GOLD STANDARD · SOIL SCIENCE · HOW X WORKS
Find the decision hidden inside the model
Distributional balance is useful only if the selection mechanism remains a probability design. Equal inclusion probabilities, a fixed sample size and random block selection connect the optimised auxiliary ordering to design-unbiased estimation.
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Chapter contents and reading routes
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Orient the probability design
Follow the block mechanism
Build design-unbiased estimates
Transfer to fieldwork
Use evidence and next routes
CHAPTER 1 OF 12 · ORIENT THE DECISION
1. A representative-looking subset is not yet a sample design
An optimiser can find a set of soil sites whose auxiliary distribution resembles the frame. If the same best set is always chosen, some units have inclusion probability one and most have zero. That supports a purposive design, not automatically design-unbiased survey estimation.
A probability design specifies every possible sample and its probability. Distributionally balanced designs preserve that structure by optimising an ordering, then applying a random selection rule rather than directly choosing one favoured subset.
CHAPTER 2 OF 12 · ORIENT THE DECISION
2. First-order inclusion probabilities anchor the weights
The first-order inclusion probability is the chance that a population unit enters the sample under repeated use of the design. With population size N and fixed sample size n, equal inclusion probabilities are n divided by N.
The Horvitz–Thompson total weights each observed value by the inverse of its inclusion probability. Under the stated design and complete response, this estimator is design-unbiased even though the ordering was optimised using auxiliary information.
CHAPTER 3 OF 12 · BUILD THE METHOD
3. How a circular fixed-size block works
Arrange all N units on a circle, choose a random start and take n consecutive units. Every start is equally likely. Because each unit appears in exactly n of the N possible starting blocks, its inclusion probability is n/N.
Optimisation changes the circular order so every possible block better resembles the auxiliary population. It must not change the random-start rule after outcomes are inspected.
CHAPTER 4 OF 12 · BUILD THE METHOD
4. A worked eight-site frame
Place eight soil sites on an optimised circle and select three consecutive sites from one of eight equally likely starts. Each site belongs to three possible blocks, so its inclusion probability is 3/8.
If the observed carbon values in one selected block are 20, 24 and 28, the Horvitz–Thompson total is their sum divided by 3/8: 72 times 8/3, or 192. The corresponding population mean is 192/8 = 24. The arithmetic is design-based; the realised estimate can still differ from the true total.
| Quantity | Value | Meaning |
|---|---|---|
| Population size N | 8 | Units on the circle |
| Fixed sample size n | 3 | Consecutive units selected |
| Inclusion probability | 3/8 | Each unit appears in three blocks |
| Illustrative HT total | 192 | Weighted observed sum |
CHAPTER 5 OF 12 · BUILD THE METHOD
5. Fixed size protects more than logistics
Every possible sample contains exactly n units, so field budget and laboratory load are predictable. Fixed size also constrains dependence: selecting one unit changes the chance of selecting nearby positions in the circular order.
The order should place auxiliary-similar units so each contiguous block covers the distribution rather than clustering one regime. Verify this across all starts, not just the realised sample.
CHAPTER 6 OF 12 · TEST THE RESULT
6. Second-order probabilities govern variance
For variance estimation, the chance that two units appear together matters. In a circular block design, nearby positions can have positive joint inclusion while sufficiently separated pairs may never co-occur. Zero second-order probabilities complicate standard unbiased variance formulas.
Do not import a simple-random-sampling variance. Use theory supported for the design, repeated randomisation for planning or a clearly stated conservative or model-assisted approach. The previous variance article explains those alternatives in depth.
CHAPTER 7 OF 12 · TEST THE RESULT
7. Keep optimisation outside the outcome
The circular order can use elevation, spectra, soil maps and access variables known before sampling. Optimising after seeing carbon or contamination outcomes makes the design outcome-dependent and can invalidate the advertised inference.
Version the frame, transformations, distance metric, optimisation seed and stopping rule. Then freeze the order before the random start is drawn.
| Stage | Allowed information | Locked output |
|---|---|---|
| Frame preparation | Known auxiliaries and eligibility | Target population |
| Order optimisation | Scaled auxiliary geometry | Circular sequence |
| Random selection | Random start only | Fixed-size sample |
| Estimation | Observed outcomes and design weights | Total, mean and uncertainty |
CHAPTER 8 OF 12 · TRANSFER AND GOVERN
8. Changed-context transfer: soil to forest inventory
A forest inventory can optimise the circle using canopy height, terrain and disturbance age, then draw a fixed number of plots. The stable mechanism is equal inclusion, random start and inverse-probability estimation.
Transfer requires a new frame and auxiliary geometry. Missing forest units or access substitutions can alter inclusion probabilities even when the mathematical ordering is correct.
CHAPTER 9 OF 12 · TRANSFER AND GOVERN
9. Handle ineligible and nonresponding sites explicitly
If a selected site is outside the target population, correct the frame definition. If it belongs but cannot be observed, follow a pre-specified reserve or nonresponse adjustment. Convenient replacement by the nearest accessible site changes the design.
Record attempts, reasons and any adjusted probabilities. A fixed nominal sample size does not guarantee a fixed number of usable observations.
CHAPTER 10 OF 12 · TRANSFER AND GOVERN
10. Limits and common misconceptions
Equal inclusion probabilities do not make observations independent, and design-unbiasedness does not guarantee a small variance. Precision gain depends on how smoothly outcomes vary over the auxiliaries used to build the order.
Distributional balance also cannot correct an outdated frame or unmeasured hotspot process. Rare hazards may need strata, unequal probabilities or a targeted component.
CHAPTER 11 OF 12 · CHECK AND CONTINUE
11. Questions soil teams often ask
Can the best-looking block be chosen after randomisation? No; that replaces the probability draw with selection. Are weights all equal? Under the first equal-probability implementation, yes, but field nonresponse or future unequal-probability extensions can change them.
Does every pair need positive joint probability? Not for unbiased point estimation, but zeros affect available variance estimators. Report that limitation instead of hiding it.
CHAPTER 12 OF 12 · CHECK AND CONTINUE
12. Evidence, specialist routes and the next check
Distributionally Balanced Sampling Designs presents the equal-inclusion, optimised circular order with a random contiguous block and explains why direct deterministic optimisation can distort inclusion probabilities and bias Horvitz–Thompson estimators. Its PubMed record summarises the design and variance-reduction results.
Continue with randomisation replicates and optimisation uncertainty, energy distance and nonlinear coverage, and balanced-sampling variance checks. The next field check is to enumerate all random starts and confirm each unit appears in exactly n blocks.
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