Lincoln-Petersen Mark-Recapture Estimate

Catch some animals, mark them, let them mix back in, then catch a second sample and see what fraction carries a mark. If 15 of the 120 you caught the second time were marked, then marked animals are one in eight of the population, and since you marked 100 of them, the population is about 800. Frederick Lincoln formalised it for duck banding in 1930 and C. G. J. Petersen had used it for plaice in the 1890s, but the logic is older than either and it is the same trick used to audit fish stocks, count homeless populations, estimate software bugs from two independent test teams, and check how complete a historical archive is.

The assumptions are not fine print. They are the method. Four of them, and every one fails in a direction you can predict. First, the population must be CLOSED between the two samples: no births, deaths, immigration or emigration. Deaths of marked animals and immigration of unmarked ones both cut the recapture fraction, which inflates the estimate. Second, every animal must be equally catchable. This is the assumption that fails most and hurts most — trap-happy animals that come back for the free bait raise the recapture fraction and shrink the estimate, while trap-shy ones do the reverse. Heterogeneity in catchability, where some individuals are simply easier to catch than others, biases the estimate DOWNWARD, sometimes badly. Third, marks must not be lost and must not be missed on inspection; a lost tag is counted as an unmarked animal and inflates the estimate. Fourth, marking must not change survival or behaviour.

There is also a purely statistical problem, which is why this page is not the version a field study reports. The Lincoln-Petersen estimator has no defined expectation when the recapture count R could have been zero, and it is biased upward whenever R is small. Chapman's 1951 correction fixes both: N̂ = (M+1)(C+1)/(R+1) − 1. For the case above it gives 762.8 rather than 800, and it is essentially unbiased when M + C is at least as large as N. Use Chapman for real work. This page flags it in the answer whenever your recapture count is under about eight, and computes it for you there.

One last piece of honesty: none of this gives you a confidence interval, and the interval on a small mark-recapture study is wide enough to change what you would do about the result. Two samples give one number; the modern practice is multiple recapture occasions and a model that estimates catchability rather than assuming it away.

If you only have the two samples, there is a rough guide worth carrying: the estimate is usable when you expect at least seven or eight recaptures, and it is close to worthless below three. Since the expected recapture count is MC/N, you can plan for that before you set a single trap — this page will solve for R, so put in your best guess of the population and see how many animals you need to mark to get a recapture count that means anything. That is by far the most useful direction to run this formula in.