NOREMARK SOFTWARE


                   NOREMARK: Population estimation from
                          mark-resighting surveys
                                     
                                     
                                     
                               Gary C. White
                                     
                                     
Author address: Department of Fishery and Wildlife, Colorado State University, Fort
Collins, CO 80523, USA.

Key Words: closure, Lincoln-Petersen, mark-resight, maximum likelihood, population
estimation, sighting surveys
                                     
	Estimation of population size of a geographically and demographically closed but
free-ranging population is a common problem encountered by wildlife biologists. The
earliest approaches to this problem were developed by Petersen in 1896 and later by
Lincoln in 1930, where capture-recapture techniques were applied. Extensions to the
simple 2 occasion Lincoln-Petersen estimator were developed for multiple occasions
(Schnabel 1938, Darroch 1958), for removal experiments (Zippin 1956, 1958), and for
heterogeneity of individual animals (Burnham and Overton 1978, 1979, Chao 1988).
For the capture-recapture technique, Otis et al. (1978) and White et al. (1982) 
summarized available methods, and others (White et al. 1978, Rexstad and Burnham 1991)
described the program CAPTURE for computing these estimators of population size.
	More technologically advanced approaches to abundance estimation have
incorporated animals marked with radio transmitters. The initial sample of animals is
captured and marked with radios, but recaptures of these animals are obtained by
observation, not actually recapturing them. The limitation of this procedure is that
unmarked animals are not marked on subsequent occasions. The advantage of this
procedure is that resightings are generally much cheaper to acquire than physically
capturing and handling the animals. The mark-resight procedure has been tested with
known populations of mule deer (Odocoileus hemionus; Bartmann et al. 1987) and
used with white-tailed deer (O. virginianus; Rice and Harder 1977), mountain sheep
(Ovis canadensis; Furlow et al. 1981, Neal et al. 1993), black bear (Ursus americanus)
and grizzly bear (U. arctos; Miller et al. 1987), and coyote (Canis latrans; Hein 1992).
Arnason et al. (1991) described a method in which the number of marked animals is not
known, whereas the mark-resight estimators described here assume the number of
marked animals is known.
	Program NOREMARK computes 4 mark-resight estimators of population abundance,
modeling variation of sighting probabilities across time, individual heterogeneity of
sighting probabilities, or immigration and emigration from a fixed study area (Eberhardt
1990). For all 4 estimators, the marked animals are assumed to have been drawn
randomly from the population (i.e., marked animals are a representative sample from
the population).

                       Joint hypergeometric maximum
                           likelihood estimator

	The first estimator in NOREMARK is the joint hypergeometric maximum likelihood
estimator (JHE; Bartmann et al. 1987, White and Garrott 1990, Neal 1990, Neal et al.
1993). This estimator assumes that each animal in the population has the same
sighting probability on an occasion as every other animal (no individual heterogeneity),
but sighting probabilities can vary across occasions. JHE is the value of N which
maximizes the joint hypergeometric likelihood for k occasions. The estimate N-hat can
be found by iterative numerical methods, and confidence intervals are determined with
the profile likelihood method (Hudson 1971, Venzon and Moolgavkar 1988). This
estimator assumes that all marked animals are on the area examined during each
survey (i.e., that the population is geographically closed). Hence, the number of marked
animals (M) is constant for each survey, although the sighting probability is not
assumed to be constant for each survey. Sighting probability is assumed to be the
same for all animals on any particular survey, and animals are assumed to be sampled
without replacement (i.e., each animal is observed > or =1 time on a survey). An extension
in NOREMARK allows additional animals to be marked between sighting occasions.

                       Immigration - emigration JHE

	The JHE estimator has been extended to accommodate immigration and emigration
(Neal et al. 1993) through a binomial process. This modified estimator, IEJHE, does not
assume a geographically closed population; rather, it assumes that the total population
with any chance of being observed on the study area is N-bar and that at the time of
the ith sighting survey, N(i) animals occur on the study area. I want to estimate the
mean number of animals on the study area, and possibly N*. At the time of the ith
sighting occasion, a known number of the marked animals (M[i]) are on the study area
of the possible T(i) animals with transmitters. The probability that an individual is on the
study area on the ith occasion can be estimated as M(i)/T(i), or in terms of the
parameters of interest as N(i)/N*. The likelihood function for this model that includes
temporary immigration and emigration from the study area is a product of the binomial
distribution for the probability that the animal is on the study area times the joint
hypergeometric likelihood. The parameters N* and N(i) for i = 1 to k can be estimated
by numerical iteration to maximize this likelihood, with the constraints that N(i) > M(i) +
u(i), where u(i) is the number of unmarked animals observed on occasion i, and N* >
N(i) for i =1 to k. Profile confidence intervals can be obtained for the k + 1 parameters. I
was not interested in the k population estimates for each sighting occasion, but rather
wanted the mean of the N(i) estimates. Therefore, I reparameterized the likelihood to
estimate the total population and mean population size on the study area directly and
their profile likelihood confidence intervals. The assumptions of this estimator are the
same as the JHE (i.e., sighting homogeneity and sampling without replacement).

                        Minta and Mangel estimator

	Minta and Mangel (1989) suggested a bootstrap estimator (MM) of population size
based on the sighting frequencies of the marked animals, f(i). The estimator does not
assume that sighting probabilities are the same for each animal on a particular
occasion, but does assume a closed population. This model assumes a sample drawn
with replacement, so that marked animals might be seen more than once on a survey.
For unmarked animals, sighting frequencies are drawn at random from the observed
sighting frequencies of the marked animals until the total number of sightings equals
the number of unmarked animal sightings. The number of animals sampled estimates
the number of unmarked animals in the population, so M plus the number sampled
estimates N. Only bootstrap samples where the number of sightings was exactly equal
to the number of unmarked animal sightings were used (i.e., cases where cumulative
sightings were >u were excluded). Minta and Mangel (1989) accepted the first value
where the cumulative sightings equaled or exceeded the number of unmarked animal
sightings. The stopping rule I used results in less bias than the rule used by Minta and
Mangel (1989). Minta and Mangel (1989) suggested the mode of the bootstrap
replicates as the population estimate. Confidence intervals were computed as
probability intervals with the 2.5th and 97.5th percentiles from the bootstrapped sample
of estimates. White (1993) demonstrated that the MM estimator is basically unbiased,
but that the confidence interval coverage was not the expected 95% for a = 0.05. A
modified procedure was suggested, but coverage still was not satisfactory.

                             Bowden's estimator

	Bowden (1993) suggested an estimator for the Minta-Mangel model where the
confidence intervals on the estimate were computed based on the variance of
resighting frequencies of marked animals. He approached the problem from a sampling
framework, where each animal in the population has sighting frequency f(i). Values of
f(i) are known for the marked animals, and the sum of the f(i)'s are known for the
unmarked animals. Bowden (1993) presented an unbiased estimator and its variance
and suggested that confidence intervals should be computed using a log
transformation. Animals are not assumed to have the same sighting probability on any
particular occasion, and the sample can be drawn with or without replacement.

                              Design options

	NOREMARK contains a design option to assist the user with determining the number
of resighting occasions, proportion of the population to mark, and proportion of the
population to resight on each occasion to achieve a specified level of precision. This
design routine uses simulated results from the JHE estimator.
	The 4 estimators also can be simulated with NOREMARK. Output from the simulations
includes expected bias, confidence interval length, and coverage.  
	Program NOREMARK is written to be used interactively, but with options to store data 
and to save results to a file or printer. The program provides context-sensitive help at 
any time and the ability to back up and re-enter or verify previous entries.

                      Program availability and system
                               requirements

	Copies of the program and related documentation are available on the Bird Monitor
Bulletin Board at (301) 498-0402 or via WWW at 
http://www.cnr.colostate.edu/~gwhite/software.html. NOREMARK is written in CA-Clipper
(user interface; Computer Associates International, Inc., Islandia, N.Y. ) and Microsoft
FORTRAN (numerical optimization procedures; Microsoft Corporation, Redmond,
Wash.) and runs on the MSDOS operating system for personal computers (PC). The
program (executable files and source code) is accompanied by an electronically stored
manual and by auxiliary files, including data files containing the mountain sheep
observations described by Neal et al. (1993).
	The system requirements are minimal: a PC with 640k of base memory and
approximately 1M of disk space will suffice. Simulation of estimators can be very time
consuming, particularly for the immigration-emigration estimator, so I recommend a
math coprocessor on a high-speed 80486 or Pentium machine.

	Acknowledgments. This work was partially supported by a contract from the Colorado
Division of Wildlife. Ideas to improve and debug the code were provided by W. Andelt,
D. Freddy, K. Burnham, D. Anderson, and the students in FW663. D. Bowden graciously provided 
the details of his estimator so that I could implement the procedure in NOREMARK.

                             Literature cited                     

ARNASON, A. N., C. J. SCHWARZ, AND J. M. GERRARD. 1991. Estimating closed
population size and number of marked animals from sighting data. J. Wildl. Manage.
55:716-730.

BARTMANN, R. M., G. C. WHITE, L. H. CARPENTER, AND R. A. GARROTT. 1987.
Aerial mark-recapture estimates of confined mule deer in pinyon-juniper woodland. J.
Wildl. Manage. 51 :41 -46.

BOWDEN, D. C. 1993. A simple technique for estimating population size. Dep. of Stat.,
Colorado State Univ., Fort Collins. 17pp.

BURNHAM, K. P., AND W. S. OVERTON. 1978. Estimation of the size of a closed
population when capture probabilities vary among animals. Biometrika 65:625-633.

BURNHAM, K. P., AND W. S . OVERTON . 1979. Robust estimation of population size
when capture probabilities vary among animals. Ecology 60:927-936.

CHAO, A. 1988. Estimating animal abundance with capture frequency data. J. Wildl.
Manage. 52:295-300.

DARROCH, J. N. 1958. The multiple recapture census: I. Estimation of a closed
population. Biometrika 45:343-359.

EBERHARDT, L. L. 1990. I Using radio-telemetry for mark-recapture studies with edge
effects. J. Appl. Ecol. 27:259-271.

FURLOW, R. C., M. HADERLIE, AND R. VAN DEN BERGE. 1981. Estimating a
bighorn sheep population by mark-recapture. Desert Bighorn Council Trans.
1981:31-33.

HEIN, E. W. 1992. Evaluations of coyote attractants and a density estimate on the
Rocky Mountain Arsenal. M.S. Thesis, Colorado State Univ., Fort Collins. 58pp.

HUDSON, D. J. 1971. Interval estimation from the likelihood function. J. Royal Stat.
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MILLER, S . D., E. F. BECKER, AND W. H. BALLARD. 1987. Black and brown bear
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MINTA, S., AND M. MANGEL. 1989. A simple population estimate based on simulation
for capture-recapture and capture-resight data. Ecology 70:1738-1751.

NEAL, A. K. 1990. Evaluation of mark resight population estimates using simulations
and field data from mountain sheep. M.S. Thesis, Colorado State Univ., Fort Collins.
198pp.

NEAL., A. K., G. C. WHITE, R. B. GILL, D. F. REED, AND J. H. OLTERMAN. 1993.
Evaluation of mark-resight model assumptions for estimating mountain sheep numbers.
J. Wildl. Manage. 57:436-450.

OTIS, D. L., K. P. BURNHAM, G. C. WHITE, AND D. R. ANDERSON. 1978. Statistical
inference from capture data on closed animal populations. Wildl. Monogr. 62. 135pp.

SCHNABEL., Z. E. 1938. Estimation of the size of animal populations by marking
experiments. U.S. Fish and Wildl. Serv. Fish. Bull. 69:191-203.

REXSTAD, E., AND K. BURNHAM. 1991. Users guide for interactive program
CAPTURE. Colo. Coop. Fish and Wildl. Res. Unit, Colo. State Univ., Fort Collins. 29pp.

RICE, W. R., AND J. D. HARDER. 1977. Application of multiple aerial sampling to a
mark-recapture census of white-tailed deer. J. Wildl. Manage. 41:197-206.

VENZON, D. J., AND S. H. MOOLGAVKAR. 1988. A method for computing
profile-likelihood based confidence intervals. Appl. Stat. 37:87-94.

WHITE, G. C. 1993. Evaluation of radio tagging marking and sighting estimators of
population size using Monte Carlo simulations. Pages 91-103 in J.-D. Lebreton and P.
M. North, eds. Marked individuals in the study of bird population, Birkhauser Verlag,
Basel, Switzerland.

WHITE, G. C., K. P. BURNHAM, D. L. OTIS, AND D. R. ANDERSON. 1978. User's
manual for program CAPTURE. Utah State Univ. Press, Logan. 40pp.

WHlTE, G. C., D. R. ANDERSON, K. P. BURNHAM, AND D. L. OTIS. 1982.
Capture-recapture and removal methods for sampling closed populations. Los Alamos
National Lab. LA-8787-NERP. Los Alamos, N. M. 235pp.

WHITE, G. C., AND R. A. GARROTT. 1990. Analysis of wildlife radio-tracking data.
Academic Press, New York, N.Y. 383pp.

ZIPPIN, C. 1956. An evaluation of the removal method of estimating animal
populations. Biometrics 12:163-169.

ZIPPIN, C. 1958. The removal method of population estimation. J. Wildl. Manage.
22:82-90.

Software Editor: Rexstad.