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Choose a probability sample that balances trusted auxiliary totals, while keeping inclusion probabilities, landing residuals and final weights visible.
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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 balance
Route 2 · Work the design
Route 3 · Estimate and transfer
Route 4 · Respect the limits
Route 5 · Verify and continue
CHAPTER 1 OF 12 · DEFINE BALANCE
1. Why balanced sampling is more than spreading points on a map
A soil survey may have strong auxiliary information before a single new core is collected: land use, terrain, remote-sensing indices, legacy texture estimates and mapped area. A spatially even sample can cover the map well yet still miss the population totals of those variables. Balanced sampling targets those totals directly.
The cube method chooses a probability sample so that weighted sample totals of selected auxiliary variables equal, or nearly equal, their known population totals. If those auxiliaries explain the soil property of interest, the resulting estimator can be more precise without pretending that the auxiliaries are the response itself.
Use it beside the Soil Science root, spatially balanced sampling guide and adaptive two-phase sampling guide. The cube balances auxiliary totals; it does not automatically balance geographic distance.
CHAPTER 2 OF 12 · DEFINE BALANCE
2. The balance equation keeps the design honest
Let unit k have inclusion probability πₖ and auxiliary vector xₖ. For selected sample s, exact balance requires
Σₖ∈s xₖ/πₖ = Σₖ∈U xₖ.
The right side is the known population total. Including a constant auxiliary controls sample size. Other components may represent total mapped area, clay-class indicators, elevation bands or remote-sensing summaries. Each πₖ must be positive for a unit that the design may need to represent.
The Horvitz–Thompson estimator of a soil total is t̂ᵧ=Σₖ∈s yₖ/πₖ. Its design validity comes from the probability sample, not from a regression model being exactly true. Balance improves precision when y is well related to x.
CHAPTER 3 OF 12 · WORK THE DESIGN
3. Worked population: four sites balance clay exactly
Consider eight equal-area candidate sites with mapped clay values 10, 15, 20, 25, 30, 35, 40 and 45. Their total is 220. Give every site π=0.5 and select four, so the constant is balanced when the sample size is four. Exact clay balance requires the selected clay sum to be 110 because weighting by 1/π=2 must recover 220.
One balanced sample is {10, 25, 35, 40}. Suppose measured soil-carbon stocks at those sites are 1.2, 2.1, 2.8 and 3.0 units. The estimated population total is 2×9.1=18.2.
| Selected site | Clay auxiliary | Inclusion probability | Design weight | Soil carbon y | Weighted y |
|---|---|---|---|---|---|
| A | 10 | 0.5 | 2 | 1.2 | 2.4 |
| D | 25 | 0.5 | 2 | 2.1 | 4.2 |
| F | 35 | 0.5 | 2 | 2.8 | 5.6 |
| G | 40 | 0.5 | 2 | 3.0 | 6.0 |
| Total | 110 | — | — | 9.1 | 18.2 |
CHAPTER 4 OF 12 · WORK THE DESIGN
4. Flight and landing turn probabilities into a sample
The cube method begins with the vector of inclusion probabilities. During the flight phase it moves that vector within the space allowed by the balance constraints while keeping every component between zero and one. Each move pushes more components to zero or one without changing the required auxiliary totals.
If a small number of fractional components remain, the landing phase rounds them to a final zero-one sample while controlling the balance error. Exact balance may be impossible because of discrete units, tight constraints or incompatible inclusion probabilities.
Record the achieved residual for every balance variable. Calling a sample balanced without showing those residuals hides the most important numerical outcome of the algorithm.
CHAPTER 5 OF 12 · WORK THE DESIGN
5. Inclusion probabilities still define representation
Balancing does not choose πₖ for you. Equal probabilities suit equal-area units when no group needs deliberate oversampling. Unequal probabilities may reflect area, cost, risk, a rare soil class or a second-phase allocation. Those choices determine the design weights dₖ=1/πₖ.
Very small probabilities create very large weights. A cube sample can match auxiliary totals and still yield an unstable soil estimate if one selected unit represents too much of the population. Inspect the minimum π, maximum design weight, weight coefficient of variation and influence of each selected unit.
Never set πₖ=0 merely because a location is inconvenient. That changes the target population unless the exclusion is declared in advance and the estimand is rewritten.
CHAPTER 6 OF 12 · ESTIMATE AND TRANSFER
6. Model assistance explains the precision gain
Suppose yₖ≈bᵀxₖ+eₖ. Exact balance removes the sample error in the bᵀx part of the total, leaving uncertainty mainly from residuals eₖ. The design remains the basis for inference, while the working model explains why the chosen auxiliaries can reduce variance.
Useful auxiliaries should be known for the whole frame, measured consistently and plausibly related to the target property. A post-treatment laboratory variable known only for sampled units cannot balance the initial selection. A legacy map with systematic age bias may satisfy the algebra while doing little for current soil carbon.
Limit the balance dimension. Too many weak or collinear auxiliaries can make landing difficult and produce fragile samples.
CHAPTER 7 OF 12 · ESTIMATE AND TRANSFER
7. Changed context: a second balanced sample gives another estimate
Another exact-clay sample is {15, 20, 30, 45}; its clay sum is also 110. Suppose its soil-carbon measurements are 1.0, 1.5, 2.4 and 3.3. The weighted total is 2×8.2=16.4 rather than 18.2.
| Sample | Selected clay values | Clay sum | Weighted clay | Carbon sum | HT carbon total |
|---|---|---|---|---|---|
| First | 10, 25, 35, 40 | 110 | 220 | 9.1 | 18.2 |
| Second | 15, 20, 30, 45 | 110 | 220 | 8.2 | 16.4 |
| Unbalanced illustration | 10, 15, 20, 25 | 70 | 140 | not used | not used |
The two valid balanced samples differ because carbon is not a deterministic function of clay. Replication, design-based variance approximation or repeated-selection simulation is still needed. Balance is a precision device, not a guarantee of identical estimates.
CHAPTER 8 OF 12 · ESTIMATE AND TRANSFER
8. Calibration weights are a separate adjustment
After data collection, calibration can adjust starting design weights wₖ near dₖ so that Σₖ∈s wₖxₖ equals known totals. When the cube sample already achieves exact balance and there is no nonresponse, the design weights may already satisfy the calibration equations.
Nonresponse, frame corrections or additional trusted totals can require recalibration. Inspect the ratios gₖ=wₖ/dₖ, negative weights, extreme weights and the distance measure used. Trimming weights breaks exact calibration unless the calibration is repeated under explicit bounds.
Do not describe calibrated weights as new inclusion probabilities. The πₖ describe the selection design; the wₖ are estimation weights after adjustment. Keeping those roles separate makes audit trails possible.
Spatial balance and auxiliary balance can be combined
Two samples can satisfy the same cube equations while having very different geographic patterns. One may cluster along an accessible road. If residual soil variation is spatially correlated, that clustering can reduce precision and leave parts of the domain poorly diagnosed.
Add spatially meaningful auxiliaries, use a spatially balanced design within calibration constraints, or select among feasible balanced samples using a declared spatial criterion. Do not smuggle coordinates into dozens of arbitrary indicators without checking rank and landing stability.
Map selected and nonselected units, nearest-neighbour distances, strata and access exclusions. Algebraic balance and geographic coverage are complementary checks.
CHAPTER 9 OF 12 · RESPECT THE LIMITS
9. Limits and common misconceptions
Exact auxiliary balance does not remove bias from frame undercoverage, informative nonresponse or measurement error. It does not make an inaccessible unit represented if that unit had zero chance of selection. It also does not validate a wrong population total.
The method cannot guarantee balance on the unknown soil response. It reduces variance only to the extent that selected auxiliaries predict that response. Balancing a large set of fashionable remote-sensing bands can be worse than balancing a smaller, stable set.
Standard simple-random-sampling variance formulas are not automatically valid for a cube design. The algorithm’s unequal probabilities, balance constraints and landing approximation must be reflected in variance assessment.
CHAPTER 10 OF 12 · RESPECT THE LIMITS
10. Evidence, software and reproducibility
Deville and Tillé introduced the method in Efficient balanced sampling: the cube method. Deville and Särndal developed the linked estimation framework in Calibration Estimators in Survey Sampling.
Archive the frame version, target population, auxiliary definitions and totals, π vector, random seed, flight and landing settings, balance residuals and final sample indicators. Save the initial and calibrated weights separately. Re-run the selection to confirm that the same seed and frame reproduce the sample.
For the worked example, verify two equations independently: selected count divided by 0.5 equals eight, and selected clay divided by 0.5 equals 220. Those simple checks catch many coding and scaling errors.
CHAPTER 11 OF 12 · VERIFY AND CONTINUE
11. Frequently asked questions
Is the cube method the same as quota sampling?
No. It is a probability-sampling algorithm with declared inclusion probabilities and balance constraints. Quota sampling commonly lacks known selection probabilities.
Must balance be exact?
Not always. Discreteness can force landing residuals. Report the residuals and judge whether they are material for the estimator.
Can soil class indicators be included?
Yes, if class totals are known on the frame. Each additional constraint consumes flexibility and should have a clear purpose.
Does calibration repair nonresponse automatically?
No. It helps only when response differences are captured by the calibration variables and positivity remains plausible. Sensitivity checks are still needed.
A practical soil-survey protocol
Define the target population and estimand. Freeze a frame with one row per selectable unit. Choose π values from area, cost and precision goals, then select a small set of reliable auxiliaries whose population totals are known.
Run the cube algorithm, save the random seed and inspect landing residuals, weights and the map. Predefine substitution and revisit rules; uncontrolled replacement destroys the design. After measurement, calculate the Horvitz–Thompson estimate and a design-appropriate variance estimate.
If calibration is used, publish the distance function, bounds, benchmark totals and g-weight distribution. Compare estimates before and after calibration and explain the difference rather than presenting one unexplained final number.
CHAPTER 12 OF 12 · VERIFY AND CONTINUE
12. Final transfer check and connected reading
For the eight-site frame, find a third four-site sample whose clay values sum to 110. Assign four plausible carbon values, calculate the design-weighted total, and explain why exact clay balance does not force the result to equal 18.2 or 16.4.
Continue through the Soil Science root, spatially balanced sampling guide, adaptive two-phase sampling guide, informative-nonresponse guide and area-of-applicability guide. Together they connect representative selection, estimation weights and honest limits on where a soil result transfers.
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