3. 5.7 Mark/Recapture Data Analysis Methods

There are numerous ways of analysing data from a MRR program. The level of confidence placed in any estimator is largely dependent upon sample size, sample effort and how well the assumptions of the analysis methods are met in the field. Some common methods of analyses found in the literature are summarized below to provide some background information. Many sophisticated and robust methods of analyzing MRR data are available as part of the programs CAPTURE, JOLLY and MARK; all of these are discussed in Species Inventory Fundamentals, manual No.1.

Minimum-Number-Alive Estimator (MNA)

One of the easiest ways of estimating the abundance of a population from a mark-release-recapture (MRR) program is called the minimum number alive method (MNA). MNA (also called the calendar count or enumeration) is an estimate based on the sum of all individuals known to be alive during a particular capture (trapping) session. An individual is known to be alive during a given capture session if it was captured during that session, or if it was captured before and after that capture session. For example, if an individual is captured during capture session #1 and #3, it can be accurately stated that it was missed (but alive) during session #2.

Although the MNA method is simple to use, this estimator has been criticized as being negatively biased in most situations. For this reason, in a summary, Ritchie and Sullivan (1989) suggest that the MNA estimate should only be used when the trappability of animals is >70%. Several articles have been written on the use of the MNA estimator (Hilborn et al. 1976; Jolly and Dickson 1983; Nichols and Pollock 1983; Boonstra 1985; Efford 1992; Hilborn and Krebs 1992). Most of these papers recommend the use of the Jolly-Seber estimator over MNA if trappability is low or unknown.

This approach, also referred to as "saturation trapping" or "enumeration" is generally not the best means of achieving a statistically valid estimate, and is not recommended. The reasons for this are:

• In many cases, a large amount of effort is needed to fully trap a population. In contrast, a valid estimate of population can be gained with less effort by using a ratio estimator, or a closed CAPTURE mark-recapture estimator.
• The assumption that an entire population is trapped or marked cannot be validated and therefore population estimates can be negatively biased (Pollock et al. 1990).
• To get an unbiased estimate of density, a population should be geographically closed. To minimize violation of this assumption, sampling should occur in a relatively small amount of time. Saturation trapping usually takes long periods of time, and therefore closure assumptions will be violated unless the researcher is working on an entirely closed system, such as an island.

Estimation by Asymptotic Capture

Population abundance can be estimated by intensively trapping and marking a population until no new (unmarked) individuals are captured. This method is essentially a modified (i.e., non-lethal) version of kill trapping where animals are removed until no animals remain. It is generally not recommended as it is subject to criticisms similar to those described above.

Ratio estimators

The Lincoln, Petersen, and Schnabel estimators are based on the ratio of marked to unmarked individuals within a population. These estimators assume that the population is "closed" to immigration and emigration. The formulas are based on the assumption that the population size is related to the number of marked and released animals in the same way that the total caught at a subsequent time is related to the number recaptured (Davis and Winstead 1980). White et al. (1982) offer excellent discussion of closed models which many be calculated using the program CAPTURE.

The Petersen (or Lincoln-Petersen) estimate is the most basic MRR method. It is based on two sample periods only (i.e., one period of marking animals, followed by a single period of recapture). It is described using the following formulas:

(1)

therefore:

(2)

where:

N = Population Estimate

M = Number of marked and released animals

C = Total number of animals captured

R = Number of marked animals that were recaptured

Lincoln-Peterson estimates are easy to calculate, and the estimator has been shown to be robust to time variation in capture probabilities. However, there are important assumptions associated with this estimator such as equal probabilities of capture between animals, population closure, and no net loss of animal marks between samples. If relative abundance is the objective then violations of assumptions may not be as significant provided that the degree to which assumptions are violated is similar between studies and over time, and therefore the estimator will show a consistent, comparable bias. If absolute abundance is the objective of methods, and animals can be marked individually then the use of the estimators in program CAPTURE is recommended.

Numerous variations on the Petersen Estimate have been developed. The Petersen Estimate is biased in that it tends to overestimate the actual population, especially if the sample is small. In response to this bias, Seber (1982) offers a variation on Petersen's formula that is less biased, and nearly unbiased if there are at least seven recaptures of marked animals. Another variation, the Schnabel estimate was developed to allow investigators to analyze data from multiple (>2) marking sessions.

The Jolly-Seber Estimator

Like the Lincoln, Petersen, and Schnabel estimators (above), the Jolly-Seber estimator is also based on the ratio of marked to unmarked individuals within a population. However, the Jolly-Seber estimate differs from others in that it recognizes, and attempts to incorporate, the fact that biological populations are generally not "closed". This "open" model will not provide a true estimate of density, but rather of abundance, as the population is not defined in terms of area. This estimator requires that at least three sampling periods be carried out in order to calculate certain variables. Pollock et al. provide good discussion of Jolly-Seber models, and the program JOLLY is very useful for simulating MRR or analyzing data.

The formula for the Jolly-Seber estimate of population size is given below.

(3)

where:

N_t = Population estimate just before sample t

t = Sample period (1,2,3,4,5,......t th sample)

__t = proportion of animals marked

(4)

m_t = Number of marked animals that were recaptured during sample t

n_t = Total number of animals captured during sample t

M_t = Estimated number of marked animals just before sample t

(5)

s_t = Number of animals released

s_t = (n_t - accidental deaths)

R_t = Number of animals released during sample t, or s_t that were recaptured during a later sampling period

Z_t = Number of animals that were not captured during sample t, but were captured before and after sample t

The Jolly Seber model is also susceptible to biases if unequal capture probabilities are exhibited in the trapped population; however, the survival rate estimate of the Jolly Seber is robust to most forms of capture probability variation, and is therefore a useful alternative for monitoring populations. In addition, there are many modifications to the Jolly-Seber to accommodate age-specific capture probabilities and survival rates (program JOLLY JOLLYAGE and POPAN). If the robust design is used then program RDSURVIV can be used to estimate temporary emigration, and allow more precise survival estimates. Also, the Jolly Seber approach to survival modelling has been modified to allow the testing of biological hypothesis using various model fitting procedures as documented in programs SURGE, and MARK.

However, all of the programs mentioned require advanced statistical knowledge, and project biologists are urged to seek the advice of a qualified biometrician. A summary of useful software is available in Species Inventory Fundamentals, No. 1, Appendix G.

Estimates for population size and coefficients of variation were calculated using the Jolly-Seber model in Nichols et al. (1981).