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A positive-count model estimates units absent because their count was zero; a second audit can reveal units missed despite being positive. This guide calibrates both mechanisms before turning a register into a population total.
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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 · Separate the missing mechanisms
Route 2 · Build the dual-frame audit
Route 3 · Calibrate the total
Route 4 · Validate and challenge
Route 5 · Check and transfer
CHAPTER 1 OF 12 · SEPARATE THE MISSING MECHANISMS
1. Two kinds of absence must be separated
A positive-count register omits units whose count is zero. It may also miss units that had a positive count but never reached the register. A zero-truncated model addresses the first absence through a fitted probability p0. It cannot, by itself, identify the second absence because those positive-but-missed units leave no row.
This distinction matters whenever a fitted population total is used for planning. If the positive-capture probability is c, a simple expansion weight is one divided by c times one minus p0. Setting c to one is a design assumption, not a consequence of fitting the count distribution.
The preceding zero-truncated GIGP population guide explains the missing-zero boundary. The covariate-selection guide shows how visible-count fit can coexist with total-size instability. Here an independent audit supplies information about capture.
CHAPTER 2 OF 12 · SEPARATE THE MISSING MECHANISMS
2. A dual frame creates observable overlap
Let frame A be the operational positive-count register and frame B be an independently assembled audit list for the same population, period and event definition. Matching the frames produces units found in both, A only and B only. Under strong assumptions, the overlap estimates the probability that eligible positive units appear in A.
The U.S. Census Bureau describes dual-system estimation as a capture–recapture method using two systems to estimate units captured by neither. Its 2020 Post-Enumeration Survey methodology also shows why independence, matching and domain-specific probabilities require explicit modelling.
For a positive-count register, the audit is not a second measurement of p0. It estimates coverage among units that satisfy the positive eligibility rule. The count model and the dual frame answer different questions and should remain visible as separate components.
CHAPTER 3 OF 12 · SEPARATE THE MISSING MECHANISMS
3. Worked example: zero absence and capture absence
Suppose frame A contains 900 positive units. An independent audit frame contains 300 positive units, of which 240 match frame A. The estimated A-capture probability among audit-eligible positive units is 240 divided by 300, or 0.80. If the count model estimates an average p0 of 0.25, the implied total is 900 divided by 0.80 times 0.75, or 1,500.
| Component | Observed or estimated value | Role in expansion |
|---|---|---|
| Positive units in frame A | 900 | Numerator |
| Audit units in frame B | 300 | Independent coverage evidence |
| Matches in both frames | 240 | Estimates A capture |
| Positive-capture probability c | 0.80 | Corrects positive omissions |
| Modelled zero probability p0 | 0.25 | Corrects zero-count omissions |
| Implied population total | 1,500 | 900 / (0.80 × 0.75) |
If complete positive capture were assumed, the same register would imply only 1,200 units. The audit changes the total because it exposes a missing mechanism that the positive counts alone cannot reveal.
CHAPTER 4 OF 12 · BUILD THE DUAL-FRAME AUDIT
4. Matching errors can dominate the audit
A unit found in both frames must be linked correctly. False matches inflate overlap and make coverage look better than it is; missed matches reduce overlap and make coverage look worse. Use stable identifiers where permitted, document deterministic and probabilistic matching rules, and clerically review uncertain links under appropriate privacy controls.
Duplicates within either frame must be resolved before overlap is counted. Align reference periods, residence or eligibility rules, and event definitions. A person recorded twice in A and once in B is not three capture opportunities unless the model explicitly treats repeated encounters.
Report a matching-sensitivity range. Recalculate coverage after assigning ambiguous pairs first as matches and then as nonmatches. If this range is wider than the statistical interval, improving linkage matters more than refining the count mixture.
CHAPTER 5 OF 12 · BUILD THE DUAL-FRAME AUDIT
5. Independence is a testable concern, not a convenience
The simplest dual-system estimator assumes that capture by A and B is independent within homogeneous domains. Positive dependence—units easy for A are also easy for B—leaves too few observed cross-frame disagreements and tends to understate the unseen population. Negative dependence can move the estimate in the other direction.
Design B to be operationally independent of A: different collection route, staff, timing logic and source records. Then post-stratify or model capture using covariates that explain heterogeneity. Independence remains an approximation, so include scenario analyses or a third source when the stakes justify it.
Böhning and van der Heijden’s covariate adjustment shows how observed heterogeneity can enter zero-truncated population estimation. It does not remove dependence created by two frames sharing the same access mechanism.
CHAPTER 6 OF 12 · CALIBRATE THE TOTAL
6. Calibrate capture by relevant covariates
Capture may vary by location, unit size, observation opportunity or subgroup. Fit a capture model using the audit frame, where the response indicates whether each audited positive unit appears in A. Use covariates defined consistently for matched and unmatched audit units.
Avoid variables recorded only after A capture, because they are unavailable for A-missed units. Pool sparse cells carefully; a cell with zero matches does not prove zero capture. Regularisation or partial pooling can stabilise estimates while preserving meaningful differences.
Carry each fitted c(x) into the total alongside p0(x). The product c(x) times one minus p0(x) is the overall probability that a target unit contributes a positive row to A under the model.
CHAPTER 7 OF 12 · CALIBRATE THE TOTAL
7. Changed capture assumptions change the total
Keep frame A at 900 positive units and p0 at 0.25. Vary the audit-calibrated capture probability. The same visible counts support materially different totals.
| Positive capture c | Zero probability p0 | Overall inclusion | Implied total |
|---|---|---|---|
| 1.00 | 0.25 | 0.750 | 1,200 |
| 0.90 | 0.25 | 0.675 | 1,333 |
| 0.80 | 0.25 | 0.600 | 1,500 |
| 0.70 | 0.25 | 0.525 | 1,714 |
Repeat the table by important domains. An average capture rate can hide a group whose weight is extreme. A robust total should not be driven by a small cell with both high p0 and low c unless external evidence supports that extrapolation.
CHAPTER 8 OF 12 · CALIBRATE THE TOTAL
8. Combine uncertainty without blending meanings
Bootstrap the count model and the audit design together. Resample the appropriate independent units, repeat record linkage when feasible, refit p0(x) and c(x), and recalculate the total. If B is a probability sample, carry its survey weights and clustering through the process.
Keep separate sensitivity layers for count-family choice, frame dependence, matching error and audit nonresponse. A narrow bootstrap interval under one model does not cover these structural uncertainties. Present a primary estimate only when the assumptions are defensible, then show how the total moves under named alternatives.
Measurement error in capture covariates can also bias totals. The zero-truncated measurement-error study demonstrates why noisy covariates should not be treated as exact simply because the model converges.
CHAPTER 9 OF 12 · VALIDATE AND CHALLENGE
9. Audit the audit
Check whether B reaches units that A systematically misses. Compare B responders and nonresponders, examine overlap by domain, and trace how many cases are unresolved at each linkage stage. If B is drawn from the same source as A, calling it independent does not make it so.
Hold out audit clusters or periods and assess capture calibration. Compare predicted and observed match rates, not only overall accuracy. Inspect whether high weights concentrate in a few units, and cap or redesign only with a documented estimand consequence.
Where a third source exists, use it to diagnose dependence rather than automatically fitting a complicated model. Sparse multi-list tables can be unstable, and structural zeros must be distinguished from unobserved combinations.
CHAPTER 10 OF 12 · VALIDATE AND CHALLENGE
10. Evidence boundaries and common mistakes
The Census sources support the logic of an independent coverage survey and dual-system estimation. The flexible ratio-regression approach and zero-truncated covariate work support model-based unseen-zero estimation. None guarantees that a locally assembled audit is independent or complete.
Common mistakes are treating p0 as the only missingness, estimating capture from A alone, counting duplicate encounters as independent units, ignoring uncertain matches, and applying one capture rate to heterogeneous groups. Another is multiplying two point estimates and reporting a standard error from only one.
The method estimates a target population defined by its eligibility rule and period. It does not discover people or units outside both the operational definition and the audit frame.
CHAPTER 11 OF 12 · CHECK AND TRANSFER
11. Questions readers usually ask
Is the audit frame required to contain zero-count units?
No for calibrating positive capture. It should independently identify eligible positive units. Estimating zero-count units remains the role of the count model or another frame with a broader population definition.
Can one small audit estimate capture for the whole population?
Only if its design and covariate support justify that transfer. Sparse or unrepresented domains require wider uncertainty, pooling or additional fieldwork.
What if A and B are dependent?
Use design changes, stratification, sensitivity scenarios or additional sources. Do not hide the assumption inside a single dual-system number.
Should extreme expansion weights be trimmed?
Only with an explicit bias–variance analysis and reported estimand change. First ask whether the cell definition, model or audit needs repair.
CHAPTER 12 OF 12 · CHECK AND TRANSFER
12. A reusable calibration routine
Define the target population and positive event, fit the truncated count model, then build an operationally independent audit frame. Resolve duplicates and matches, model positive capture by defensible covariates, combine c(x) with p0(x), and propagate design, model, linkage and dependence uncertainty.
Keep the route connected to the Probability root, the Gold Standard of Education and the How X Works hub. The transferable principle is to measure every route by which a target unit can disappear before trusting the total.
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