Find your next learning step
A probability sample can begin well and still lose representativeness when selected soil sites cannot be observed. The key question is whether access failure is unrelated to the missing soil value after using the information recorded for every planned site.
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 · Define the missingness
Route 2 · Work the response ledger
Route 3 · Correct and stress the model
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE MISSINGNESS
1. A selected site and an observed site are different things
A spatially balanced probability sample gives each eligible soil unit a known chance of selection. Fieldwork then creates a second process: some selected sites are measured, while others are inaccessible, unsafe, denied, disturbed or lost.
If response is unrelated to the missing soil value after conditioning on recorded design and access variables, adjustment may be defensible. If high contamination, steep terrain, wetness or land use affects both access and the missing outcome, nonresponse is informative.
This guide extends the Soil Science root, spatially balanced sampling and preferential-sampling correction.
CHAPTER 2 OF 12 · DEFINE THE MISSINGNESS
2. Define selection, response and the target population
Let Iᵢ indicate selection under the original design and Rᵢ indicate successful observation after selection. The final inclusion indicator is IᵢRᵢ. A design weight 1/P(Iᵢ=1) does not correct a response process that depends on site conditions.
Record auxiliary variables for every planned site: slope, road distance, ownership, land cover, predicted wetness, scheduled visit, reason for failure and replacement status. These variables can support a response model πᵢ = P(Rᵢ=1|auxiliary data).
Do not redefine the target population after seeing failures unless the scientific estimand truly changes. Calling inaccessible land “out of scope” can hide a coverage error rather than solve it.
CHAPTER 3 OF 12 · WORK THE RESPONSE LEDGER
3. Worked response ledger
Consider six equal-area planned cells with invented soil values and response probabilities that decline as values rise. The complete-population mean is 38/6 = 6.333.
| Cell | Soil value | Response probability π |
|---|---|---|
| 1 | 2 | 0.90 |
| 2 | 4 | 0.80 |
| 3 | 5 | 0.80 |
| 4 | 7 | 0.60 |
| 5 | 9 | 0.40 |
| 6 | 11 | 0.30 |
The expected respondent mean, approximated by Σπᵢyᵢ/Σπᵢ, is 20.1/3.8 ≈ 5.289. High cells contribute less often, so the responding set is expected to understate the equal-area mean.
CHAPTER 4 OF 12 · WORK THE RESPONSE LEDGER
4. Response weighting repairs only measured causes
If πᵢ can be estimated from auxiliary variables known for respondents and nonrespondents, inverse-response weighting multiplies the design weight by 1/πᵢ. Low-response sites receive more weight because they represent similar planned sites that often fail.
Weights can become unstable when π is small. Inspect their range, effective sample size and influence. Trimming may reduce variance but introduces bias; calibration to known area totals can improve balance but cannot guarantee removal of outcome-dependent nonresponse.
Estimate response probabilities using the fieldwork sequence available at decision time. A variable measured only after successful sampling cannot explain failures without extra modelling.
CHAPTER 5 OF 12 · WORK THE RESPONSE LEDGER
5. A realised failure pattern needs sensitivity analysis
Suppose cells 1–4 respond and cells 5–6 do not. The respondent mean is (2+4+5+7)/4 = 4.5. Let the missing-site mean equal the respondent mean plus δ.
| Sensitivity shift δ | Assumed missing mean | Completed six-cell mean |
|---|---|---|
| 0 | 4.5 | 4.500 |
| 2 | 6.5 | 5.167 |
| 4 | 8.5 | 5.833 |
| 6 | 10.5 | 6.500 |
The table does not estimate δ. It makes the untestable part visible and shows when a decision—such as exceeding a management threshold—would change.
CHAPTER 6 OF 12 · CORRECT AND STRESS THE MODEL
6. Selection models connect response to the latent soil value
A selection model writes P(R=1|Y,auxiliary data) and a model for Y jointly. For example, logit P(Rᵢ=1) = a + bᵀxᵢ + γYᵢ. The coefficient γ represents residual outcome dependence after observed covariates.
Y is missing when R=0, so γ is weakly or non-identified from observed data alone. Fit a range of scientifically plausible γ values, or use informative prior knowledge, and report how estimates and maps respond.
This is analogous to preferential sampling but occurs after planned selection. One model explains where sites were chosen; the other explains which chosen sites yielded observations. Both processes can operate together.
CHAPTER 7 OF 12 · CORRECT AND STRESS THE MODEL
7. Changed context: safety exclusions become permanent
Suppose a temporary flood prevents access this season. A revisit may recover the selected site, preserving the original estimand. If a new safety policy permanently excludes steep terrain, the frame and operational population may diverge.
Report both: the intended target and the reachable domain. Predictions for permanently inaccessible terrain are model-based extrapolations unless other evidence enters.
Use the area-of-applicability guide to show whether inaccessible areas lie outside the observed covariate domain.
CHAPTER 8 OF 12 · CORRECT AND STRESS THE MODEL
8. Replacements do not erase nonresponse
Replacing an inaccessible site with a nearby accessible one changes inclusion probabilities and can move the sample toward roads, gentle slopes or cooperative landholders. Record every attempted site and the replacement rule.
A pre-specified probability replacement drawn from the remaining frame is different from walking to the nearest convenient point. The latter may be useful operationally but does not preserve the original design without adjustment.
Analyse the intended sample, realised sample and replacement sample separately before pooling. Map their covariate distributions and distances to access routes.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
“Missing at random” does not mean random in everyday language. It is a conditional statement: after observed information, missingness no longer depends on missing values under the specified model.
High response rate does not guarantee low bias. A small, systematically excluded subgroup can matter greatly. Low response rate does not prove severe bias if strong auxiliary information supports adjustment.
Weights and sensitivity models cannot replace safe fieldwork, permissions, quality control or a clear target population. Decisions about contaminated land require applicable professional and regulatory procedures.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and reproducibility route
Rubin gives the foundational conditions under which a missing-data process may be ignored in Inference and missing data. Diggle, Menezes and Su show how a spatial observation-location process can require joint modelling in Geostatistical inference under preferential sampling.
Archive the original sample, inclusion probabilities, all attempted coordinates, response status, dated failure reason, replacements, auxiliary variables, response model, weight adjustments and every sensitivity parameter. Preserve maps before and after adjustment.
Report the estimate under several defensible assumptions rather than presenting one nonresponse model as observed fact.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Is an inaccessible site automatically missing not at random?
No. The mechanism may be explained by observed access variables. Outcome dependence after conditioning is the concern.
Can a nearby replacement stand in for the failed point?
Only under a declared design or model. Convenience proximity does not guarantee equivalent soil conditions.
Should extreme response weights be trimmed?
Possibly, but trimming trades variance for bias. Show both the rule and sensitivity.
What if failure reasons were never recorded?
Use available auxiliary data, reconstruct cautiously from field records and widen sensitivity analysis rather than inventing certainty.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
Recalculate the complete mean 6.333 and expected respondent mean 5.289. Then use the realised four-site pattern to find the δ at which a management threshold of six is crossed. Explain which quantities are observed and which are assumptions.
Continue through the Soil Science root, spatially balanced sampling, preferential sampling and area-of-applicability masks. Together they keep planned inclusion, realised response and prediction transfer in separate ledgers.
Top · Contents · Previous · How X Works