Capture-Recapture: Closed Capture Abundance - Marked Individuals • MDChelp

Closed population abundance estimation methods- marked individuals

As mentioned in the overview, capture-recapture models for closed populations are strictly used for estimating the abundance of critters. Though many abundance estimation methods have, at their core, some kind of closed population capture-recapture framework, these methods below are generally applied to strictly estimate abundance \(\hat{N}\). There have been limited applications of these methods beyond obtaining a point estimate for a location, at a single point in time, and none that robustly address abundance-habitat relationships, though expanding the model development to address these questions using Bayesian methods (or spatially explicit capture-recapture) are possibilities.

Below are several common capture-recapture abundance estimation methods suitable for closed populations, often shortened as “closed capture methods”. Demographic closure means that, while animals are being observed, there are no births, deaths, immigration, or emigration within the study area. The methods below are appropriate for “marked” individuals. That is, these assume you are physically capturing and marking individuals. Another document covers abundance estimation in closed populations for individuals you can uniquely identify, but are not physically marking them.

Fundamentally, we’re trying to solve this equation: \[ N=\frac{n}{p} \] Where \(N\) is the unknown abundance,\(n\) is the number of unique individuals we know about, and \(p\) is the capture probability.

Probably the most important thing to recognize about these models is, aside from needing to meet the closure assumption, all methods are very, very, sensitive to violations of assumptions about capture probability and closure. If capture/recapture probability is not addressed correctly, your estimates of abundance can be very, very wrong.

Typically, the use of the methods below will require you have some form of trap grid or transects (think small mammal traps) or can at least define some spatial extent you are trapping (e.g., sampling defined stream reaches, capturing waterfowl within a spatially defined breeding population). Again, having demographic closure is of key importance here; if animals are moving in and out of the study area or the size of the study area keeps changing (due to sampling effort) then your estimate of \(N\) becomes somewhat useless.

The original Lincoln-Petersen (LP) estimator (Petersen 1896¹, Lincoln 1930²) is a simple 2-sample technique to estimate abundance. By 2-sample technique, I mean you only need 2 capture occasions: the first capture occasion when you mark and release animals, and an additional occasion when you recapture them. That’s it. The LP estimator calculates abundance as: \[N=\frac{n_1\times \ n_2}{k}\]

\(n_1\) the number of animals encountered on the first occasion. All encountered animals are marked and released.
\(n_2\) the total number of animals encountered on the second occasion.
\(k\) is the number of marked individuals from \(n_1\), captured again and included in \(n_2\).

This is a very simple equation, but it’s not the one actually used in practice. If you do end up using a LP, you will most likely use the Chapman version/ estimator. It’s what’s implemented in MDChelp::chapman().

Multiple variations and extensions of the LP method exist. If you have multiple encounters where you mark, release, and recapture individuals, an alternative known as the Schnabel method/estimator is appropriate (after Zoe Emily Schnabel). There are also methods that combine the Schnabel method with simultaneous removals, but are not discussed here and aren’t generally used anymore. The extensions beyond the simple LP framework have largely been supplanted by methods that use uniquely marked animals, primarily because of the development of better marking techniques (e.g. RFID tags). Still, it’s important to highlight the conceptual contributions of these individuals. It’s also a helpful reminder of some less common methods that are sill viable when you can’t meet the data collection needs of more contemporary alternatives.

Important Assumptions/ Limitations/ Tidbits

The population is closed between the first and second capture events!!!!
Tags or markings do not need to be unique among individuals, but there are no tag losses between the following the release of marked individuals.
All individuals have the same recapture probability.
- The capture probability at the first event can be different than the probability at the second event, but the capture probability between unmarked and marked individuals is the same.
  - This means that the methods used to capture individuals during the first event for marking can be different than those used to capture individuals during the second event.
    - The most extreme example of this is when the LP method is used to estimate abundance when the initial capture period is (obviously) a live capture, marking and release, but the recapture is a dead recovery (e.g. waterfowl band recovery, tagged fish).
LP methods perform poorly when fewer than 50 individuals are marked ( ideally, mark > 100) or \(p\) is small. This is important if there is group-level heterogeneity in capture probability. If capture probability differs between male/ female or adult/juvenile, you “bin” individuals in each group separately which can lead to small \(n_1\) for any group

Why still use it ?

While “better” methods do exist that deal with some very common violations of the assumptions that LP relies on (capture heterogeneity), they typically require you find more populations to sample, have more capture events, etc. The fact that the alternatives might be difficult to implement is not an excuse to just use LP if you cannot adequately meet the assumptions!!! It is worth remembering though, that it can be advantageous to alter your study design to meet the assumptions of a simpler approach than to address logistical or animal welfare constraints of methods that promise more flexible or relaxed assumptions.

There are times when the LP or other simpler approaches won’t cut it. Usually it’s because the assumption of a homogeneous encounter probability for marked and unmarked animals isn’t met. If you cannot meet that assumption, then it’s time to switch to a likelihood approach that can!

Doing so requires some changes. Mathematically, we now start using likelihood based methods. This just means we start using probability theory and distributions to estimate abundance rather than algebra like above. The more practical, real-world implication is, we now need to have individuals marked with unique markings and we need to follow them. You are likely going to also have > 2 encounter events, or opportunities to capture and recapture marked animals. Information about whether an individual(s) was captured at each encounter event is called an encounter history. We now have several parameters that get used to describe these encounter or capture histories.

\(\hat{N}\) the estimated abundance
\(p\) the capture and/or recapture probability.
\(c\) the recapture probability, if different from \(p\)

These are not the only parameters you may come across if you end up getting further into closed-capture models, but they’re the foundation on what all other variations are built and what can be used to help discriminate among the different variations.

Variations on closed-capture models

A big focus of the historical development around closed-capture models is to account for capture heterogeneity. This has led to a host of variations in the literature. Below are some more common variations in closed capture models that might help clarify things if you get more into these methods, but also represent some important practical issues you should consider with your own data. Unless noted, these variations exist for both full-likelihood and Huggins methods.

Behavioral response to trapping: Use \(p\) and \(c\) to account for trap happiness or trap shyness
Temporal variation in capture probability: \(p\) and/or \(c\) varies among encounter events
Heterogeneity via “mixtures”: There are groups of individuals with different \(p\) or \(c\), but you can’t identify them in the field. We can use a “mixture model” that introduces another parameter \(\pi\) that represents the probability an individual belongs to one unobservable group or the other. Groups have different \(p/c\) parameters, but individuals within groups are assumed homogeneous.
Heterogeneity as individual random effects: Is only feasible with the Huggins model. Rather than grouping individuals, or assuming individuals within groups are homogeneous, we can treat all critters as being different from one another via an individual random effect. Like random effects in linear models, there is some “average” \(p/c\) but individuals deviate from this average. Those deviations follow a normal distribution.

These variations are an important extension. As mentioned before, estimates of abundance are sensitive to assumptions about \(p/c\), whether they are the same among individuals, vary among individuals, vary across capture events, etc. Appropriately accounting for these possibilities is critical to obtaining accurate and unbiased estimates of abundance.

Criticisims/ Complications

There are a few common criticisms or complications that arise with the use of likelihood-based closed-capture methods

\(N\) is only an estimate of the population size vulnerable to capture. If you sample more area (intentionally or unintentionally), your estimate of \(N\) increases. Other methods focus on estimating density try to address this problem (e.g. spatially-explicit capture-recapture; SECR).
Most applications do not try to address or estimate what causes variation in \(N\)-it’s possible, but likely logistically challenging (trapping effort in multiple areas) and still suffers from the above issue.
Because of this, it’s typically suggested to use a SECR approach if possible. It results in an exactly defined spatial extent, making \(N\) more reliable and can more easily incorporate/ test for environmental variables affecting abundance.
Other methods have you sample smaller, but still defined areas to estimate density in a model-based approach. It’s much simpler to incorporate information about
If you can exactly define the extent of the space being sampled, \(N\) represents a realized estimate of abundance, that is the estimate of the number of critters actually running around at that time. Other methods may provide model-based estimates of \(N\) or both. Model-based estimates provide the expected abundance, given information about things like habitat or other environmental covariates. These are not the same quantity and which one you should use will depend on your question.

The above covers more of the technical details of closed-capture methods. It’s important to understand them and they cannot be ignored. But if you are in the processing of identifying appropriate methods for your particular question, below are some general, practical points about closed-capture methods that may be helpful.

The closed capture methods described above are best suited when getting an estimate of \(N\) is of interest, not understanding various factors responsible for variation in \(N\).
Because of many of the strict assumptions, it’s probably best not to think of \(N\) as a “true” estimate of abundance, unless you can guarantee all individuals are vulnerable to capture. If you can at least ensure a representative sample, it provides a very good index.
In practice, the best uses of these methods independently (e.g. not part of a robust design or other integrated method) will probably be for simple monitoring and status and trend assessment.
Some specifics like distinction between \(p/c\) are specific to physically capturing animals, but conceptually, but closed-capture methods are a very general concept that most other capture-recapture methods are based on.