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A soil survey rarely serves one outcome. This guide shows how to balance a sample for several maps without pretending that one auxiliary set is best for carbon, texture and contamination at once.
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 · Frame the multi-outcome problem
Route 2 · Find the precision frontier
Route 3 · Choose a feasible compromise
Route 4 · Estimate and challenge
Route 5 · Check and transfer
CHAPTER 1 OF 12 · FRAME THE MULTI-OUTCOME PROBLEM
1. One sample, several soil outcomes
A soil campaign may need maps of organic carbon, clay fraction, pH, nutrient status and contaminant risk from the same field visits. These outcomes respond to different terrain, geology, land use and management variables. A design tuned to one response can be inefficient for another.
Multi-outcome balanced sampling makes that conflict visible before fieldwork. It treats precision as a vector rather than hiding every objective inside one average. The aim is not a magical design that is best for all outcomes. It is a defensible compromise whose losses and gains are stated in the units decision-makers care about.
CHAPTER 2 OF 12 · FRAME THE MULTI-OUTCOME PROBLEM
2. What balance guarantees—and what it does not
A balanced sample makes Horvitz–Thompson estimates of chosen auxiliary totals equal, or very close, to the known population totals. The cube method can select such samples with prescribed inclusion probabilities and many auxiliaries. When an outcome is well predicted by those auxiliaries, balancing its predictors can reduce residual variation and improve precision.
The guarantee concerns the auxiliaries, not every soil outcome. If clay is related to parent material while carbon is related to vegetation and management, balancing only vegetation may help carbon far more than clay. The primary cube-method paper by Deville and Tillé explicitly links variance reduction to correlations between outcomes and controlled variables.
CHAPTER 3 OF 12 · FRAME THE MULTI-OUTCOME PROBLEM
3. Worked comparison: neither auxiliary set wins everywhere
Suppose a planning study predicts design effects relative to simple random sampling for three outcomes. Lower is better. Design A balances elevation and vegetation index. Design B balances geology and land use. The numbers are a transparent teaching forecast, not field results.
| Candidate design | Carbon design effect | Clay design effect | Contaminant design effect | Immediate reading |
|---|---|---|---|---|
| A: elevation + vegetation | 0.25 | 0.80 | 1.10 | Strong for carbon; weak for contamination |
| B: geology + land use | 0.65 | 0.45 | 0.70 | Strong for clay and contamination |
| C: expanded compromise | 0.40 | 0.55 | 0.78 | Moderate gains across all three |
Design A does not dominate B because it is better for carbon but worse for clay and contamination. Design C may be attractive, yet it also does not dominate A: carbon precision is forecast to be lower under A. The choice needs priorities, constraints and uncertainty, not a universal ranking.
CHAPTER 4 OF 12 · FIND THE PRECISION FRONTIER
4. Pareto precision frontiers
A design is Pareto dominated if another feasible design is no worse for every outcome and strictly better for at least one. Dominated designs can usually be removed before discussing policy weights. The remaining non-dominated set forms a precision frontier.
The frontier separates two questions. Statistical design identifies feasible trade-offs. Governance chooses where on that frontier to operate. A team can change outcome weights without rerunning every candidate if the frontier and its uncertainty have been preserved.
Plot forecast variance or design effect for pairs of outcomes, then retain the full outcome vector in the audit table. A two-dimensional plot is a view, not the whole decision when there are three or more responses.
CHAPTER 5 OF 12 · FIND THE PRECISION FRONTIER
5. Compromise auxiliaries are a model portfolio
An auxiliary set is a portfolio of predictions about what will matter. Start with variables available wall to wall at the design stage: terrain, remote-sensing indices, mapped geology, land use, legacy measurements and administrative strata. For each outcome, model residual variance after using candidate auxiliaries.
Avoid adding every available variable. Too many balance equations can make the landing phase difficult, increase sensitivity to mapping error and spend field flexibility on weak predictors. Group redundant variables, use scientifically interpretable summaries and test whether a smaller portfolio stays near the frontier.
The outcome-specific balance-diagnostics guide shows how auxiliary correlation and forecast error should be evaluated separately for each response.
CHAPTER 6 OF 12 · CHOOSE A FEASIBLE COMPROMISE
6. Inclusion probabilities also carry priorities
Balancing variables and inclusion probabilities play different roles. Balance aligns sample-weighted auxiliary totals with known totals. Unequal inclusion probabilities allocate more sampling effort to units expected to be informative or variable. Optimal probabilities for one outcome can be suboptimal for another.
Chauvet, Bonnéry and Deville’s work on optimal inclusion probabilities for balanced sampling notes that target probabilities depend on the variable of interest and may need estimated inputs. For several outcomes, calculate candidate allocations under each outcome model, then test compromise rules such as a capped weighted maximum. Keep minimum inclusion probabilities high enough to prevent extreme weights and unsafe undercoverage.
CHAPTER 7 OF 12 · CHOOSE A FEASIBLE COMPROMISE
7. Changed priorities can move the chosen design
Use the worked design effects and score a design with a weighted sum. With equal weights, the mean design effects for A, B and C are about 0.72, 0.60 and 0.58, so C narrowly leads. If carbon receives weight 0.70 and the other outcomes 0.15 each, A scores about 0.46, B about 0.67 and C about 0.48, so A leads. If contamination becomes the priority, B can lead.
| Priority rule | Weight on carbon | Weight on clay | Weight on contamination | Lowest teaching score |
|---|---|---|---|---|
| Equal outcomes | 0.333 | 0.333 | 0.333 | C, about 0.58 |
| Carbon programme | 0.700 | 0.150 | 0.150 | A, about 0.46 |
| Contaminant response | 0.150 | 0.150 | 0.700 | B, about 0.69 |
| No agreed weights | Not fixed | Not fixed | Not fixed | Report the Pareto set |
The transfer check is to vary plausible weights and outcome models. If the selection flips easily, the honest conclusion is a near tie or a need for a robust compromise, not false precision.
CHAPTER 8 OF 12 · CHOOSE A FEASIBLE COMPROMISE
8. Feasibility, cube flight and landing
The cube method has a flight phase that moves inclusion indicators while preserving balance equations and a landing phase that reaches a sample when exact balance is no longer possible. The final design should report residual imbalance for every auxiliary, not just declare “balanced.”
Field constraints can remove mathematically attractive samples. Access, travel time, minimum spatial spread, laboratory capacity and safety must enter before selection. Run many randomised selections to see whether certain sites or regions carry fragile responsibility for balance. A design that is optimal only under one landing path is operationally brittle.
The model-assisted cube-selection guide develops the mechanics and calibration checks, while spatially balanced sampling adds geographic spread and legacy-site constraints.
CHAPTER 9 OF 12 · ESTIMATE AND CHALLENGE
9. Variance forecasts and post-sample estimation
Planning design effects are forecasts based on auxiliary–outcome relationships, often from legacy data. After sampling, use a design-consistent estimator and an appropriate balanced-sampling variance approximation. Do not report the planning forecast as the achieved standard error.
Residual-based approximations, local differences or replication methods can be useful, but each has assumptions. The balanced-sampling variance guide compares these checks. Preserve selection probabilities, balance variables, random seeds and any replacement rules so inference matches the design actually implemented.
CHAPTER 10 OF 12 · ESTIMATE AND CHALLENGE
10. Evidence checks, limits and failure modes
The empirical relationship between auxiliaries and outcomes may drift across seasons or land-use changes. Map error can make apparent balance differ from balance on the true covariate. Missing laboratory results can destroy the planned compromise. Stress-test the frontier under model refits, noisy auxiliaries, nonresponse and cost changes.
Common mistakes are to average design effects before checking domination, choose weights after seeing the preferred design, balance on unavailable field measurements, ignore extreme weights, or treat exact auxiliary balance as proof of unbiased outcome maps. The official Statistics Canada treatment of balanced sampling and anticipated variance is a useful design-based reference for how model information enters planning without replacing probability sampling.
CHAPTER 11 OF 12 · CHECK AND TRANSFER
11. Questions readers usually ask
Must every outcome have its own auxiliary?
No. One auxiliary can predict several outcomes, but its outcome-specific residual relationship should be checked rather than assumed.
Is a weighted sum the only compromise rule?
No. Teams can minimise the worst relative loss, impose precision floors, use lexicographic priorities or keep the full Pareto set. The rule should be chosen before final design selection.
Can the design include legacy sites?
Yes, if their inclusion probabilities and constraints are handled explicitly. Forced sites can narrow the feasible balance region and should be included in simulations.
What if there are too many auxiliaries for the sample size?
Reduce or combine them, relax balance tolerances, increase the sample, or reserve some objectives for model-based estimation. Exact balance on a crowded weak set is not a free gain.
CHAPTER 12 OF 12 · CHECK AND TRANSFER
12. A transfer routine and connected reading
List the outcomes, define an interpretable precision measure for each, generate feasible auxiliary and inclusion-probability portfolios, and forecast the outcome-specific variance vector. Remove dominated designs, show the frontier, choose a declared compromise rule, and rerun it under model and cost perturbations. After fieldwork, estimate variance from the realised design rather than from the plan.
Continue with balanced-sampling design effects, two-phase soil sampling and preferential-sampling correction. The general lesson transfers to any multi-purpose survey: the best defensible design often lies on a frontier, not at a single outcome’s optimum.
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