This file was created by scanning the printed publication.
Errors identified by the software have been corrected;
however, some errors may remain.
Sampling Problems in
Estimating Small Mammal
Population Size1
George E. Menkens, Jr.2 and Stanley H.
Anderson3
Species conservation and management or analysis of environmental
impacts require accurate estimates of
population size. Because censusing
entire populations is difficult, if not
impossible, a sampling program is
generally employed to estimate animal abundance. In small mammal
studies, sampling is frequently performed using live traps placed in
grids. Numerous approaches have
been used to estimate animal abundance on trapping grids (eg., catchper-uni t effort, removal methods) but
capture-mark-recapture techniques
are the most commonly used (Seber
1986).
Four sources of error may influence an estimator's bias and precision (Cochran 1977, McDonald 1981).
Two, missing data and gross errors
(e.g., misreading tag numbers) are
"human" errors and can be avoided
by using careful field and laboratory
techniques. The remaining sources,
'Paper presented at symposium, Management of Amphibians, Reptiles, and
Small Mammals in North America (Flagstaff,
AZ,July 79-21, 1988).
2George E. Menkens, Jr., is a Research
Associate with the Wyoming Cooperative
Fish and Wldlife Research Unit,4Laramie,
WY 82071.
3StanleyH. Anderson is Leader, Wyoming Cooperative Fish and Wildlife Research
Laramie, WY 8207 1.
4Cooperatorsin the Wyoming Cooperative Fish and Wildlife Research Unit include:
the Department of Zoology and Physiology,
University of Wyoming; Wyoming Game
and Fish Department; and the U.S. Fish and
Wildlife Service.
Abstract. -Estimates of populationsize are
influenced by four sources of error: measurement,
sampling, missing data, and gross errors.
Measurement error can be reduced by using the
correct estimator, reducing variation in capture
probabilities, and by increasing sample size and trap
period length. Sampling error can be decreased by
increasing the number of grids trapped.
measurement and sampling error,
may, in many cases greatly affect an
estimate (McDonald 1981). Measurement error is the error resulting from
the use of imprecise or biased (or a
combination of these) data collection
methods (McDonald 1981). In markrecapture studies, measurement error influences the bias and precision
of an estimate for any single grid.
Sampling variance is considered to
be a measurement error in mark-recapture studies (White et al. 1982).
Sampling error is error introduced
by natural variation between sampling units, i.e., trap grids.
Potentially large sources of measurement error in mark-recapture
studies may result from capture
probability variation and model selection. All mark-recapture estimators make specific assump tions about
capture probability variation within
and among animals and trapping
days. Three factors influencing individual capture probability variation
have received attention (Burnham
and Overton 1969, Otis et al. 1978,
Pollock 1981, Seber 1982) and are
time, behavior, and individual
heterogeneity. Models assuming time
variation allow all animals to have
the same capture probability on a
given day, but this probability may
change between days. Models allowing behavioral responses to trapping
assume all animals initially possess
identical capture probabilities, but
these probabilities may change upon
first capture. Capture probabilities
may increase (animals become trap
happy) or decrease (animals become
trap shy) after initial capture. Models
assuming that individual heterogeneity is present allow each animal to
have a unique capture probability
that does not change over time. Combinations of these factors may also
occur. For example, an animal's capture probability may be influenced
by both time and behavioral effects.
Model selection is another source
of measurement error. Selection of an
inappropriate or incorrect model for
data analysis results in estimates
with unknown degrees of bias and
unacceptably large or unrealistically
small standard errors (Otis et al.
1978, White et al. 1982). CAPTURE
(Otis et al. 1978) is a widely used
computer program for estimating
population size using mark-recapture
data that also provides an objective
method for selecting the correct
model when any of the above
sources of capture probability variation are present.
In this paper, we investigate the
effects that variation in capture
probabilities due to time, behavior,
and individual heterogeneity have on
estimates of animal abundance and
model selection. We also discuss improvement of an estimate using data
pooling. We use these results to
show how reducing trap period
length influences estimator bias,
standard error, and confidence interval coverage rate, and discuss how
this may help reduce the number of
grids required to detect a given difference between yearly estimates of
population sizes.
Material and Methods
To investigate effects of both capture
probability variation and trap period
reduction, we used program CAPTURE (Otis et al. 1978) to randomly
generate and analyze data sets with
known population characteristics
(see Menkens 1987 for details). CAPTURE contains eight models, five
with estimators, for estimating population size for closed populations
when capture probabilities do not
vary (model M(o)),or when they
vary with time (model M(t)),behav-
ioral response (model M(b)),individual heterogeneity (model M(h)) or a
combination of the behavioral and
individual heterogeneity models
(model M(bh)).Using CAPTURE, we
specified the number of trapping periods, population size, and capture
probabilities, and patterns of variation. CAPTURE was then used to
analyze each data set.
We analyzed the same data sets
using Chapman's unbiased version
of the Lincoln-Petersen estimator and
its variance estimator (Seber 1982).
Because the Lincoln-Petersen estimator uses data from only two periods,
each data set was split prior to estimation. Thus in.a 5 day trapping
study, the first 3 days constituted the
marking period, and the second 2
days was the recapture period. In
studies 10 days long, the first 5 days
were the marking period, the second
5 days the recapture period.
Data were generated for a wide
range of conditions. We used trap
periods of 5 and 10 days, population
sizes of 50 and 100 and a wide variety of capture probability patterns
(table 1).One thousand data sets
were generated for each combination
of these conditions. In this paper, we
only generated data meeting the assumptions of one of the five models
with estimators in CAPTURE. For
each data set, CAPTURE was forced
to perform the analysis using the correct model. For example, if data were
generated under the assumption of
time variation, CAPTURE was forced
to use model M(t) for the analysis.
Simulations were also performed using the same, and additional, capture
probabilities (table I), with CAPTURE being allowed to select an estimator using its model selection procedure.
Results
Performance of both the Lincoln-Petersen estimator and CAPTURE is
dependent upon the size and magnitude of the variation in capture
probabilities (table 2). Estimators
have lower degrees of bias, smaller
standard errors, and higher confidence interval coverage rates when
capture probabilities are high and
their variation is low (tables 1 and 2)
over all population sizes. When capture probability variation is constant,
the estimator's bias tends to decrease
and confidence interval coverage
rates increase with increasing population size (table 2). Although this
pattern is evident for standard errors, patterns of change with increasing sample size are not as clear (table
2).
In general, the estimator's bias decreases and confidence interval coverage rates increase as trapping period length increases (table 2). This
pattern is not as obvious for standard
errors, although they d o tend to improve with increasing trap period
length (table 2). In most cases the
magnitude of change in bias is
smaller for good capture probabilities than for poor capture probabilities when the trapping period increases (table 2). Although estimated
standard errors tend to decrease
with lengthening trap periods (more
so with good capture probabilities),
the magnitude of this change is generally smaller than is change in bias
(table 2). As with bias and standard
error, confidence interval coverage
rates improve as trapping period increases; the magnitude of change
tends to be larger when capture
probabilities are poor (table 2).
Except when data were generated
under model M(o), CAPTURE selected the correct model less than
11% of the time (table 3). The Lincoln-Petersen estimator failed to provide an estimate at most 7% of the
time (table 3).
Discussion
In small mammal studies, measurement errors can significantly influence an estimator's bias and precision. This study shows the importance of both reducing capture
probability variation and increasing
the size of those probabilities on
measurement error. Decreasing capture probability variation reduces the
estimate's bias and coefficient of
variation, and increases its confidence interval coverage rate. This
result has also been stressed by
Burnham and Overton (1969), Menkens (19871, Menkens and Anderson
(in press), Otis et al. (1978), and
White et al. (1982). Of particular significance is the need to reduce variation due to behavioral responses
(i.e., trap-happiness and shyness)
and individual heterogeneity, especially when these factors act in concert (Menkens and Anderson in
press, Otis et al. 1978, White et al.
1982). Reduction of time variation,
particularly if the Lincoln-Petersen
estimator is used, is important, but
not as critical (Menkens 1987, Menkens and Anderson in press). Again,
reducing variation in capture probabilities leads to estimates that have
lower bias and increased precision.
Methods for reducing variation in
capture probabilities are numerous
(see Otis et al. 1978, Seber 1986,
White et al. 1982). Behavioral responses may be reduced by the use
of different capture and recapture
techniques. For example, animals
could be captured using live traps
and marked, and then "recaptured"
visually using spotting scopes
(Fagerstone and Biggins 1986). In
addition, use of traps not avoided by
animals, and use of non-intrusive
marking techniques (e.g., ear tags
instead of toe clipping) may also help
reduce behavioral responses. Use of
traps not avoided by animals may
help increase capture probabilities. If
sample sizes are large, heterogeneity
may be reduced by stratifying the
data into sex and age groups with
separate analyses performed on each
group (Otis et al. 1978, White et al.
1982). If data are stratified however,
the effects of small sample size on the
estimator's properties must be considered.
Capture probabilities may be increased and their variation reduced
after study completion by pooling
individual trap periods into single
marking and recapture periods as
was done in our simulations (Menkens 1987, Menkens and Anderson in
press). When data are pooled in this
way and the.Lincoln-Petersen estimator used, capture probabilities are 20
to 25% higher than those for individual days (Menkens 1987). In most
cases, data pooling results in estimates with improved properties.
Use of the wrong model for analysis leads to estimates with unknown
degrees of bias and unacceptably
large or unreasonably small standard
errors (Otis et al. 1978, White et al.
1982), thus contributing significantly
to measurement error. In this study,
we forced CAPTURE to use the correct model for analysis. This, is unrealistic however, in that biologist
never know which model is appropriate. CAPTURE provides a objective model selection procedure, however this procedure works poorly
with the small sample sizes typically
encountered in many field studies
(Menkens 1987, Menkens and Anderson in press, Otis et al. 1978, White et
al. 1982).In most cases, the LincolnPetersen estimator is a valid alternative to CAPTLJRE when sample sizes
are small, except when capture
probabilities are influenced by severe
behavioral responses or large degrees of individual heterogeneity
(Menkens 1987, Menkens and Anderson in press). Because use of the most
appropriate model is critical, CAPTURE should be used to determine
the type and magnitude of capture
probability variation in a data set,
and if variation is low, the LincolnPetersen estimator should be used in
analysis (Menkens and Anderson in
press).
Many additional factors contribute
to measurement error. Eliminating
these requires detailed knowledge of
species behavior and ecology, and
use of compatible techniques. For example, baits identical to, or that
closely approximate natural f d
items, should be used (Dobson and
Kjelgaard 1985). Tags that are easily
lost will lead to severe overestimates
of population size and should not be
used. Other factors that could contribute to measurement error include
use of traps or other activities that
decrease survival or increase emigration or immigration, and use of improper traps for the species.
Sampling error is the error that
results from natural variation between sampling units; the larger this
variation, the larger the number of
units that must be sampled to detect
a difference in population size. For
example, when environmental impacts are being assessed, sampling
error would be decreased by increasing the number of grids in the control
and experimental groups. Reducing
variation in capture probabilities allows decreasing the number of days
each grid is trapped without large
increases in bias or standard errors
or decreases in confidence interval
coverage rates. By reducing the trap
period, more grids can be sampled in
a shorter period of time, thereby reducing sampling error and improving the estimate of overall population
size. Trapping in as short a time
interval as possible will also decrease
variation caused by temporal population effects.
One approach to reducing Sampling error is to reduce intergrid
variation by using a stratified Sampling approach. In this case, investigators could stratify the habitat
based on some characteristic that is
correlated with animal density and
trap within these strata. Sample sizes
would be estimated for each strata.
Conclusions
Reduction of capture probability
variation and maximizing their mag-
nitude are critical to obtaining unbiased and precise estimates of population size, and also allow the selection
of the proper model for use in analysis. Although we have concentrated
on small mammals, our points concerning reduction of both variation in
capture probabilities and of measurement and sampling error, pertain to
other studies using mark-recapture
techniques (e.g., papers in Ralph et
al.). Our conclusions will hopefully
force investigators to realize that
their techniques, particularly in
poorly designed and carelessly performed studies, may not provide as
detailed and profound conclusions as
they might expect. We reiterate that
care in designing a study can minimize many (but not all) of the
sources of measurement and sampling errors we have discussed.
Acknowledgments
We thank Mark Boyce, Marc Evans,
Richard Greer, Lyman McDonald,
and Brian Miller for comments on
earlier versions of this manuscript.
The Department of Zoology and
Physiology, the University of Wyuming provided the computer time
used for the simulations reported
here.
Literature Cited
Burnham, Kenneth P. and William S.
Overton. 1969. A simula tion study
of live-trapping and estimation of
population size. Oregon State University, Department of Statistics,
Technical Report 14.40 p. and appendix.
Cochran, William G. 1977. Sampling
Techniques. John Wiley and Sons,
New York.
Dobson, F. Stephen and Julia D.
Kjelgaard. 1985. The influence of
food resources on population dynamics in Columbian ground
squirrels. Canadian Journal of Zoology 6332095-2104.
Fagerstone, Katheleen A. and Dean
E. Biggins. 1986. Comparison of
capture-recapture and visual
count indices of prairie dog densities in black-footed ferret habitat.
Great Basin Naturalist Memoirs
8:94-98.
McDonald, Lyman L. 1981. Summarizing remarks: Observer
variability, In Estimating numbers
of birds, Studies in Avian Biology
Number 6, p. 426-435. J.M. Scott
and C.J. Ralph, editors. The Cooper Ornithological Society.
Menkens, George E., Jr. 1987. Temporal and spatial variation in whitetailed prairie dog (Cynomys
leucurus) populations and life histories in Wyoming. Unpublished
Ph.D. Dissertation, University of
Wyoming, Laramie.
Menkens, George E., Jr. and Stanley
H. Anderson. in press. Estimating
small mammal population sizes.
Ecology.
Otis, David L., Kenneth h). Burnham,
Gary C. White, and David R. Anderson. 1978. Statistical inference
from capture data on closed animal populations. Wildlife Monographs 62:l-135.
Pollock, Kenneth H. 1981. Capturerecapture models: a review of current methods, assumptions and
experimental design, In Estimating
numbers of birds, Studies in Avian
Biology Number 6, p. 426-435. J.M.
Scott and C.J. Ralph, editors. The
Cooper Ornithological Society.
Ralph, C. John and J. Michael Scott
(eds). 1981. Estimating numbers of
birds, Studies in Avian Biology
Number 6, p. 426-435. The Cooper
Ornithological Ssciety.
Seber, G.A.F. 1982. The estimation of
animal. abundance. MacMillan
Publishing Company, New York.
Seber, G.A.F. 1986. A review of estimating animal abundance. Biometrics 42:267-292.
White, Gary C., David R. Anderson,
Kenneth P. Burnham, and David
L. Otis. 1982. Capture-recapture
and removal methods for sampling closed populations. Los
Alamos National Laboratory, LA8787-NERP.