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eduKateSG · Why Mathematics? · Delete-One Jackknife: Variance, Pseudovalues and Influence
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–34
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when we want uncertainty from a statistic without specifying a full probability distribution? The delete-one jackknife recomputes the statistic after removing each observation in turn. The spread of these leave-one-out values estimates variance and reveals how strongly each case affects the answer.
Did You Know? For the sample mean, jackknife pseudovalues are exactly the original observations. That is why the jackknife variance of a mean reproduces the familiar sample-variance-divided-by-n formula.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–34 · Study and verify
A statistic can look stable until one influential observation is removed. Recomputing n versions exposes local sensitivity and converts it into a variance estimate under smoothness and independence conditions.
If theta_(i) is the estimate without case i and bar-theta is their average, the jackknife variance is (n−1)/n times the sum of squared deviations from bar-theta. Read the SIAM chapter summary.
For values (2,4,6,8), the full mean is 5. The leave-one-out means are 6, 16/3≈5.333, 14/3≈4.667 and 4.
| Data | Full mean | Leave-one means |
|---|---|---|
| 2,4,6,8 | 5 | 6, 5.333, 4.667, 4 |
Their average is 5. Squared deviations sum to 1+1/9+1/9+1=20/9. Multiplying by (4−1)/4 gives 5/3≈1.667.
| Squared sum | Scale | Variance |
|---|---|---|
| 20/9 | 3/4 | 5/3 |
The sample variance of (2,4,6,8) is 20/3. Dividing by n=4 gives 5/3, matching the jackknife variance for this linear statistic.
Deleting one case changes the effective sample size and shrinks variation among leave-one-out estimates. The factor (n−1)/n rescales that spread to first-order sampling variance.
Define P_i=n theta-hat−(n−1)theta_(i). Pseudovalues place each deletion effect on the scale of an observation-level contribution.
With n=4 and theta-hat=5, the four pseudovalues are 20−3(6)=2, 20−3(16/3)=4, 20−3(14/3)=6 and 20−3(4)=8.
If deleting a case lowers the estimate, that case pulls the full estimate upward. The sign depends on whether one reports deletion change, pseudovalue or influence, so conventions must be named.
Means, smooth regression coefficients and many estimating-equation solutions respond approximately linearly to small empirical-weight changes, supporting jackknife variance estimation.
Medians, extreme quantiles, maxima and selection rules can change discontinuously. Delete-one jackknife variance may be inconsistent or unstable for these targets.
The standard delete-one construction assumes observations are independent units. Clustered data require deleting clusters; dependent time series need blocks or another dependence-aware method.
Leave out each independent row, refit the model and track a coefficient or prediction. High-leverage points can dominate the jackknife spread.
For a ratio of means, deleting one case changes numerator and denominator together. The method preserves their pairing, but denominators near zero produce instability.
Deleting one survey unit without recalibrating design weights can target the wrong variance. Replicate-weight jackknife schemes are designed around the actual sample design.
A normal interval can use the jackknife standard error, but skewness, boundaries and small n can make symmetric endpoints poor. The interval rule needs separate validation.
Both leave cases out, but cross-validation estimates predictive performance while jackknife deletion estimates bias, variance or influence of a statistic.
Jackknife uses n systematic deletions and is deterministic once data are fixed. Bootstrap uses many random resamples and can approximate more of the sampling distribution.
Ordinary jackknife recomputes the statistic on delete-one datasets. Jackknife-after-bootstrap reuses bootstrap replicates that happened to omit each case.
Reproduce a linear statistic with known variance, inspect deletion plots, check smoothness, use cluster or design deletions where needed and compare with bootstrap simulation.
Delete-one jackknife can fail for nonsmooth statistics, dependent rows, tiny samples, unstable fits, boundaries or data whose selection process is not represented by deletion.
Place one omitted index per row with leave-one estimate, deviation, square and pseudovalue. Sum squares and verify the scaling factor.
Loop over n deletions, unit-test the four means, variance 5/3 and pseudovalues (2,4,6,8). Record failed fits and sign convention.
Jackknife diagnostics appear in statistics, econometrics, medicine and engineering. Careers require influence reasoning, numerical care and design awareness.
Compare mean, median and maximum on samples with one outlier. Plot deletion values and compare jackknife with bootstrap variance.
Leave-one-out does not automatically mean cross-validation, influential does not mean erroneous, and n rows need not equal n independent units.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The jackknife-after-bootstrap guide explains reuse of bootstrap omissions. The cluster-bootstrap guide explains group units.
Students should calculate all four leave-one means and pseudovalues. Parents can ask what one independent unit means in the study.
- Calculate the full mean.
- Calculate four deletions.
- Recompute 5/3.
- Derive pseudovalues.
- Diagnose nonsmoothness.
### How many recomputations? n.
### Is it random? Not normally.
### Does it work for every statistic? No.
### What should be deleted under clustering? Independent clusters.
Report statistic, independent unit, n, deletion rule, leave-one values, scaling, standard error, pseudovalue convention, failures, influence plot and validity checks.
For the mean, leave-one changes are algebraically tied to centred observations, making the jackknife variance equal to the usual variance of the sample mean.
Confirm independent deletion units, exact scaling, stable recomputation, suitable smoothness, transparent sign convention and no automatic deletion of influential cases.
Continue through the Mathematics Learning Hub. Recalculate one deletion, pseudovalue or infinitesimal-weight example by hand before trusting software output.
Extended practice laboratory
Practice case 1: List data
Setup: Start from the article’s delete-one-jackknife mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Calculate full statistic
Setup: Start from the article’s delete-one-jackknife mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Delete case 1
Setup: Start from the article’s delete-one-jackknife mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Delete case 2
Setup: Start from the article’s delete-one-jackknife mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Delete case 3
Setup: Start from the article’s delete-one-jackknife mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Delete case 4
Setup: Start from the article’s delete-one-jackknife mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Average deletions
Setup: Start from the article’s delete-one-jackknife mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Calculate deviations
Setup: Start from the article’s delete-one-jackknife mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Square deviations
Setup: Start from the article’s delete-one-jackknife mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Apply scaling
Setup: Start from the article’s delete-one-jackknife mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Calculate SE
Setup: Start from the article’s delete-one-jackknife mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Compute pseudovalues
Setup: Start from the article’s delete-one-jackknife mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Plot influence
Setup: Start from the article’s delete-one-jackknife mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Add outlier
Setup: Start from the article’s delete-one-jackknife mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Use median
Setup: Start from the article’s delete-one-jackknife mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Use maximum
Setup: Start from the article’s delete-one-jackknife mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Use regression
Setup: Start from the article’s delete-one-jackknife mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Delete clusters
Setup: Start from the article’s delete-one-jackknife mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Compare bootstrap
Setup: Start from the article’s delete-one-jackknife mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Compare cross-validation
Setup: Start from the article’s delete-one-jackknife mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Increase n
Setup: Start from the article’s delete-one-jackknife mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Audit failed fits
Setup: Start from the article’s delete-one-jackknife mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit sign
Setup: Start from the article’s delete-one-jackknife mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s delete-one-jackknife mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s delete-one-jackknife mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the independent unit, verify the deletion or derivative identity, state one diagnostic, record units and compare one defensible alternative with clear reasoning. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.