Concept Architecture
Concept
Theoretically, Jackknife Standard Error is a resampling-based estimate of the sampling variability of a statistical estimator obtained using the jackknife procedure. It is founded on non-parametric estimation theory and influence function methodology, providing an estimate of precision without requiring analytical variance formulae. The concept exists to quantify uncertainty for complex estimators whose sampling distributions may be difficult to derive.
Mathematically, the jackknife standard error is calculated from the variability of leave-one-out estimates obtained by systematically omitting each observation from the sample. The method assumes that the collection of jackknife replicates approximates the sampling variability of the estimator. The standard error is obtained as the square root of the jackknife variance estimator.
In practice, the statistic of interest is recalculated after omitting each observation or cluster, producing a set of jackknife replicate estimates. These estimates are combined to calculate the jackknife variance and standard error. In health economics, jackknife standard errors are used to quantify uncertainty in cost-effectiveness estimates, regression coefficients, survival statistics and other complex estimators when analytical standard errors are unavailable or difficult to derive.
Purpose
Used to estimate the sampling uncertainty of statistical estimators through leave-one-out resampling, supporting robust inference for complex health economic analyses.
Mathematical Formulae
Primary Formula
Jackknife standard error:
SEJ = �[((n ? 1) / n) ? ????� (??(?i) ? ??J)�]
where:
- n = sample size
- ??(?i) = estimate after omitting observation i
- ??J = average of the jackknife replicate estimates
Supporting Formulae
Average jackknife estimate:
??J = (1/n) ? ????� ??(?i)
Jackknife variance:
VarJ(??) = ((n ? 1) / n) ? ????� (??(?i) ? ??J)�
Pseudo-value:
PV? = n?? ? (n ? 1)??(?i)
Related Mathematical Methods
- Jackknife
- Bootstrap
- Leave-One-Out Resampling
- Non-Parametric Variance Estimation
- Influence Function
- Bias Estimation
Example
A health economist estimates the incremental cost-effectiveness ratio using data from 50 patients.
Leave-one-out estimation produces 50 replicate estimates with an average:
??J = 1.245
The sum of squared deviations is:
?(??(?i) ? ??J)� = 0.080
The jackknife standard error is:
SEJ = �[((50 ? 1) / 50) ? 0.080]
SEJ = �0.0784
SEJ = 0.280
This value is used to construct confidence intervals and assess the precision of the estimated treatment effect.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:B51) | Calculates the mean of the jackknife replicate estimates. |
| SUMSQ | =SUMSQ(B2:B51-$B$52) | Calculates the sum of squared deviations from the jackknife mean. |
| SQRT | =SQRT(((A1-1)/A1)*C2) | Calculates the jackknife standard error. |
| COUNT | =COUNT(B2:B51) | Determines the number of jackknife replicates. |
VBA (Optional)
Automate leave-one-out resampling and calculate jackknife standard errors for multiple estimators within health economic models.
Sources
- Quenouille MH. Problems in Plane Sampling. Annals of Mathematical Statistics. 1949.
- Tukey JW. Bias and Confidence in Not Quite Large Samples. 1958.
- Efron B, Tibshirani RJ. An Introduction to the Bootstrap.
- Shao J, Tu D. The Jackknife and Bootstrap.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Bayesian Methods in Health Economics — Gianluca Baio, 1st Edition ed., 2012 (Chapman & Hall / CRC Press)
An overview of Bayesian statistical methods for the analysis of health economic data, covering economic evaluation concepts, statistical cost-effectiveness analysis, Bayesian computation and MCMC, and applied health economic evaluation.
BookView source →
Frequently Asked Questions (6)
What is a jackknife standard error?
An estimate of a parameter's statistical uncertainty calculated using the jackknife technique of omitting one observation at a time and examining resulting variability.
Source: Tukey 1958
What does a jackknife standard error quantify?
A jackknife standard error quantifies the uncertainty of a statistic using the jackknife's leave-one-out procedure, deriving the spread of the statistic from how much it shifts as each observation is omitted in turn. It offers a way to attach a standard error to a quantity without a neat formula, much as the bootstrap does, and is often quicker since it requires only as many recalculations as there are observations. It works less well for unstable statistics that jump when a single point is removed. Measuring uncertainty by systematic omission is its approach. Efron and Tibshirani (1993) describe this.
Source: Efron & Tibshirani 1993
How is a jackknife standard error calculated?
A jackknife standard error is calculated by computing the statistic on each leave-one-out subsample, formed by omitting one observation at a time, and then combining the variability of these values through the jackknife formula, which scales the spread of the leave-one-out estimates to give the standard error. So a jackknife standard error is calculated from the dispersion of the statistic across the leave-one-out recomputations, applying the jackknife scaling to convert this variability into an estimate of the standard error, which provides a resampling-based measure of uncertainty that can be applied when analytical standard error formulas are unavailable or difficult to derive.
Source: Tukey 1958
When is a jackknife standard error useful?
A jackknife standard error is useful when the standard error of a statistic is not easily available from a formula, offering a simple resampling estimate through leave-one-out recomputation. It is convenient for reasonably smooth statistics. So a jackknife standard error is useful for quantifying uncertainty where analytical formulas are complex or lacking, providing a straightforward alternative, though for statistics that are not smooth it may be unreliable, and the bootstrap standard error is often preferred as a more general method, with the jackknife standard error remaining a handy option for simpler statistics and where its assumptions hold.
Source: Tukey 1958
How does a jackknife standard error compare with a bootstrap standard error?
A jackknife standard error is computed from leave-one-out subsamples, a fixed number equal to the sample size, while a bootstrap standard error is computed from many resamples drawn with replacement. The bootstrap is more general and often more accurate, especially for complex or non-smooth statistics, while the jackknife is simpler and deterministic. So a jackknife and a bootstrap standard error both estimate uncertainty by resampling but differ in method, with the jackknife using systematic omission and the bootstrap random resampling, and the bootstrap standard error is generally preferred for its broader applicability, while the jackknife standard error suits smoother statistics and simpler settings.
Source: Efron 1979
What are the limitations of a jackknife standard error?
The limitations of a jackknife standard error include that it can be inaccurate for statistics that are not smooth functions of the data, such as the median, where leaving out single observations fails to capture the variability well; and that it may be less reliable than the bootstrap for complex statistics. So a jackknife standard error is used with awareness that it suits smooth statistics and can mislead for non-smooth ones, which is why it is applied where appropriate and the bootstrap is preferred for statistics where the jackknife performs poorly, with the jackknife standard error remaining a simple and useful estimate of uncertainty in suitable cases.
Source: Tukey 1958
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British health economist
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Verification date: 16 Dec 2025
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