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Estimate the uncertainty left after auxiliary balance, while keeping residual approximation, landing effects and replicate design visible.
Open the chapter contents · Soil Science root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Define residual uncertainty
Route 2 · Work the approximation
Route 3 · Replicate and transfer
Route 4 · Respect the limits
Route 5 · Verify and continue
CHAPTER 1 OF 12 · DEFINE RESIDUAL UNCERTAINTY
1. Why a balanced estimate still needs uncertainty
A cube sample can reproduce known auxiliary totals exactly and produce a precise-looking soil total. That algebra does not make sampling variance zero. The unmeasured soil property still departs from its relationship with clay, terrain, land use or remote sensing, and different balanced samples can contain different residuals.
Variance estimation is difficult because joint inclusion probabilities under balanced designs can be unavailable or numerically awkward. Residual-substitution approximations and carefully constructed replicate weights provide practical routes, but each rests on assumptions that must be visible.
Start with the Soil Science root and the cube-selection guide. Then connect the uncertainty calculation to adaptive two-phase sampling. Balance improves a design; it does not remove the need for an interval.
CHAPTER 2 OF 12 · DEFINE RESIDUAL UNCERTAINTY
2. Residuals carry the uncertainty left after balance
Suppose the soil variable y is approximated by a working model m(x). Define residual eₖ=yₖ−m(xₖ). If the sample is balanced on x, the modelled component of the total is reproduced by design, so uncertainty is driven mainly by the residual component.
Deville and Tillé’s variance approximation formalizes this idea for unequal-probability balanced sampling. In practice, coefficients are estimated from the sample and population quantities are replaced by design-weighted estimates. The approximation is not an ordinary regression standard error.
The working model is a precision device, not the basis for declaring the design unbiased. Poor residual behaviour warns that the chosen balance variables do not explain the target well.
CHAPTER 3 OF 12 · WORK THE APPROXIMATION
3. Worked residual ledger for four balanced sites
Reuse the balanced sample with clay values 10, 25, 35 and 40, each selected with π=0.5. Let measured soil carbon be 1.2, 2.1, 2.8 and 3.0. Use the transparent working prediction m(x)=0.5+0.05x.
Predictions are 1.00, 1.75, 2.25 and 2.50, leaving residuals 0.20, 0.35, 0.55 and 0.50. Their mean is 0.40 and sample variance is 0.025.
| Clay x | Observed y | Working prediction 0.5+0.05x | Residual e | Residual minus mean |
|---|---|---|---|---|
| 10 | 1.20 | 1.00 | 0.20 | −0.20 |
| 25 | 2.10 | 1.75 | 0.35 | −0.05 |
| 35 | 2.80 | 2.25 | 0.55 | 0.15 |
| 40 | 3.00 | 2.50 | 0.50 | 0.10 |
| Total | 9.10 | 7.50 | 1.60 | 0.00 |
CHAPTER 4 OF 12 · WORK THE APPROXIMATION
4. A teaching proxy shows what residual substitution does
For orientation only, treat the four equal-probability sites as a simple random sample of four from eight. With sampling fraction f=0.5 and residual variance sₑ²=0.025, the residual-based standard error of the population mean is
SE(mean)=√[(1−f)sₑ²/n]=0.0559.
Multiplying by N=8 gives a total standard error of about 0.447. This is a teaching proxy, not the cube-design variance formula. It shows how a smaller residual spread translates into a narrower uncertainty estimate after the balanced component has been removed.
For real cube samples, use a variance approximation derived for the actual inclusion probabilities, balance variables and landing behaviour.
CHAPTER 5 OF 12 · WORK THE APPROXIMATION
5. Joint inclusion probabilities may not be available
The exact Horvitz–Thompson variance uses first- and second-order inclusion probabilities πₖ and πₖℓ. Cube selection produces the sample through flight and landing moves rather than a simple closed-form design, so πₖℓ can be hard to obtain.
Approximations replace the unknown covariance structure with coefficients depending on π values and a residualized outcome. Their accuracy depends on the design, sampling fraction, balance dimension and how closely the sample lands on its targets.
Report the approximation chosen and its formula, not just a software function name. Compare it with repeated selection on the frozen frame when simulation is feasible.
CHAPTER 6 OF 12 · REPLICATE AND TRANSFER
6. Replicate weights must respect the original design
Replication methods create perturbed weight sets, recalculate the estimator and use the variation among replicate estimates. A useful replicate system should preserve key strata, unequal probabilities and balance constraints as closely as its method requires.
Deleting one unit and rescaling ordinary weights can destroy auxiliary balance and exaggerate variance. Conversely, forcing every replicate to satisfy too many constraints can hide real uncertainty. The replicate construction must match the linearization or martingale argument being used.
Audit each replicate: weighted auxiliary totals, weight range, number of active units and whether the estimator can be computed. Publish the replicate factor and variance formula alongside the weights.
CHAPTER 7 OF 12 · REPLICATE AND TRANSFER
7. Changed context: another balanced sample has wider residuals
Consider the second clay-balanced sample with x={15,20,30,45} and y={1.0,1.5,2.4,3.3}. Under the same working prediction, residuals are −0.25, 0, 0.40 and 0.55. Their sample variance is 0.1342.
The same teaching proxy gives SE(mean)=√[0.5×0.1342/4]=0.1295 and SE(total)=1.036, more than twice the first sample’s value.
| Balanced sample | Residual variance | Proxy SE of mean | Proxy SE of total | Clay balance achieved? |
|---|---|---|---|---|
| 10, 25, 35, 40 | 0.0250 | 0.0559 | 0.447 | Yes |
| 15, 20, 30, 45 | 0.1342 | 0.1295 | 1.036 | Yes |
| Ratio, second/first | 5.37 | 2.32 | 2.32 | — |
CHAPTER 8 OF 12 · REPLICATE AND TRANSFER
8. Landing residuals belong in the variance audit
When exact balance is impossible, the cube landing phase leaves residual differences between weighted sample auxiliary totals and known totals. Those differences can reintroduce uncertainty in the modelled component.
Store the landing residual vector and scale each component by a meaningful population total or standard deviation. A tiny raw error in one unit can be large on another scale. Examine whether samples with larger landing errors also show larger estimation errors in repeated-selection studies.
If calibration is applied after selection, report the variance before and after calibration and the resulting weight changes. Calibration can repair benchmark totals while increasing weight variability.
CHAPTER 9 OF 12 · RESPECT THE LIMITS
9. Diagnostics should test the entire uncertainty pipeline
Use repeated cube selections on the fixed frame when computationally possible. For a study variable known on a historical or synthetic population, compare empirical sampling variance with the proposed variance estimator and check interval coverage.
Stratify diagnostics by balance quality, weight extremity and residual pattern. Inspect leverage from units with small π or large residuals. One average coverage number can hide failure in rare soil classes.
Compare linearization, residual approximation and replicate estimates. Disagreement is evidence to investigate, not a reason to select the smallest standard error.
CHAPTER 10 OF 12 · RESPECT THE LIMITS
10. Limits and common misconceptions
Balance on auxiliary variables does not make the variance of every response small. Precision gains depend on the relationship between y and the balance variables. A poor working model leaves large residuals.
The residual regression is not proof that the soil mechanism is linear. It is an approximation used for variance. Nonlinearity, interactions and spatial residual dependence can require richer diagnostics.
Replicate weights are not interchangeable across designs. A generic jackknife or bootstrap can violate unequal probabilities, fixed sample size or cube balance. Small samples and high-dimensional balance can also make replicate construction unstable.
Evidence, algorithms and reproducibility
Deville and Tillé developed residual-based approximation in Variance approximation under balanced sampling. Chauvet and Tillé proposed a martingale-based improvement in Improved variance estimation for balanced samples drawn via the Cube method. The selection design itself is established in Efficient balanced sampling: the cube method.
Archive the frame, π values, balance matrix, sample indicators, flight and landing settings, response data, working model, residuals, variance coefficients and every replicate weight. Save random seeds for both selection and replication.
Reproduce the two worked residual variances independently. Then simulate many balanced selections from a small complete population and compare the average reported variance with the empirical sampling variance.
CHAPTER 11 OF 12 · VERIFY AND CONTINUE
11. Frequently asked questions
Can I use an ordinary regression standard error?
Not as a substitute for design variance. The sample was selected by a probability design with balance and possibly unequal weights.
Does exact balance imply zero variance?
Only for a response that is exactly a linear combination of the balanced variables under the relevant estimator. Real soil responses retain residual variation.
Are replicate weights always better than linearization?
No. Their quality depends on how they reproduce the design and balance constraints. Compare methods and validate coverage.
What if landing residuals are large?
Investigate constraint incompatibility, inclusion probabilities and balance dimension. Report the achieved rather than intended balance.
A practical reporting protocol
Publish the point estimate, variance method, standard error, confidence interval, degrees-of-freedom rule, balance residuals and weight diagnostics. Provide replicate weights or sufficient code to reconstruct them when permitted.
CHAPTER 12 OF 12 · VERIFY AND CONTINUE
12. Final transfer check and connected reading
Keep n=4 and f=0.5, but suppose the residual variance is 0.08. Calculate the teaching proxy standard errors for the mean and the total of eight units. Then explain why this proxy still cannot be labelled the exact cube-design variance.
Continue through the Soil Science root, cube-selection guide, spatially balanced sampling guide, adaptive two-phase sampling and informative-nonresponse guide. Together they connect selection, weighting, uncertainty and transfer limits.
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