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by: Jordane Kemmer


Jordane Kemmer
GPA 3.79

K. Pollock

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K. Pollock
Class Notes
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This 33 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 506 at North Carolina State University taught by K. Pollock in Fall. Since its upload, it has received 25 views. For similar materials see /class/223941/st-506-north-carolina-state-university in Statistics at North Carolina State University.




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Date Created: 10/15/15
LECTURE 8 CLOSED CAPTURERECAPTURE MODELS More on Closed Models Today OverVieW of Models Review Cell Structures for Simple Models More on Heterogeneity Models CATURE amp MARK LincolnPetersen example More general examples Table 31 Capturerecapture models for closed populations that allow for unequal capture probabilities Mono 0 Source of variation CS capture Modelquot Heterogeneity Mo Mh Xa Mb Mbh X M Mh X Mb Mtbh X Trap response X X Time gtltgtltgtlt X probability Estimator availability yes yes yes yes yes yes yes 1 10 gtl This set of 8 models comes om Otis et al 1978 aXs denote the sources of variation in capture probability incorporated in the models 96 2 M0 THE EQUAL CATCHABILITY MODEL The simplest model but usually unrealistic There are two parameters in this model N the population size P the probability of capture which is constant over all animals over all periods ML Estimators found iteratively using the programs CAPTURE or MARK Estimators can be highly biased if heterogeneity or trap response is occurring Variation in capture probabilities due to time are less troublesome 3 Mb TRAP RESPONSE MODEL This model makes the following assumptions 1 Every unmarked animal in the population has the same probability of capture p for all samples 2 Every marked animal in the population has the same probability of recapture c for all samples after it has been captured once There are three parameters in the model N the population size p the probability of capture for unmarked c the probability of capture for marked ML estimators found iteratively using programs CAPTURE or MARK 6 Mt THE TIME MODEL This is the traditional Schnabel model that only allows for time variation in capture probabilities The parameters in the model are N the population size p1 p2 pk the unmarked capture probability of all animals in each sample Programs CAPTURE or MARK provides the MLEs of N and the ps These estimators are not robust to heterogeneity and trap response Lets stop for a minute and make sure we understand the cell structures Mo Mt Mb I will write down the capture histories and cell structures of expected values for each model on the board The cell structures will de ne the likelihoods in MARK or CAPTURE 4 THE HETEROGENEITY MODEL Thismodel allows capture probabilities to Vary by animal due to heterogeneity but there is no trap response or time variation The parameters inthe model are N thepopulatiOZn size pj thecapture probability of animal j for j l N pj s areassumed to come from distribution Fp otherwise the model is overparameterized I Estimators include 39 Burnham s J ackknife39 Lee and ChaOquots Coverage Estimator Norris nonparametric MLE Mixture Model Burnham s estimator is widely used but it has a questionable theoretical basis It is given by program CAPTURE x MORE Thismod Ol 61 all I THE HETEROGENEIT ows icaptdre probabilities to Vary heterog The paramete N the melt popi capu pj s are rs inthe model are llatioin siz JI39C probal assumed to 00 model IS overparamet C ility erize Some of What I saylwill also of a but there is no trap IGSpOnSIe or t iima jforj me from d appb distri I to t bution F 1 and 39b39h r Yl by air ime 1 P 01 node 9 AODEL limal due to I variation herwise the ls as well 1 Q I I i 5 Mbh THE HETEROGENEITY AND TRAP RESPONSE MODEL This model allows capture probabilities to vary due to heterogeneity and trap response but not time The parameters in the model are N the population size pj the unmarked capture probability for the jth animal cj the marked capture probability for the jth animal The pj cj are assumed to come from some bivariate distribution Gpc The Heterogeneity And Trap Response Model continued Program CAPTURE provides a generalized removal estimator Basically the rst second third samp1es are ignored consecutively to try to reduce the in uence of the heterogeneity Unfortunately the precision of the estimators gets worse as more samples are ignored An alternative jackknife estimator given by Pollock and Otto 1983 is also in CAPTURE Norris has provided a nonparametric MLE of N and G 7 OTHER TIME DEPENDENT MODELS 39 Model Mtb now has estimator available in CAPTURE 39 Model Mth now has estimator available in CAPTURE Model Mtbh is only a conceptual model and has no estimators of N available Reminder of Last Lecture Example 9 EXAMPLE Meadow vole study by James Nichols Five sampling periods Traps prebaited with corn Will show Model Selection Output Will show Model Mh The Heterogeneity Model output because it was the chosen model Precision of the estimator is quite good because of the high capture probabilities MODEL SELECTION CAPTURE There is an old procedure in CAPTURE which is quite complex also it only works well if data are very good ie high capture probabilities The method is based on a whole series of tests which are summarised into one overall criteria between 0 and 1 39Reduce the number of models to be chosen from if possible Sometimes there may be biological reasons to eliminate some models eg trap response We will see this in the taXi cab example MARK If one is using ML models in MARK then one can use AIC methods to chose among models Table 33 Model selection procedure from program CAPTURE for the meadow vole data collected by JD Nichols at Patuxent Wildlife Research Center Laurel Maryland in October 1981 Model M0 Mh Mb Mbh Mt Mth Mtb Mtbh Criteria 080 100 038 059 000 032 052 098 This suggests one should use the Mh estimator although there is some evidence of trap response and time being present as well In the interests of simplicity and getting an estimate we need to use Mh estimator Table 34 Selected statistics and parameter estimated from program CAPTURE for meadow vole data collected at Patuxent Wildlife Research Center Laurel Maryland in October 1981 by JD Nichols Model Mh the heterogeneity model is used Frequencies of capturea 139 1 2 3 4 5 F I 29 1 5 1 5 16 27 Number of animals captured 102b Average PHAT 044 Interpolated population estimate 139 with Standard Error 1085 Approximate 95 Con dence Interval from 177 to 161 3 These are the numbers of animals caught from 1 to 5 times bThis is the number of distinct animals captured at least once p 044 is a very high probability 44 of animals were captured on each occasion MANY COMPUTER PROGRAMS MARK CAPTURE JOLLY J OLLYAGE POPAN RELEASE SURVIV MS SURVIV RDSURVIV TMSURVIV CAPTURE Classic closed population models of Otis et al 197 8 but with just some updating for new estimators Contains a model selection procedure not V good Can be run on the web from Patuxent software archive Copy of program can also be downloaded from Patuxent software archive Can also run from MARK MARK User friendly windows based program for capture recapturetelemetry and band return models Many options under statement Can run CAPTURE and POPAN from MARK Uses AIC for model selection Allows multiple groups age classes multistate extension covariates Can download from their web siteCan also download an online book and other resources httpwwwphidotorgso waremarkdocsbook USE of CAPTURE on the Web httpwwwmbrpwrcusgs gov software I will demonstrate how to do this in class today See hardcopy handout from class on the input formats and output for three examples Rabbit dataDarroch Mt Microtus dataJacknife Mh Removal dataZippen Related to MbI will discuss later Use of CAPTURE from MARK I will demonstrate this in Class using the taXi cab data as an example You have to access the output in an unusual way by going to the tests tab in a MARK output Window It will be more natural to look at MARK rst Use of MARK For Closed Models Starting Today and continuing Tuesday I will demonstrate MARK for closed population capturerecapture models First for the rabbit data 2 periods LP and then for the taXi cab data 10 periods I will emphasize the input format and syntax I will show you how to use the parameter index matrices PIMS to create specific models to runMO Mt Mb I will show you how the AIC is used to compare models and select the best one I will show you how to look at the output les for a chosen model I will show how to switch to looking at CAPTURE output when you are in a MARK analysis if you want to use both at once For Tuesday I will try and compile some detailed summary notes on some key points of using the program but you will also be learning by trying it Taxi Cab Example from Edinburough Carrothers1973 CAN ACCESS CAPTURE FROM MARK CAN USE MARK DIRECTLY Closed Population N420 k10 occasions on 10 days close together No trap response Constant sampling effort so perhaps no time variation either Heterogeneity likely Model selection criteria Model selected has maximum value Model Mo Mh Mb Mbh Mt Mth Mtb Mtbh Criteria 091 100 045 061 000 051 039 06 Appropriate model probably is Mh Suggested estimator is J ackknife or Chao Estimator for Mh Model Mh Suggested for use here J acknife 471 with standard error 3632 Chao 407 with standard error 2742 Finite Mixture approach did not work here Huge SE Model MO Null Model not to be used MLE 368 with standard error 144896 Always underestimates when there is heterogeneity Closed CaptureRecapture Analysis and Use of Programs Usually best to Use MARK 1 Use CAPTURE from MARK Old program but allows heterogeneity models to be fit in one analysisDoes not use AIC and does not do multiple groups Use MARK closed options directly good interface AIC multiple groups Standard Closed Captures the non heterogeneity models Huggins model covariates approach to tting heterogeneity models not shown in class but related to logistic regression Norris and Pollock Pledger fmite mixture models for heterogeneity Summary Slides Will revisit Summary Closed CaptureRecapture Design Issues Precision Issues 0 Need adequate capture probabilities and numbers of samples to estimate standard errors that are small enough ie RSE 20 0 Look at Tables in Otis et al 1978Note that good model selection requires much larger capture probs than just estimation under one assumed correct model 0 Full Simulation Study 0 Use Expected Values for guesses of What the data might be like and do analysis on that data using MARK or CAPTURE Summary Closed CaptureRecapture Design Issues Minimise Model Bias Satisfy Assumptions 1 Closure Short studiesno mortality no recruitment no immigration or emigration Check With telemetry sometimes 2 Equal Catchability Heterogeneityoften hard to avoid unless one can use different methods of capture in each sample Which is not usually feasible Rerandomise trap locations each time Collect covariate data for Huggins method or to stratify on Trap Response often hard to avoid unless one can use different methods of capture in each sample Which is not usually feasible Time Variationtry to eliminate so that simpler models can be used 3 No Tag Loss Obviously avoid check out in pilot studies Use double tagging method to estimate tag loss if it is a problem Summary Closed CaptureRecapture Programs CAPTURE old but still useful run from web or inside MARK Which is preferred depends on the situation to access from MARK look under the tests tab illogical but that s What they did MARK Very powerful but complex to learn to use There are many procedures and we will only cover a few Key issues input format PIMS and their manipulation Key References Williams et al 2002 Analysis and Management of Vertebrate Populations Academic Press Amstrup et al 2005 Handbook of Capture Recapture Methods Princeton University Press Pollock et al 1990 Statistical Inference for CaptureRecapture Models Wildlife Society Monograph pdf available Old but still useful for the basics


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