature of the statistic being estimated.
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
- Persistent URI
- https://healtheconomics.wiki/concept/bootstrapping
- Term code
- HE-ES-SA-017
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