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Test whether balancing variables actually predict each soil outcome, and separate anticipated precision from realised design evidence.
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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 outcome value
Route 2 · Forecast efficiency
Route 3 · Diagnose and transfer
Route 4 · Respect the limits
Route 5 · Verify and continue
CHAPTER 1 OF 12 · DEFINE OUTCOME VALUE
1. Why a balanced sample is not equally efficient for every soil outcome
A soil sample can be exactly balanced on elevation, land use and mapped texture while delivering excellent precision for one outcome and little gain for another. Balance controls auxiliary totals; it does not guarantee that every laboratory response is predictable from those auxiliaries.
The missing bridge is outcome-specific residual variation. If soil organic carbon is strongly explained by the balancing variables, its residuals are small and balance can reduce variance. If a contaminant hotspot is nearly unrelated to them, the same design may offer little precision gain.
This distinction prevents a useful sample design from being praised with one universal efficiency number.
CHAPTER 2 OF 12 · DEFINE OUTCOME VALUE
2. Balance equations protect auxiliary totals, not unknown responses
For inclusion probabilities π_k and auxiliary vectors x_k, exact balance makes the Horvitz–Thompson estimate of the auxiliary total equal the known population total. The design reproduces what is known before fieldwork.
For an outcome y, write y_k=x_k^Tβ+e_k. Under a useful working model, the variance left after balancing is driven mainly by residuals e_k rather than the full outcome variation.
The result is conditional on the relationship between y and x. Change y, measurement depth or laboratory method, and β and the residuals can change.
CHAPTER 3 OF 12 · FORECAST EFFICIENCY
3. Anticipated variance is a planning forecast
Anticipated variance averages design variance over a working outcome model. It answers a pre-fieldwork question: if the assumed relationship between auxiliaries and outcome is approximately right, how precise should this design be?
It is not the realised design variance and not a guarantee for the observed sample. Use it to compare candidate auxiliary sets, inclusion probabilities and spatial-spread constraints before collection.
After collection, use design-consistent variance estimation or justified replicates. Compare the realised evidence with the anticipated forecast rather than substituting one for the other.
CHAPTER 4 OF 12 · FORECAST EFFICIENCY
4. Worked ledger: one design, three outcome-specific gains
Imagine a 100-site balanced design. Under simple random sampling, the planned variance of each standardised mean is 0.040. Residual forecasts under the balanced design differ by outcome.
| Outcome | Forecast balanced variance | Design effect versus 0.040 | Effective sample size 100/DEFF |
|---|---|---|---|
| Soil organic carbon | 0.010 | 0.25 | 400 |
| Clay fraction | 0.030 | 0.75 | 133 |
| Trace contaminant | 0.050 | 1.25 | 80 |
Effective sample size is an interpretation of estimator variance, not a claim that 100 field sites became 400 physical observations. Report the comparator and outcome beside it.
CHAPTER 5 OF 12 · FORECAST EFFICIENCY
5. Auxiliary correlation is useful only when it transfers
An auxiliary map can correlate strongly with an outcome in legacy data and weakly in the current population. Land-use change, new management, depth mismatch or laboratory drift can break the relationship.
Audit correlation by region, depth, season and important subpopulation. A strong pooled relationship can hide a reversal or a low-slope group.
Prefer auxiliaries measured consistently over the target frame. A spectacular but stale predictor can create a confident precision forecast that does not survive fieldwork.
CHAPTER 6 OF 12 · DIAGNOSE AND TRANSFER
6. Balance diagnostics should be reported in outcome space
First verify the actual auxiliary imbalance, including any landing-phase residual after the cube method. Then examine outcome-model residuals using pilot, legacy or cross-fitted information.
Useful diagnostics include weighted residual variance, leverage by inclusion probability, residual maps, leave-one-region-out prediction and sensitivity to dropping each auxiliary. Avoid selecting auxiliaries with the final outcome values from the same sample and then calling the result a design-stage forecast.
A short diagnostic table should connect each auxiliary to the outcome, the expected mechanism and the evidence window.
CHAPTER 7 OF 12 · DIAGNOSE AND TRANSFER
7. Changed context: a good national predictor can fail in one soil region
Suppose elevation and vegetation index explain carbon well nationally, producing forecast variance 0.010. In a reclaimed lowland zone, management history dominates and the local residual forecast rises to 0.045.
| Deployment domain | SRS comparator variance | Forecast balanced variance | Design effect |
|---|---|---|---|
| National frame | 0.040 | 0.010 | 0.25 |
| Reclaimed lowland | 0.040 | 0.045 | 1.125 |
The response is not to discard balance. Add relevant auxiliaries, allocate a local booster sample, or report domain-specific uncertainty.
CHAPTER 8 OF 12 · DIAGNOSE AND TRANSFER
8. Too many balancing variables can make the design brittle
Every exact balance constraint consumes design freedom. Highly collinear or weakly relevant auxiliaries can make the flight phase difficult, enlarge landing residuals or force undesirable selections.
Use subject knowledge and outcome forecasts to rank constraints. Preserve variables required for coverage or policy domains, then test whether extra predictors materially reduce anticipated residual variance.
Dimension reduction can help, but principal components are not automatically interpretable or stable. Keep the mapping from transformed constraints back to field coverage visible.
CHAPTER 9 OF 12 · RESPECT THE LIMITS
9. Replicate checks compare forecast with realised precision
When the design supports it, generate replicate weights or repeated balanced samples that respect the actual selection mechanism. Recompute the target estimator and any calibration step in every replicate.
Compare replicate variance with the anticipated variance and with the simple-random comparator. Large disagreement can signal poor outcome transfer, landing effects, influential weights or a variance approximation outside its useful range.
Save seeds, inclusion probabilities, balancing tolerances and the exact replicate algorithm. Reproducibility includes the selection mechanism, not only the final site coordinates.
CHAPTER 10 OF 12 · RESPECT THE LIMITS
10. Limits, misconceptions and evidence checks
Balance does not remove selection bias caused by an incomplete frame, inaccessible sites or informative nonresponse. An auxiliary variable unrelated to the outcome may protect coverage but need not improve precision. Conversely, a strong outcome predictor can improve precision without repairing bias from missing domains.
Deville and Tillé show why balanced-sampling variance depends on regression residuals in Variance approximation under balanced sampling. The official Statistics Canada discussion of anticipated variance and balanced designs explains why model expectations matter at design time.
Pre-register the comparator, estimator, outcome family and deployment domains. Report auxiliary imbalance, residual forecasts, realised variance and sensitivity to inaccessible sites separately.
CHAPTER 11 OF 12 · VERIFY AND CONTINUE
11. Frequently asked questions
Can one design effect summarise the whole soil survey?
No. It is specific to an estimator, outcome, domain and variance comparator.
Should every available covariate be balanced exactly?
No. Prioritise coverage requirements and variables with defensible outcome relevance; test stability and feasibility.
Is high correlation enough?
No. It must transfer to the target population, depth, season and measurement process.
Can effective sample size exceed the physical sample size?
As a variance interpretation, yes. It does not create more independent specimens.
CHAPTER 12 OF 12 · VERIFY AND CONTINUE
12. Final transfer check and connected reading
A 120-site design has forecast variance 0.018 for pH and 0.054 for a metal concentration; the simple-random comparator is 0.036 for both. Compute each design effect and effective sample size. Explain why the same balance constraints produce opposite conclusions.
Continue through the Soil Science root, cube-selection guide, balanced-sampling variance guide, design-effect guide and spatially balanced sampling guide. Together they connect auxiliary balance, residual uncertainty, efficiency interpretation and spatial coverage.
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