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eduKateSG · Why Mathematics? · Fay’s BRR: Partial Weight Perturbation and Stable Survey Replicates
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 traditional balanced repeated replication deletes half the primary units and makes nonlinear estimates unstable? Fay's BRR keeps every PSU in every replicate but shifts their weights toward a high and low value. The gentler perturbation improves replicate stability, while a precise correction rescales the smaller movement into a variance estimate.
Did You Know? If the Fay coefficient is 0.5, paired PSU multipliers are 0.5 and 1.5 instead of 0 and 2. Because the perturbation is only half as large, squared replicate movements are divided by (1−0.5)^2=0.25.
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
Traditional BRR can remove every observation from a rare category in one half-sample or cause a model to fail. Fay's method retains both paired PSUs and reduces the size of each weight perturbation.
With Fay coefficient epsilon between zero and one, the low multiplier is epsilon and the high multiplier is 2−epsilon. When epsilon=0, the method reduces to traditional zero-and-two BRR.
The R survey documentation states that Fay's method gives observations weight rho in one half-sample and 2−rho in the other, rather than removing half the sample. Read the documentation.
Let epsilon=0.5. The low multiplier is 0.5 and the high multiplier is 1.5. Their average is 1, preserving the paired centre across complementary sign patterns.
Suppose the full estimate is 100 and four Fay replicate estimates are 99, 101.5, 99.5 and 100. Their deviations are −1, 1.5, −0.5 and 0.
| epsilon | Low | High |
|---|---|---|
| 0.5 | 0.5 | 1.5 |
Squared deviations sum to 1+2.25+0.25+0=3.5. Divide by R(1−epsilon)^2=4×0.25=1, giving variance 3.5 and standard error approximately 1.871.
For a locally linear estimator, shrinking every contrast by 1−epsilon shrinks replicate deviations by that factor. Variance uses squared deviations, so it shrinks by (1−epsilon)^2.
| Full | Replicates | Variance | SE |
|---|---|---|---|
| 100 | 99, 101.5, 99.5, 100 | 3.5 | 1.871 |
SAS gives the Fay variance as [R(1−epsilon)^2]^−1 times the sum of squared differences between replicate and full estimates. Read the official formula.
A value closer to one creates gentler perturbations but amplifies the variance correction and can magnify numerical error. The survey producer chooses a coefficient with its estimator set and stability needs in mind.
CDC documentation for NHANES III describes Fay's method, its corrected variance formula and balanced orthogonal replications. Read the official methodology.
Because low-weight PSUs remain present, categorical denominators and model cells are less likely to vanish. Small effective weight can still make a replicate influential or nearly separated.
Quantiles can jump when observations are removed. Partial weight perturbation may improve numerical stability, but discreteness and few effective observations still limit interval accuracy.
Traditional BRR may produce complete separation in a replicate. Fay weights can keep both outcome patterns represented, though separation may persist if the full data are sparse.
Public-use files normally provide final replicate weight columns. Analysts should not multiply them again by epsilon and 2−epsilon; rho is needed for the variance scaling and software declaration.
Set epsilon=0: multipliers become zero and two, while the correction denominator becomes R. This algebraic limit should reproduce ordinary BRR.
As epsilon approaches one, replicate weights approach full weights, deviations approach zero and the correction denominator approaches zero. Finite precision and model noise then become important.
Both use the same balanced sign patterns. Traditional BRR makes a full zero-and-two switch; Fay BRR makes a partial switch and rescales the resulting deviations.
Fay BRR perturbs every paired stratum in each replicate. JRR deletes one PSU in one stratum or deletion unit at a time, using jackknife coefficients.
Confirm epsilon from technical documentation, inspect low and high multipliers, plot replicate deviations, compare estimates under rho declarations and flag near-identical replicates that may expose precision problems.
Reproduce the epsilon=0 and epsilon=0.5 identities, verify variance 3.5, compare Fay and traditional BRR on a linear estimator and test sparse nonlinear fits.
Fay's method improves numerical stability; it does not guarantee valid inference for every nonsmooth statistic, few-PSU design, rare domain or misspecified model.
Store epsilon, R, full estimate, replicate estimates, deviations and squares. Make the denominator formula visible and test epsilon=0 as a control.
Read the supplied replicate weights without re-perturbing them, unit-test multipliers 0.5 and 1.5, denominator 1, variance 3.5 and standard error 1.871.
Fay BRR supports analysts working with national health, education, household and labour surveys. It rewards mathematical care with software metadata and diagnostics.
Use one toy survey to compare epsilon values 0, 0.3, 0.5 and 0.8. Track replicate movement, corrected variance and failed nonlinear fits.
A larger epsilon is not automatically better, gentler weights do not remove design assumptions, and supplied replicate weights must not be perturbed a second time.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The BRR article develops Hadamard balance. The delete-d jackknife article explains why perturbation size matters for nonsmooth estimators.
Students should derive the squared correction from a linear scaling argument. Parents can ask whether the software knows the survey's Fay coefficient rather than guessing from the weight columns.
- Calculate the two multipliers.
- Derive denominator 1.
- Recompute variance 3.5.
- Check epsilon=0.
- Explain the near-one warning.
### Does Fay delete a PSU? No; it gives paired PSUs low and high weights.
### What happens at epsilon=0? Traditional BRR.
### Why divide by (1−epsilon)^2? The replicate contrast was shrunk before squaring.
### Does Fay guarantee stable models? No.
Report survey, epsilon, paired variance design, R, provided weight columns, scale, centring, software declaration, failed fits, sparse domains and stability comparison.
Fay's method deliberately shrinks the replicate perturbation, observes a smaller movement and divides squared movement by the square of that shrinkage.
Confirm the published coefficient, no double perturbation, correct squared correction, balanced signs, stable replicates and cautious interpretation for nonlinear targets.
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: Choose epsilon
Setup: Start from the article’s Fay-BRR 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: Calculate low multiplier
Setup: Start from the article’s Fay-BRR 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 high multiplier
Setup: Start from the article’s Fay-BRR 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: Apply sign pattern
Setup: Start from the article’s Fay-BRR 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: Calculate replicate 1
Setup: Start from the article’s Fay-BRR 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: Calculate all replicates
Setup: Start from the article’s Fay-BRR 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: Find deviations
Setup: Start from the article’s Fay-BRR 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: Square deviations
Setup: Start from the article’s Fay-BRR 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: Calculate correction
Setup: Start from the article’s Fay-BRR 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: Calculate variance
Setup: Start from the article’s Fay-BRR 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 Fay-BRR 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: Set epsilon zero
Setup: Start from the article’s Fay-BRR 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: Set epsilon 0.3
Setup: Start from the article’s Fay-BRR 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: Set epsilon 0.8
Setup: Start from the article’s Fay-BRR 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 sparse category
Setup: Start from the article’s Fay-BRR 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: Use quantile
Setup: Start from the article’s Fay-BRR 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: Use logistic model
Setup: Start from the article’s Fay-BRR 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 BRR
Setup: Start from the article’s Fay-BRR 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 JRR
Setup: Start from the article’s Fay-BRR 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: Check supplied weights
Setup: Start from the article’s Fay-BRR 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 precision
Setup: Start from the article’s Fay-BRR 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 Fay-BRR 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 Fay-BRR 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 Fay-BRR 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 Fay-BRR 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.