Understanding Single Mark Recapture Population Estimation Methods

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Understanding Single Mark-Recapture Population
Estimation Methods

• WFS 560

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The practice of capturing and marking, releasing,
and recapturing both marked and unmarked animals
to estimate population size began with Carl G.
Petersen. Petersen was a fisheries biologist who
was studying fish movements, but when a third of
his marked fish were harvested by fishermen he
realized he could estimate population size. His
estimation method was published in 1896.
Frederick Lincoln developed the Lincoln Index to
estimate the size of waterfowl populations in
North America based on banding and recovery data.
Because not all banded birds are reported the
Lincoln Index would over-estimate population
size, but if reporting rates were constant over
time the Lincoln Index could be useful for
monitoring population trends. The simple estima
tor based on the method is commonly called the
Lincoln-Petersen estimator although the
approach was used by LaPlace in the 17th century
and probably should be referred to as the
Petersen-Lincoln estimator!
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So, what is this estimator? The naïve estimator
is
Think of it this way. Capture and mark n1
animals and release them back into the
population. Then collect another sample of n2
animals, of which m were marked and released in
the n1 sample. Now m/n2 is an estimate of the p
roportion of marked animals in the population.
For example, if 50 of the population is marked,
and we marked 100 animals, then N 200 animals.
Thus, a slightly more intuitive way to write the
estimator might be
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Or, if p m/n2 proportion of marked animals in
the population
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But before we go any further with this math,
lets discuss the assumptions associated with
this estimator of population size
The population is closed geographically and demo
graphically. That is, the population does not
change during the period in which samples are
taken. No deaths, births, immigration, nor
emigration. Every member of the population has th
e same probability of being captured and marked.
Marking does not change the capture probability
of an animal. The second sample is a simple rando
m sample (again, all animals have the same
probability of capture). Marks are not lost and a
ll marks are recognizable and recorded upon
recapture. Some assumptions are more important
than others, and violation of any of these
assumptions have different effects on the
resulting population estimates.
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Violation of the assumption of closure
If only births or immigration occur between
marking and recapture then the population
estimate is the number of animals in the
population at the time of recapture.
If only deaths or emigration occur between
marking and recapture then the population
estimate is the number of animals at the time of
marking. If random immigration/emigration occurs
then the population estimate is unbiased, but
precision suffers. If movement is non-random (e.g
., Markovian) estimates will be biased.
A more complete evaluation of this issue was
published by Bill Kendall Kendall, W. L. 1999
. Robustness of closed capture-recapture methods
to violations of the closure assumption. Ecology
80(8) 2517-2525.
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Violation of the assumption of equal catchability
If marked animals exhibit a response to trapping
(become trap-happy or trap-shy) then population
estimates will be negatively or positively
biased, respectively. If there is heterogeneity i
n capture probabilities among animals then the
population estimates will be biased (unknown
direction of bias, except under very special
situations). Variance will likely be underestimat
ed.
Note this applies to assumptions 2 and 3.
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Violation of the assumption that marks not lost
If marks are lost or not recorded then the
recapture probability will be under-estimated.
This will result in a positive bias in population
estimates.
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The naïve estimator is biased for small samples
sizes, and a correction for small sample sizes is
This is unbiased if n1n2 N and m 7. The
variance can be estimated as
George Seber provides a detailed discussion of
the Lincoln-Petersen estimator and issues
associated with its use. Seber, G. A. F. 1982.
The estimation of animal abundance and related
parameters. 2nd edition. C. Griffin, London,
U.K.
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Although the correction to the naïve estimator
addresses small sample bias, in many cases this
problem is the least of your worries.
Trap response is particularly problematic, espec
ially if the same method of capture and recapture
is used. For example, bears captured in a snare
are less likely to be recaptured using a snare.
Thus, when snares are used to capture and
recapture bears, m/n2 is underestimated and
population size is overestimated.
Similarly, fish that are electroshocked and mark
ed may be more likely to have left the reach of
stream when the recapture occurs. Thus,
population size will be overestimated.
Other species may become trap-happy. In winter,
rabbits become stressed and are attracted to
bait. They may quickly learn that being captured
means food. Here, m/n2 is overestimated and
population size is then underestimated.
Capture homogeneity is probably the most difficu
lt assumption to meet with this estimator. Why
should we expect all individuals to have the same
probability of capture?
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The following few slides illustrate heterogeneity
associated with harvest probabilities of black
bear. In Pennsylvania, bears are captured by a
variety of means (dart guns, snares, barrel
traps) and recaptured in the hunting season as
dead bears. Thus, m/n1 represents a harvest
rate. These slides illustrate that capture prob
ability varies spatially, demographically, and
according to environmental conditions. Not
surprising (I hope), yet despite the fact that
any biologist can name any number of variables
that can affect probability of capture, the
Lincoln-Petersen index is probably one of the
most commonly used estimators in fisheries and
wildlife.
Diefenbach, D. R., J. L. Laake, G. L. Alt. 2004.
Spatio-temporal and demographic variation in the
harvest of black bears implications for
population estimation. Journal of Wildlife
Management 68947-959.
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Violating assumptions can result in substantial
bias in resulting population estimates, which is
a much greater problem than the issue of
correcting for small sample size bias. Violating
an assumption is essentially saying that you are
using the wrong model. So, in the bear example,
we have 2 options (1) eliminate the assumption
violation, or (2) use a better model. Although
changing field methods might solve an assumption
violation in some situations, clearly it is not
possible to change inherent characteristics of
bears that make them more (or less) susceptible
to harvest. We need to look at modifications to
relax some model assumptions.
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Let h n2, hc m, and nc n1. This is a
slightly different parameterization of an earlier
slide. Essentially, we are conditioning on the
sample of marked bears to estimate capture
probability. In this simple example this does
not help us resolve the capture heterogeneity
problem, but what if we could model p as a
function of variables (e.g., age, sex, etc.)?
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If we can model p then we basically have a
Horvitz-Thompson type estimator. What we have
done is relaxed the assumption that every
individual has the same probability of capture
we have incorporated (some forms of) capture
heterogeneity into the model. Note that if we
only use an intercept term in p(zi) then this
model reduces to the Lincoln-Petersen estimator.
We can say that the Lincoln-Petersen is a special
case of this model
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We can use standard logistic regression to model
capture probabilities, in which the response
variable is harvested (1) or not harvested (0).
The independent variables must be categorical,
and many types of model selection tools can be
used to identify a parsimonious model.
With the Horvitz-Thompson estimator we are sayin
g that each bear, associated with specific
characteristics, has an estimable probability of
being harvested. For example, if a given bear
has a probability of 0.5 of being harvested,
every harvested bear of this type represents two
bears in the population. Summing 1/p(zi) over
all bears harvested gives us an estimate of
population size.
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This approach to estimating population size, in
which we condition on sampled bears, was
developed by Huggins (1989) for multiple
capture-recapture models and is incorporated into
the closed capture models of MARK.
Unfortunately, we will not have any time in this
course to study these models. Huggins, R. M. 1
989. On the statistical analysis of capture
experiments. Biometrika 76133140.
Huggins, R. M. 1991. Some practical aspects of
a conditional likelihood approach to capture
experiments. Biometrics 47725732
Also, the Horvitz-Thompson type estimator has
been used with aerial surveys, termed
sightability models (Samuel et al. 1987,
Steinhorst and Samuel 1989). Steinhorst, R. K
., and M. D. Samuel. 1989. Sightability
adjustment methods for aerial surveys of
wildlife populations. Biometrics 45412-425.
Samuel, M. D., E. O. Garton, M. W. Schlegel, R.
G. Carson. 1987. Visibility bias during aerial
surveys of elk in northcentral Idaho. Journal
of Wildlife Management 51622-630.
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In sightability models the idea is that you spend
time and effort up front developing a predictive
model of variables that affect detection of
wildlife using some type of survey method,
usually aerial surveys. This effort begins with
marking individual animals so that they are
identifiable during surveys as being marked (but
not so identifiable that their probability of
detection is different). Then a series of surve
ys are conducted in which the response variable
is 1 (observed) or 0 (not observed) of the marked
individuals. At the same time, independent
variables are measured. These independent varia
bles might be behavior of the animal (standing,
running, bedded, etc.), environmental conditions
(e.g., snow or no snow, obstructing vegetation,
etc.). One important variable associated with
big game is group size animals in larger groups
are more likely to be detected.
See http//pacfwru.cas.psu.edu/wfs560/sightabili
ty.pdf