Mark and recapture

Notation

Let

N = Number of animals in the population

n = Number of animals marked on the first visit

K = Number of animals captured on the second visit

k = Number of recaptured animals that were marked

A biologist wants to estimate the size of a population of turtles in a lake. She captures 10 turtles on her first visit to the lake, and marks their backs with paint. A week later she returns to the lake and captures 15 turtles. Five of these 15 turtles have paint on their backs, indicating that they are recaptured animals. This example is (n, K, k) = (10, 15, 5). The problem is to estimate N.

Lincoln–Petersen estimator

The Lincoln–Petersen method (also known as the Petersen–Lincoln index or Lincoln index) can be used to estimate population size if only two visits are made to the study area. This method assumes that the study population is "closed". In other words, the two visits to the study area are close enough in time so that no individuals die, are born, or move into or out of the study area between visits. The model also assumes that no marks fall off animals between visits to the field site by the researcher, and that the researcher correctly records all marks.

Given those conditions, estimated population size is:

${\displaystyle \hat{N}=\frac{nK}{k},}$

Derivation

It is assumed that all individuals have the same probability of being captured in the second sample, regardless of whether they were previously captured in the first sample (with only two samples, this assumption cannot be tested directly).

This implies that, in the second sample, the proportion of marked individuals that are caught (${\displaystyle k/K}$) should equal the proportion of the total population that is marked (${\displaystyle n/N}$). For example, if half of the marked individuals were recaptured, it would be assumed that half of the total population was included in the second sample.

In symbols,

${\displaystyle \frac{k}{K}=\frac{n}{N}.}$

A rearrangement of this gives

${\displaystyle \hat{N}=\frac{nK}{k},}$

the formula used for the Lincoln–Petersen method.

Sample calculation

In the example (n, K, k) = (10, 15, 5) the Lincoln–Petersen method estimates that there are 30 turtles in the lake.

${\displaystyle \hat{N}=\frac{nK}{k}=\frac{10\times 15}{5}=30}$

Chapman estimator

The Lincoln–Petersen estimator is asymptotically unbiased as sample size approaches infinity, but is biased at small sample sizes. An alternative less biased estimator of population size is given by the Chapman estimator:

${\displaystyle {\hat{N}}_C=\frac{(n+1)(K+1)}{k+1}-1}$

Sample calculation

The example (n, K, k) = (10, 15, 5) gives

${\displaystyle {\hat{N}}_C=\frac{(n+1)(K+1)}{k+1}-1=\frac{11\times 16}{6}-1=28.3}$

Note that the answer provided by this equation must be truncated not rounded. Thus, the Chapman method estimates 28 turtles in the lake.

Surprisingly, Chapman's estimate was one conjecture from a range of possible estimators: "In practice, the whole number immediately less than (K+1)(n+1)/(k+1) or even Kn/(k+1) will be the estimate. The above form is more convenient for mathematical purposes."(see footnote, page 144). Chapman also found the estimator could have considerable negative bias for small Kn/N (page 146), but was unconcerned because the estimated standard deviations were large for these cases.

Confidence interval

An approximate ${\displaystyle 100(1-\alpha )\mathrm{\%}}$ confidence interval for the population size N can be obtained as:

${\displaystyle K+n-k-0.5+\frac{(K-k+0.5)(n-k+0.5)}{(k+0.5)}\exp (\pm z_{\alpha /2}{\hat{\sigma }}_{0.5}),}$

where $z_{\alpha /2}$ corresponds to the ${\displaystyle 1-\alpha /2}$ quantile of a standard normal random variable, and

${\displaystyle {\hat{\sigma }}_{0.5}=\sqrt{\frac{1}{k+0.5}+\frac{1}{K-k+0.5}+\frac{1}{n-k+0.5}+\frac{k+0.5}{(n-k+0.5)(K-k+0.5)}}.}$

The example (n, K, k) = (10, 15, 5) gives the estimate N ≈ 30 with a 95% confidence interval of 22 to 65.

It has been shown that this confidence interval has actual coverage probabilities that are close to the nominal ${\displaystyle 100(1-\alpha )\mathrm{\%}}$ level even for small populations and extreme capture probabilities (near to 0 or 1), in which cases other confidence intervals fail to achieve the nominal coverage levels.

Capture probability

The capture probability refers to the probability of a detecting an individual animal or person of interest, and has been used in both ecology and epidemiology for detecting animal or human diseases, respectively.

The capture probability is often defined as a two-variable model, in which f is defined as the fraction of a finite resource devoted to detecting the animal or person of interest from a high risk sector of an animal or human population, and q is the frequency of time that the problem (e.g., an animal disease) occurs in the high-risk versus the low-risk sector. For example, an application of the model in the 1920s was to detect typhoid carriers in London, who were either arriving from zones with high rates of tuberculosis (probability q that a passenger with the disease came from such an area, where q>0.5), or low rates (probability 1−q). It was posited that only 5 out of 100 of the travelers could be detected, and 10 out of 100 were from the high risk area. Then the capture probability P was defined as:

${\displaystyle P=\frac{5}{10}fq+\frac{5}{90}(1-f)(1-q),}$

where the first term refers to the probability of detection (capture probability) in a high risk zone, and the latter term refers to the probability of detection in a low risk zone. Importantly, the formula can be re-written as a linear equation in terms of f:

${\displaystyle P=\left(\frac{5}{10}q-\frac{5}{90}(1-q)\right)f+\frac{5}{90}(1-q).}$

Because this is a linear function, it follows that for certain versions of q for which the slope of this line (the first term multiplied by f) is positive, all of the detection resource should be devoted to the high-risk population (f should be set to 1 to maximize the capture probability), whereas for other value of q, for which the slope of the line is negative, all of the detection should be devoted to the low-risk population (f should be set to 0. We can solve the above equation for the values of q for which the slope will be positive to determine the values for which f should be set to 1 to maximize the capture probability:

${\displaystyle \left(\frac{5}{10}q-\frac{5}{90}(1-q)\right)>0,}$

which simplifies to:

${\displaystyle q>\frac{1}{10}.}$

This is an example of linear optimization. In more complex cases, where more than one resource f is devoted to more than two areas, multivariate optimization is often used, through the simplex algorithm or its derivatives.

More than two visits

The literature on the analysis of capture-recapture studies has blossomed since the early 1990s. There are very elaborate statistical models available for the analysis of these experiments. A simple model which easily accommodates the three source, or the three visit study, is to fit a Poisson regression model. Sophisticated mark-recapture models can be fit with several packages for the Open Source R programming language. These include "Spatially Explicit Capture-Recapture (secr)", "Loglinear Models for Capture-Recapture Experiments (Rcapture)", and "Mark-Recapture Distance Sampling (mrds)". Such models can also be fit with specialized programs such as MARK or E-SURGE.

Other related methods which are often used include the Jolly–Seber model (used in open populations and for multiple census estimates) and Schnabel estimators (an expansion to the Lincoln–Petersen method for closed populations). These are described in detail by Sutherland.

Integrated approaches

Modelling mark-recapture data is trending towards a more integrative approach, which combines mark-recapture data with population dynamics models and other types of data. The integrated approach is more computationally demanding, but extracts more information from the data improving parameter and uncertainty estimates.