Concept Architecture
Concept
Theoretically, Jackknife is a resampling method used to estimate statistical bias, variance and standard errors by systematically omitting one or more observations from a dataset and recalculating the statistic of interest. It is founded on non-parametric estimation theory and influence function methodology. The concept exists to assess the sampling variability and bias of statistical estimators without relying heavily on distributional assumptions.
Mathematically, the jackknife constructs a series of leave-one-out estimates by repeatedly removing each observation from the sample. These pseudo-values are then combined to estimate the variance, bias and standard error of the estimator. The method is particularly useful for complex estimators whose analytical variance is difficult to derive.
In practice, the jackknife is implemented by sequentially deleting one observation (or one cluster) at a time, recalculating the estimator after each deletion and combining the resulting estimates according to established jackknife formulae. In health economics, jackknife methods are used to estimate uncertainty in cost-effectiveness statistics, regression coefficients, survival estimates and other complex estimators when analytical solutions are unavailable or difficult to obtain.
Purpose
Used to estimate bias, variance and standard errors through systematic leave-one-out resampling, supporting robust statistical inference for complex estimators in health economic analyses.
Mathematical Formulae
Primary Formula
Jackknife estimate:
??(?i)
where ??(?i) denotes the estimate obtained after deleting observation i.
Jackknife estimator:
??J = (1/n) ? ????� ??(?i)
Supporting Formulae
Jackknife variance:
VarJ(??) = ((n ? 1) / n) ? ????� (??(?i) ? ??J)�
Jackknife standard error:
SEJ = �VarJ(??)
Jackknife bias estimate:
BiasJ = (n ? 1)(??J ? ??)
Pseudo-value:
PV? = n?? ? (n ? 1)??(?i)
Related Mathematical Methods
- Bootstrap
- Cross-Validation
- Influence Function
- Leave-One-Out Estimation
- Non-Parametric Variance Estimation
- Bias Estimation
- Resampling Methods
Example
A health economist estimates the mean annual healthcare cost using data from 100 patients.
The complete sample estimate is:
?? = �4,800
Each patient is removed individually, producing 100 leave-one-out estimates.
The average jackknife estimate is:
??J = �4,790
The estimated bias is:
BiasJ = (100 ? 1)(4790 ? 4800)
BiasJ = ?�990
Using the leave-one-out estimates, the jackknife variance and standard error are then calculated to quantify uncertainty in the estimated mean healthcare cost.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| AVERAGE | =AVERAGE(B2:B101) | Calculates the average leave-one-out estimate. |
| SUMSQ | =SUMSQ(B2:B101-$B$102) | Calculates the sum of squared deviations for jackknife variance estimation. |
| SQRT | =SQRT(((A1-1)/A1)*C2) | Calculates the jackknife standard error. |
| ROW | =ROW()-1 | Identifies the observation omitted in each leave-one-out iteration. |
VBA (Optional)
Automate leave-one-out resampling, jackknife variance estimation, pseudo-value calculation and uncertainty reporting for health economic analyses.
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
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 the jackknife?
A resampling technique estimating a statistic's variability by systematically recalculating it after removing one observation at a time.
Source: Tukey 1958
How does the jackknife estimate a statistic's variability?
The jackknife estimates how much a statistic would vary by recomputing it many times, each time leaving out one observation from the dataset. Comparing these leave-one-out values shows how sensitive the statistic is to individual data points, from which its variability and bias can be gauged. It is simpler and more systematic than the bootstrap, which resamples at random, but it copes less well with statistics that change abruptly, such as the median. Recomputing after dropping each point in turn is its method. Efron and Tibshirani (1993) describe this technique.
Source: Efron & Tibshirani 1993
How does the jackknife work?
The jackknife works by creating, for a dataset of a given size, that many subsamples, each omitting one observation, computing the statistic of interest on each subsample, and then using the spread of these leave-one-out values to estimate the standard error, and their average to estimate and correct bias. So the jackknife works by systematically deleting one observation at a time and recomputing the statistic, with the variability among the resulting values yielding an estimate of the statistic's standard error and its pattern yielding an estimate of bias, providing a resampling-based approach to uncertainty that requires only repeated recalculation of the statistic.
Source: Tukey 1958
When is the jackknife used?
The jackknife is used to estimate the standard error or bias of a statistic when an analytical formula is unavailable or complex, offering a simple resampling alternative. It is applied to a range of statistics, though it works best for reasonably smooth ones. So the jackknife is used to quantify uncertainty and bias for statistics lacking easy formulas, providing a straightforward leave-one-out method, though for statistics that are not smooth, such as the median, it can perform poorly, and the bootstrap, which resamples with replacement, is often preferred as a more general resampling approach, with the jackknife remaining useful for simpler cases and bias estimation.
Source: Tukey 1958
How does the jackknife differ from the bootstrap?
The jackknife estimates variability by systematically leaving out one observation at a time and recomputing the statistic, producing a fixed number of subsamples equal to the sample size, while the bootstrap resamples with replacement to create many resamples of the original size. The bootstrap is more general and often more accurate, while the jackknife is simpler and deterministic. So the two differ in their resampling scheme, with the jackknife using leave-one-out subsamples and the bootstrap using resampling with replacement, and although both estimate standard errors, the bootstrap handles a wider range of statistics better, while the jackknife is particularly convenient for bias estimation and simpler cases.
Source: Efron 1979
What are the limitations of the jackknife?
The limitations of the jackknife include that it can perform poorly for statistics that are not smooth functions of the data, such as the median or other quantiles, where leaving out single observations does not adequately capture variability; and that it may be less accurate than the bootstrap for complex statistics. So the jackknife is used with awareness that it suits reasonably smooth statistics and can be unreliable for non-smooth ones, which is why the bootstrap is often preferred as a more general resampling method, though the jackknife remains a simple and useful tool for estimating standard errors and bias in appropriate situations.
Source: Tukey 1958
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 16 Dec 2025
Content version: 1.0.0
Canonical Identity
- Persistent URI
- https://healtheconomics.wiki/concept/jackknife
- Term code
- HE-ES-SA-088
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