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eduKateSG · Why Mathematics? · Jackknife Repeated Replication: PSU Deletion and Survey Weights
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 survey observations arrive in clusters selected through a stratified design? Jackknife repeated replication deletes one primary sampling unit at a time, raises the weights of its partners and recomputes the complete estimate. The resulting movement measures uncertainty at the sampling-unit level rather than pretending that every person is independent.
Did You Know? In a stratum with two sampled primary units, deleting one and doubling the other preserves the stratum's total analysis weight in that replicate. The simple arithmetic carries the design idea.
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
People in the same selected school, neighbourhood or clinic may share conditions and were often selected together. Variance estimation must respect the primary sampling unit, or PSU, that carried the first-stage randomness.
Statistics Canada explains that a delete-one survey jackknife gives the omitted PSU zero weight and modifies the weights of remaining PSUs in that stratum using a jackknife coefficient. Read the official formulation.
JK1 is commonly used for an unstratified design, while JKn refers to stratified deletion schemes. The scale and replicate factors depend on how many PSUs are in each stratum and on finite-population treatment.
Suppose one stratum has PSU A with weighted total 120 and PSU B with weighted total 80, so the full stratum total is 200.
| PSU A | PSU B | Full |
|---|---|---|
| 120 | 80 | 200 |
Set A's replicate weights to zero and multiply B's weights by 2. The replicate stratum total becomes 2×80=160.
Set B's replicate weights to zero and multiply A's weights by 2. The replicate stratum total becomes 2×120=240.
The two deviations from 200 are −40 and 40. With a two-PSU delete-one stratum factor of 1/2, the contribution is (1/2)(1600+1600)=1600, giving standard error 40 for this isolated toy stratum.
| Delete A | Delete B | Variance contribution |
|---|---|---|
| 160 | 240 | 1600 |
The retained PSU temporarily represents the omitted PSU's share of the stratum. This is a repeated-sampling approximation, not a claim that the two PSUs are substantively identical.
If a stratum contains m sampled PSUs, deleting one commonly increases the other m−1 PSU weights by a factor related to m/(m−1), with a matching variance coefficient. Use the survey's actual replicate specification.
When sampling fractions are material, finite-population corrections may alter variance. The R survey documentation distinguishes population size, sampling fraction and correction inputs for JK1 and JKn. Read the documentation.
A stratum with one noncertainty PSU cannot support within-stratum deletion by itself. Combining strata, certainty treatment or another documented adjustment changes assumptions and must be disclosed.
A full production replicate may rerun weighting adjustments after deletion. Merely zeroing one final-weight PSU and inflating its partners can miss nonlinear calibration effects.
A PSU may contain no sampled cases in a small domain, yet it still belongs to the full design. Domain analysis should retain the full design information rather than dropping all out-of-domain rows before variance estimation.
Fit the same weighted estimating equation for every replicate. A coefficient's replicate spread then incorporates the sample design to the extent represented by the weights and fitting procedure.
Some systems centre squared deviations at the full estimate; others use the mean of replicate estimates under an alternative mean-squared-error setting. The choice can matter when replicates are biased or unstable.
Leaving out one respondent answers a case-sensitivity question. JRR leaves out one sampled PSU because that is the independent first-stage unit represented in the design.
JRR has one replicate per deleted PSU or structured PSU deletion. Delete-a-group combines units into fewer groups, reducing the number of replicate columns at the cost of a larger deletion.
JRR deletes one PSU within a stratum per replicate. BRR coordinates half-sample selections across many paired strata using a balanced sign matrix.
Count PSUs per stratum, inspect lonely strata, confirm zero and inflated weight patterns, compare replicate totals and flag coefficients that fail after one PSU is removed.
Reproduce the 120-and-80 calculation, verify a survey manual example, compare direct Taylor linearisation with JRR for a smooth estimator and test sparse domains.
JRR cannot create information absent from a few-PSU design. It remains sensitive to miscoded strata, PSU identifiers, omitted finite-population corrections and unstable estimators.
List stratum, PSU, full weight and replicate weights. Summarise zeroed and inflated patterns before calculating any estimate.
Group rows by stratum and PSU, verify one allowed deletion per replicate, unit-test totals 160 and 240 and the 1600 variance contribution, then fit the target estimator.
Complex-survey analysts, epidemiologists, education researchers and government statisticians use JRR. The skill is to connect design metadata with correct computation.
Build three toy strata, compare leave-one-row and leave-one-PSU estimates, then add a lonely stratum and document a defensible handling rule.
PSU does not mean one respondent, doubling a partner does not double the sample size, and a software default does not replace the survey's technical notes.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The delete-a-group article explains grouped deletions. The stratified bootstrap article compares within-stratum resampling.
Students should draw the strata and PSUs before writing formulas. Parents can ask what the independent sampling unit is and whether the public-use file supplies replicate weights.
- Calculate 160 and 240.
- Derive variance 1600.
- Explain partner inflation.
- Identify a lonely PSU.
- Compare JRR with row deletion.
### What is a PSU? A primary sampling unit selected at an early stage.
### Why delete a PSU? It represents the relevant sampling variation.
### Can calibration be ignored? Not when the official method repeats it.
### Does JRR solve few-PSU problems? No.
Report stratum and PSU variables, replicate type, coefficients, finite-population handling, centring, domain method, lonely-PSU rule, failed fits and degrees of freedom.
Zeroing A and doubling B gives 160; zeroing B and doubling A gives 240. Their symmetric movement around 200 makes the sampling-unit contrast visible.
Confirm PSU-level deletion, stratum-specific factors, official replicate weights when available, calibration treatment, lonely-PSU rules and stable fitting.
Continue through the Mathematics Learning Hub. Rework one replicate-weight example by hand, then explain which sampling-design feature the replicate preserves.
Extended practice laboratory
Practice case 1: Map strata
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Map PSUs
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Calculate full total
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 PSU A
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Double PSU B
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 PSU B
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Double PSU A
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 jackknife-repeated-replication mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 jackknife-repeated-replication mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 factor
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Take square root
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Add third PSU
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Add lonely stratum
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Use domain
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 model
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Add FPC
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Compare row deletion
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Compare DAGJK
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 BRR
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Audit calibration
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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: Audit centring
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 failures
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 metadata
Setup: Start from the article’s jackknife-repeated-replication mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 jackknife-repeated-replication mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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 jackknife-repeated-replication mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the sampling unit, apply the documented replicate factor, state one diagnostic, record the result 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.