Capture–Recapture Population Estimation

Capture–Recapture Population Estimation

In a closed equally catchable population, capture-recapture estimation involves taking a first sample, marking the individuals, and then taking a second sample later. The overlap between the two samples can be used to estimate the total unknown population size.

Worked Calculation

Consider the case where we mark M = 3 on the first capture and take a second sample of size n = 4. We observe k = 1 recaptured marked units.

1. The Petersen estimate is given by M × n / k, which evaluates to 3 × 4 / 1 = 12.
2. The Chapman statistic is given by (M + 1)(n + 1) / (k + 1) - 1, which evaluates to (4 × 5) / 2 - 1 = 10 - 1 = 9.
3. The minimum possible population (observed union) is M + n - k = 3 + 4 - 1 = 6.
4. For a candidate population of N = 8, the hypergeometric likelihood is P(X = 1) = (3 choose 1) × (5 choose 3) / (8 choose 4) = 3 × 10 / 70 = 3/7.

Experiment

Follow these steps to observe the behavior of the estimators and the likelihood function:

1. Select a value for Marked first.
2. Select a value for Second sample.
3. Select a value for Recaptured. Notice that if you select 0 recaptures, the Petersen estimate is undefined (or infinite), but the Chapman statistic remains finite.
4. Select a Candidate population to evaluate its hypergeometric likelihood. The plot shows the full likelihood distribution for populations starting from the observed union size.
5. Capture–Recapture Population Estimation module

Limitations

These estimators assume a closed population (no births, deaths, immigration, or emigration between samples), equal catchability of all individuals, and that marks are not lost or overlooked. A finite Chapman statistic does not prove a finite upper bound on the population size. This model does not infer confidence intervals.

References

The concepts are derived from the following sources:

• Penn State STAT506 capture–recapture notes: https://online.stat.psu.edu/stat506/Lesson12