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Translate balanced-sampling variance into an estimator-specific design effect and effective sample size without hiding outcome dependence.
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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 the comparison
Route 2 · Work the efficiency
Route 3 · Audit and transfer
Route 4 · Respect the limits
Route 5 · Verify and continue
CHAPTER 1 OF 12 · DEFINE THE COMPARISON
1. Why a balanced sample still needs an efficiency statement
Balanced sampling deliberately matches known auxiliary totals, but readers still need to know what that balance bought. A standard error gives uncertainty in the original units. A design effect compares that uncertainty with a declared simple-random-sampling reference.
For an estimator θ-hat, define DEFF=Var_design(θ-hat)/Var_SRS(θ-hat) for the same nominal sample size and target. Then n_eff=n/DEFF is a communication device: the simple-random sample size with roughly comparable variance.
Neither number is a permanent label on the sample. It depends on the soil variable, estimator, reference design, response pattern and variance method.
CHAPTER 2 OF 12 · DEFINE THE COMPARISON
2. Design effect is an estimator-specific ratio
A DEFF below one means the actual design is more efficient than the chosen simple-random reference for that estimator. A value above one means less efficient. The standard-error ratio is sqrt(DEFF).
Balance can reduce variance when auxiliary variables predict the soil outcome. Unequal weights, landing residuals, nonresponse and weak auxiliary relationships can offset that gain.
Always name the numerator and denominator. Comparing a spatially balanced estimator with a with-replacement SRS formula answers a different question from comparing it with finite-population SRS without replacement.
CHAPTER 3 OF 12 · WORK THE EFFICIENCY
3. Worked comparison: the same hundred sites can act like 160
Suppose a soil programme samples n=100 sites. For mean clay content, the estimated design variance is 0.025. The declared SRS variance for the same target and sample size is 0.040.
DEFF=0.025/0.040=0.625. The standard-error ratio is sqrt(0.625)=0.791, and n_eff=100/0.625=160. The effective size exceeds the nominal size because the balance supplied useful information.
| Quantity | Value | Interpretation |
|---|---|---|
| Nominal n | 100 | Field observations collected |
| Design variance | 0.025 | Estimated under the balanced design |
| SRS reference variance | 0.040 | Declared comparison at n=100 |
| DEFF | 0.625 | Variance is 62.5% of the SRS reference |
| Effective n | 160 | SRS-equivalent precision, not extra observations |
This does not create sixty measurements. It translates precision under one defined comparison.
CHAPTER 4 OF 12 · WORK THE EFFICIENCY
4. Precision gain should also be shown on the standard-error scale
Variance ratios can sound more dramatic than the uncertainty a reader sees. Here the standard error falls from sqrt(0.040)=0.200 to sqrt(0.025)=0.158.
The relative reduction is about 20.9%, not 37.5%, because standard errors use the square root. Report both scales when planning or communicating gains.
A confidence interval also depends on degrees of freedom, finite-population correction and the distributional approximation. Effective n is not a substitute for a complete interval calculation.
CHAPTER 5 OF 12 · WORK THE EFFICIENCY
5. Weight-only effective size answers a narrower question
A common unequal-weight approximation is n_eff,w=(Σw_i)^2/Σw_i^2, with DEFF_w=n/n_eff,w. It describes dispersion in weights under restrictive assumptions.
Suppose fifty sites have weight one and fifty have weight two. Then Σw=150 and Σw²=250. The weight-only effective size is 150²/250=90, and DEFF_w=100/90=1.111.
This calculation ignores the outcome, balance variables and spatial arrangement. It can be useful as a weight diagnostic, but it is not the realised design effect for every soil estimate.
CHAPTER 6 OF 12 · AUDIT AND TRANSFER
6. Balance can offset unequal-weight loss
The worked clay estimate had a realised DEFF of 0.625 even though the illustrative weights alone gave 1.111. There is no contradiction. The first ratio includes the relationship between clay and the balancing variables; the second sees only weight variation.
Do not multiply a weight-only design effect by a balance benefit unless a derivation justifies the decomposition. Components can interact.
Report weight concentration separately: minimum and maximum weights, coefficient of variation, influential units and the effect of trimming or calibration. Then show estimator-specific variance evidence.
CHAPTER 7 OF 12 · AUDIT AND TRANSFER
7. Changed context: a weakly predicted soil variable loses the gain
Use the same 100 selected sites for a contaminant whose relationship with the auxiliary variables is weak. Suppose the estimated design variance is 0.050 while the SRS reference remains 0.040.
| Soil outcome | Design variance | SRS variance | DEFF | Effective n |
|---|---|---|---|---|
| Clay content | 0.025 | 0.040 | 0.625 | 160 |
| Contaminant concentration | 0.050 | 0.040 | 1.250 | 80 |
The design has not changed; the outcome–auxiliary relationship has. This transfer check prevents a clay-based precision claim from being applied to every soil property.
CHAPTER 8 OF 12 · AUDIT AND TRANSFER
8. Estimated gains should be compared with planning gains
Before fieldwork, planners may predict DEFF from pilot correlations, spatial covariance and weight variation. After fieldwork, compute realised estimator-specific design effects with the chosen variance estimator.
Compare the two. A shortfall can come from weaker correlations, nonresponse, landing imbalance, measurement error or a changed population. A larger gain may reveal better-than-expected auxiliary information.
Keep planning values labelled as assumptions. Do not silently replace them with realised values when evaluating the design.
CHAPTER 9 OF 12 · RESPECT THE LIMITS
9. Replicate estimates make the comparison auditable
Design-respecting replicate weights can generate a variance estimate for the balanced design. Apply the same estimator to every replicate, including calibration and nonresponse adjustments.
Construct the SRS reference using the same target definition and finite-population convention. If the denominator comes from a different outcome transformation or population, the ratio loses meaning.
Plot replicate estimates and identify whether a few replicates drive the DEFF. A single ratio can conceal unstable variance estimation.
CHAPTER 10 OF 12 · RESPECT THE LIMITS
10. Limits and common misconceptions
Effective sample size is not a count of independent sites. It is a variance translation for one estimator and one reference. It may exceed n under stratification or balance and may fall far below n under clustering or unequal weights.
Kish-style weight formulas do not capture outcome correlation with selection probabilities or calibration variables. Balanced-sampling variance can also be difficult to estimate because exact joint inclusion probabilities may be unavailable.
Do not compare DEFF values unless their SRS references, finite-population corrections and variance methods align. A smaller DEFF is not automatically worth higher field cost or poorer coverage of other outcomes.
Evidence and reproducibility
Deville and Tillé establish the selection design in Efficient balanced sampling: the cube method. Their residual variance framework is developed in Variance approximation under balanced sampling. Kish discusses inference and design effects in Inference from complex samples. Spencer examines unequal-weight approximations when outcomes correlate with selection probabilities in An approximate design effect for unequal weighting.
Archive the target estimator, SRS reference formula, finite-population correction, variance method, replicate weights, auxiliary variables, achieved balance, nonresponse adjustments and every DEFF/ESS calculation.
CHAPTER 11 OF 12 · VERIFY AND CONTINUE
11. Frequently asked questions
Can effective sample size exceed nominal n?
Yes. Useful stratification or balance can make the estimator more precise than SRS of the same nominal size.
Is DEFF one number for the survey?
No. It is specific to an estimator and outcome, though summaries can be reported across a planned outcome set.
Can I use n_eff in a t distribution automatically?
No. Degrees of freedom come from the variance method and design, not from n/DEFF alone.
Should I report the weight-only formula?
Report it as a diagnostic when relevant, clearly separated from realised estimator-specific DEFF.
What belongs in a minimum report?
Nominal n, estimator, design and SRS variances, DEFF, standard-error ratio, n_eff, reference convention and uncertainty method.
CHAPTER 12 OF 12 · VERIFY AND CONTINUE
12. Final transfer check and connected reading
A balanced design with n=120 has estimated variance 0.018, while the declared SRS variance is 0.027. Calculate DEFF, the standard-error ratio and n_eff. Then name two reasons the result may not transfer to soil pH.
Continue through the Soil Science root, cube-selection guide, balanced-sampling variance guide, spatially balanced sampling guide and adaptive two-phase guide. Together they connect selection, variance, efficiency and field redesign.
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