JHE, Bowden`s estimator - Warner College of Natural Resources
Contents
1. 1990 Using radio telemetry for mark recapture studies with edge effects Journal of Applied Ecology 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 Transactions 1981 31 33 Hudson D J 1971 Interval estimation from the likelihood function Journal of the Royal Statistical Society Series B 33 256 262 Leslie D M Jr and C L Douglas 1979 Desert bighorn sheep of the River Mountains Nevada Wildlife Monographs 66 1 56 Leslie D M Jr and C L Douglas 1986 Modeling demographics of bighorn sheep current abilities and missing links North American Wildlife and Natural Resources Conference Transactions 51 62 73 Miller S D E F Becker and W H Ballard 1987 Black and brown bear density estimates using modified capture recapture techniques in Alaska International Conference on Bear Research and Management 7 23 35 Minta S and M Mangel 1989 A simple population estimate based on simulation for capture recapture and capture resight data Ecology 70 1738 1751 FW663 Mark Resight Analysis with Program NOREMARK Page 9 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 for2. 1990 Neal 1990 Neal et al 1993 JHE is the value of N which maximizes the following likelihood M k m J n m QN M n m gt i R and the terms are defined for all i 1 to k sighting occasions The estimate can be found by iterative numerical methods Confidence intervals are determined with the profile likelihood method Hudson 1971 Venzon and Moolgavkar 1988 This estimator assumes that all the marked animals are on the area surveyed for 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 bees Second the JHE estimator has been extended to accommodate immigration and emigration Neal et al 1993 through a binomial process This estimator is labeled IEJHE and does not assume that the population is geographically closed Assume that the total population with any chance of being observed on the study area is N and that at the time of the i sighting survey N animals occur on the study area I am interested in estimating the mean number of animals on the study area and possibly N At the time of the i sighting occasion a known number of the marked animals M are on the study area of the possible T animals with transmitters The probability that an individual is on the study area on the 7 occasion can be FW663 Mark Resi
3. estimating mountain sheep numbers Journal of Wildlife Management 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 Wildlife Monographs 62 1 135 Rice W R and J D Harder 1977 Application of multiple aerial sampling to a mark recapture census of white tailed deer Journal of Wildlife Management 41 197 206 Venzon D J and Moolgavkar S H 1988 A method for computing profile likelihood based confidence intervals Applied Statistics 37 87 94 White 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 Laboratory 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 New York USA 383pp 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 Berlin
4. seen 10 9 3 6 7 5 5 12 2 5 8 9 8 3 12 10 4 9 4 3 6 6 6 3 and 7 There were a total of 615 unmarked sheep observed Enter these data in Program NOREMARK You will note that the program allows you to retrieve data with the F2 function key once the data entry screen appears In the subdirectory J CLASSES FW663 EXERCISE 22 are a number of examples you can try including the 2 examples you have just executed Files with the suffix HYP are for the hypergeometric estimators both closed and open models whereas files with the suffix MM are for the Minta Mangel and Bowden estimators For the lab today just explore the closed JHE and Bowden estimators As shown by White 1993 the Minta Mangel estimator does not perform properly We will work with the open population models later After you understand the input for these 2 models try exploring the design and simulation capabilities of the program A complete user s manual formatted in Word Perfect 6 1 is provided in the file J CLASSES FW663 EXERCISE 22 USERMAN WP Questions to Consider 1 How many marked sheep with 14 resighting occasions would Andrea Neal have needed to obtain a 95 confidence interval of 10 for the JHE For Bowden s estimator 2 How many resighting occasions with 25 marked sheep would Andrea Neal have needed to obtain a 95 confidence interval of 10 for the JHE For Bowden s estimator 3 What are the different assumptions of the JHE and Bow
5. FW663 Laboratory Exercise Mark Resight Analysis with Program NOREMARK Technologically advanced approaches to the problem of 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 only observing them 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 resighting occasions are generally much cheaper to acquire than when the animals must be physically captured and handled The mark resight procedure has been tested with known populations of mule deer Bartmann et al 1987 and used with white tailed deer Rice and Harder 1977 mountain sheep Furlow et al 1981 Neal et al 1993 black and grizzly bears Miller et al 1987 coyotes Hein 1993 and moose Bowden and Kufeld 1995 The objective of this exercise is to explore program NOREMARK a program to compute mark resight estimators of population abundance Three estimators of abundance are provided the joint hypergeometric maximum likelihood estimator JHE Bartmann et al 1987 White and Garrott 1990 Neal et al 1993 the joint hypergeometric maximum likelihood estimator extended to incorporate animals moving on and off the study area Neal et al 1993 White 1993 the Minta Mangel estimator Minta and Mangel 1989 and Bowden s estimator
6. ate Population Size for No Immigration Emigration with JHE provides additional detail about each of the choices and changes as you move the cursor to different selections from the 7 choices The function key F1 is the help function Context sensitive help can be obtained any time the program is waiting for input by hitting the F1 key A window will pop up on the screen displaying information about the program s request To exit this window hit the Esc key which is equivalent to selecting the Exit option from the menu choices In general the Esc key always takes you back one window to a previous menu The first choice is JHE Closed Population Model Estimation and provides the capability to compute the JHE estimator for a set of data Equivalent functions are provided with the next 3 options for computing the Immigration Emigration JHE Minta Mangel and Bowden estimators The fourth option Design Experiment Using Interpolation for Closed Population allows the user FW663 Mark Resight Analysis with Program NOREMARK Page 6 to quickly design an experiment using the simulations results from Neal 1990 These simulations were summarized in a dBase file and this option allows the user to interpolate the simulations to provide an idea of confidence interval length for a particular set of parameter values The next 4 choices in the main menu allow the user to simulate experiments These options are useful in the design of experiments that requ
7. den s estimators Which model seems more realistic for resighting mountain sheep from a helicopter 4 One scenario proposed for use of mark resight estimators is use with radio collared bears Resightings of bears takes place at bait stations where motion and infrared sensitive cameras photograph the bears What kinds of problems would arise with such surveys Pertinent Literature Arnason A N Schwarz C J and Gerrard J M 1991 Estimating closed population size and number of marked animals from sighting data Journal of Wildlife Management 55 716 730 FW663 Mark Resight Analysis with Program NOREMARK Page 8 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 Journal of Wildlife Management 51 41 46 Bowden D C and R C Kufeld 1995 Generalized mark sight population size estimation applied to Colorado moose Journal of Wildlife Management 59 840 851 Buckland S T and P H Garthwaite 1990 Algorithm AS 259 Estimation confidence intervals by the Robbins Monro search process Applied Statistics 39 413 424 Chao A 1988 Estimating animal abundance with capture frequency data Journal of Wildlife Management 52 295 300 Chapman D G 1951 Some properties of the hypergeometric distribution with applications to zoological sample censuses University of California Publication in Statistics 1 131 160 Eberhardt L L
8. for the Minta Mangel model Bowden and Kufeld 1995 First a section defining the mathematical notation will be presented followed by a section describing the details of each of the estimators Then the startup of Program NOREMARK will be shown with annotated examples Notation T Number of marked telemetered animals in the population at the time of the i survey i 1 4 When the number of marked animals is assumed constant across surveys the value is denoted as T M Number of marked animals in the population that are on the area surveyed at the time of the i sighting survey For all M constant define M M n Number of animals seen during the i sighting survey consisting of m marked animals and u unmarked animals so that n m u Number of times marked animal j was observed during the k surveys sighting frequencies k T FW663 Mark Resight Analysis with Program NOREMARK Page 2 m Total number of sightings of marked animals seen so that m m Yf u Total number of sightings of unmarked animals seen so that u u Mean capture frequency of marked animals m T sf Variance of the sighting frequencies of the marked animals T De Ie a es T Estimators Four estimators of population size for marking and sighting experiments are provided in Program NOREMARK First is the joint hypergeometric maximum likelihood estimator JHE Bartmann et al 1987 White and Garrott
9. ght Analysis with Program NOREMARK Page 3 estimated as M T or in terms of the parameters of interest as N N Then the likelihood for the model that includes immigration and emigration is a product of the binomial distribution for emigration immigration times the joint hypergeometric likelihood of Eq 1 M i M k T N N 5 i m n m A N N IT M m n aah eee i i i i1 M N N N n The parameters N and N for i 1 to k can be estimated by numerical iteration to maximize this likelihood with the constraints that N gt M u and N gt N 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 desired the mean of the N estimates Therefore I re parameterized the likelihood to estimate the total population and mean population size on the study area directly and their profile likelihood confidence intervals In the re parameterized likelihood I used N where a 0 Third Minta and Mangel 1989 suggested a bootstrap estimator MM of population size based on the sighting frequencies of the marked animals f For unmarked animals sighting frequencies are drawn at random from the observed sighting frequencies of the marked animals until the total number of captures equals u The number of animals sampled is then an estimate of the number of unmarked a
10. ion size is 2 S u E ec fe Ss with variance 2 A S E Varin AAS Program NOREMARK To run NOREMARK copy the files in the subdirectory J CLASSES FW663 NOREMARK to your local subdirectory The main reason for doing this is to have the help files readily available to the program Program NOREMARK is executed with the command NOREMARK typed at the DOS prompt The following menu appears FW663 Mark Resight Analysis with Program NOREMARK Page 5 01 05 94 Memory Available 141 10 47 57 DESIGN AND ANALYSIS OF MARK RESIGHT EXPERIMENTS Closed Po Estimation gration E el Estimation ta Mange t ion en s Mode gn Experi t terpolation for Closed Population E Closed Po t Simu ation igration Emi Simulation ta Mangel Esti i ation Bowden s Estimat tion Estimate Po ti i f o Immigration Emigration with JHE EXIT The ESC key takes you back one menu screen You can get context sensitive help with the F1 key Nine options are presented for the user to select from To select a choice the user can either specify the first letter of the choice or use the arrow keys to move the cursor to the choice After the cursor is positioned on the desired choice the user hits the Enter key to execute the option For computers with a mouse the mouse cursor can be moved to the desired option and double clicked to select the choice The line Estim
11. ire parameter values outside the ranges simulated by Neal 1990 This introduction should be enough to get you started using Program REMARK In general the program is all menu driven with additional options available via function keys When you are being asked for input and don t understand what is being asked try the F1 key to obtain additional information For your first problem try entering the following data from the keyboard Andrea Neal a former graduate student at CSU collected the following mark resight data on mountain sheep Neal et al 1993 Twenty five sheep were marked with radio collars Fourteen resighting surveys were conducted For each of these surveys the following data were observed Total Marks Total Marked Unmarked Marked Available Seen Seen Seen 25 22 40 9 31 25 21 63 11 52 25 21 66 14 52 25 20 57 11 46 25 20 52 10 42 25 20 61 9 52 25 22 87 19 68 25 18 46 8 38 25 23 63 13 50 25 18 30 8 22 25 20 69 14 55 25 17 35 6 29 25 20 51 13 38 25 21 57 17 40 Run these data in Program NOREMARK with the No Immigration Emigration Joint Hypergeometric Estimator To run the Bowden estimator that allows heterogeneity of resighting probabilities among individuals you must enter the number of times each marked animal was observed plus the total number of unmarked animals observed The following is the number of times that each of Andrea s FW663 Mark Resight Analysis with Program NOREMARK Page 7 25 marked sheep were
12. nimals in the population so that M plus the number sampled is an estimator for N Only samplings where the number of sightings was equal to u were used i e cases where the cumulative sightings exceeded u were rejected Minta and Mangel 1989 accepted the first value where the cumulative sightings equalled or exceeded u 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 5 and 97 5 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 at the expected 95 for a 0 05 He suggested a modified procedure but coverage again was not adequate Fourth Bowden and Kufeld 1995 suggested an estimator for the Minta Mangel model where the confidence intervals on the estimate were computed based on the variance of the resighting frequencies of the marked animals He approached the problem from a sampling FW663 Mark Resight Analysis with Program NOREMARK Page 4 framework where each animal in the population has the attribute f of the number of times it was resighted The values of f are known for the marked animals and the sum of the fs u are known for the unmarked animals Then an unbiased estimator of the populat
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