Brief User Guide for Program CARE-3: Analyzing Continuous
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1. 0 981 0 110 The estimates of Mtb model converged 0 yes 0 000 N and se N 172 195 1 681 phi and se phi 1 000 0 139 The estimates of Mth model Note The first reg coefficient is an intercept beta_O In landa_tau converged 0 yes 0 000 N and se N 179 221 3 259 reg coefficients 1 091 0 175 0 293 standard error 0 089 0 091 0 094 The estimates of Mbh model Note The first reg coefficient is an intercept beta_0 is the estimate of the intercept and the last coefficient is the behavioral response effect theta_ p 2 In phi converged 0 yes 0 000 N and se N 178 776 3 877 reg coefficients 1 193 0 175 0 293 0 022 standard error 0 128 0 091 0 094 0 113 Note The first reg coefficient is an intercept theta_0 ln landa_tau and the last coefficent is the behavioral response effect theta_ p 2 ln phi converged 0 yes 0 000 N and se N 206 009 15 340 reg coefficients 0 497 0 155 0 329 0 687 standard error 0 226 0 074 0 080 0 258 We interpret the results for the most complicated model Min The model assume an intensity function t 2 t exp Z Z Za aX t where Z 1 the ith individual is a male Z Z the ith individual is an adult and X 7 J the ith individual has been captured in 0 f The parameter estimates are 0 155 s e 0 08 B 0 329 s e 0 08 and amp 0 687 s e 0 253 and all co2. The behavioral response is not significantly different from 0 thus a proper model would be model Min Based on the above output under model Mn we have B 0 643 s e 0 144 B 0 072 s e 0 034 and both coefficients are significantly different from 0 Hence it implies that males are more catchable and the capture intensity decreases with weight The resulting population size estimate is 197 with an estimated s e of 9 46 based on an asymptotic formula given in Hwang and Chao 2002 Example 2 Discrete Type Data Data is stored in the file c coul dat The running procedures are similar to those in Example 1 except that Steps c and d are changed to the following a Please input the type of data 1 for continuous 2 for discrete 2 b Please input the number of distinct individuals 171 c Please input the number of occasions 10 d Please input the number of individual covariates 72 e Please input the filename containing individual covariates and the capture times c coul dat f Please input the number of bootstrap for Mtbh 200 g Please input the filename to save the output c test2 out The above specification corresponds to the following setting in the source program datatype 2 r 171 c tau 10 p 2 and nb 200 The output is shown below also saved in c test2 out converged 0 yes 0 000 N and se N 172 198 1 931 converged 0 yes 0 000 N and se N 176 833 3 388 phi and se phi
3. baseline cumulative intensity function by A fe u du gt te 0 7 Also let the conditional MLE be denoted as BGAI BB then we subsequently get 7 exp B Z The proposed Horvitz Thompson type of population size estimator in Hwang and Chao 2002 is A M A v I 8 1 1 exp 7 A X 1 1 exp f 4A 1 i l Program CARE 3 calculates the above population size estimate and its s e as well as the regression coefficients i e estimate of 2 f f 8 for the above four models in Table 1 asymptotic method except for model Mipn suggested a bootstrap procedure to obtain s e 4 RUNNING PROCEDURES BY EXAMPLES 4 1 Models With Covariates The estimated s e for all estimates are obtained by an For model Men Hwang and Chao 2002 See the above paper for details Example 1 Continuous Type Data Data is stored in the file c simeg dat We describe the procedures for analyzing the data in the file simeg dat All procedures must be executed in a GAUSS environment 1 Provoke GAUSS environment either by doubly clicking GSRUNSO on your desktop as described earlier or by clicking the executable file GSRUN exe stored in the directory GSRUNSO 2 Click File on the top menu of GAUSS and subsequently click Run Program and select the program CARE 3 gcg which is stored in a pre specified working directory The default is c program files CARE 3 It prompts you subsequently t
4. for citing CARE 3 Hwang W H and Chao A 2003 Program CARE 3 CApture REcapture Part 3 Program and User s Guide published at http chao stat nthu edu tw 1 INTRODUCTION We first distinguish continuous time and discrete time capture recapture experiments 1 Continuous Time The population is sampled over a fixed time interval 0 7 For each animal captured in the experiment the complete capture history consists of a series of capture times As an example an individual s capture history 1 4 6 5 8 9 means that the animal was caught in time unit of 1 4 6 5 8 and 9 The capture time can be any number in the interval 0 7 but an animal may be captured many times 2 Discrete Time The target population is sampled over T occasions or samples The complete capture history for each animal is expressed as a sequence of 0 s and 1 s where 0 denotes absence and 1 denotes presence For example in a five occasion capture recapture experiment each animal can be counted at most five times a history 0 1 0 0 1 means that the animal was caught in the second and fifth occasions but not in the others The maximum frequency for each animal is the number of occasions Program CARE 2 focuses on analyzing this type of data If the number of occasions is large this type of data can also be regarded as continuous type and the possible captures times are the integers 1 2 T For example in the case of T 10 a his
5. of practice Let A t be any arbitrary non negative time varying function defined in 0 z The covariates are used to model individual heterogeneity Let B f f 8 be a vector of unknown regression coefficients We use A t exp 8 Z and to model respectively the time heterogeneity and the behavioural response effects Thus a multiplicative type of model Mipn is A t A t exp Z until first capture A t exp P Z for any recapture The continuous time models featured in CARE 3 are summarized in Table 1 Table 1 Continuous Time Models Featured in CARE 3 Model Assumption Restriction in model Mwn A t exp B Z until first capture Moh A t FF PA t exp B Z for any recapture Le set t A in model M i TE A exp f Z until first capture Ai ii n i gA exp p Z for any recapture Man 4 t A t exp B Z i e set 1 in model Mun Mi yae A t until first capture R Canmore Ma Aa A t for any recapture A until first capture i e B 0 4 t Ain model Min My A t QA for any recapture M A t A t i e set o f B 0 in model Men Let exp q and X t I the ith animal has been captured in 0 denotes the prior capture history where Z is the usual indicator function For the most general model Mon the intensity of the ith individual can be rewritten as A t Ay thexp B Z aX 1 Let n exp Z and denote the
6. similar to those in Example 2 except for the following changes d Please input the number of individual covariates 0 e Please input the filename containing individual covariates and the capture times c coul2 dat Please input the number of bootstrap for Mtbh 0 g Please input the filename to save the output c test4 out The above specification corresponds to the following setting in the source program datatype 2 r 171 c tau 10 p 0 and nb 0 The output is shown below converged 0 yes 0 000 N and se N 172 198 1 931 converged 0 yes 0 000 N and se N 176 833 3 388 phi and se phi 0 981 0 110 converged 0 yes 0 000 N and se N 172 195 1 681 phi and se phi 1 000 0 139
7. September 2003 Brief User Guide for Program CARE 3 Analyzing Continuous Time Capture Recapture Data Wen Han Hwang and Anne Chao Program CARE 3 for CApture REcapture is written in GAUSS language and the program calculates population size estimates for various closed continuous time capture recapture models with without covariates Relevant covariates such as environmental variables or individuals characteristics can be incorporated into the models to assess the effect of each covariate on the capture probabilities Program CARE 3 can be downloaded from Anne Chao s website at http chao stat nthu edu tw softwareCE html The source files along with illustrative data sets will be stored automatically in a specified directory in your computer You do not need to purchase GAUSS software to run this program A working environment of Gauss is provided by the following procedures First doubly click the downloaded file care 3 exe to unzip all files to the specified folder Then doubly click the GRTM exe to unzip all files of the Gauss Run Time Module GRTM which is GUASS free ware for non commercial redistribution Then doubly click the setup exe to install the GRTM The GRTM allows licensee to redistribute licensee s compiled GAUSS programs free of charge to other users who do not have GAUSS so long as licensee s GAUSS program is distributed free of charge Then restart your computer after completing the installat
8. efficients are significant Hence it implies that females are more catchable and adults have higher intensity than do non adults The resulting population size estimate is 206 s e 14 2 based on 200 bootstrap replications 4 2 Models Without Covariates Example 3 Continuous Type Data Data is stored in the file c simeg2 dat The procedures are similar to those in Example 1 except for the following changes e Please input the number of individual covariates 20 f Please input the filename containing individual covariates and the capture times c simeg2 dat g Please input the number of bootstrap replications for obtaining s e under model Mtbh 20 h Please input the filename to save the output c test3 out Then wait for a while for executing the program the output will be shown in the output window in GAUSS and also will be saved in the designated file The above specification corresponds to the following setting in the source program datatype 1 r 161 c 7 p 0 tau 2 and nb 0 The results in output window are shown below also saved in c test3 out converged 0 yes 0 000 N and se N 185 560 6 418 converged 0 yes 0 000 N and se N 212 078 21 194 phi and se phi 1 514 0 324 The estimates of Mtb model converged 0 yes 0 000 N and se N 183 657 12 535 phi and se phi 0 960 0 236 Example 4 Discrete Type Data Data is stored in the file c coul2 dat The procedures are
9. he following input steps 3 Then proceed the following steps a Please input the type of data 1 for continuous 2 for discrete 21 b Please input the number of distinct individuals 161 c Please input the maximum frequency 7 d Please input the end time of study period 2 e Please input the number of individual covariates 72 f Please input the filename containing individual covariates and the capture times c simeg dat g Please input the number of bootstrap replications for obtaining s e under model Mtbh 200 Note This step specifies the number of bootstrap replications for calculating variance estimator for model tbh A suggested number of bootstrap replications is 500 Here we use 200 is just for demonstration You may specify a larger number but it will take longer execution time h Please input the filename to save the output c test out Then wait for a while for executing the program the output will be shown in the output window in GAUSS and also will be saved in the designated file The above specification corresponds to the following setting in the source program datatype 1 r 161 c 7 p 2 tau 2 and nb 200 The results in output window are shown below also saved in c test out converged 0 yes 0 000 N and se N 185 560 6 418 converged 0 yes 0 000 N and se N 212 078 21 194 phi and se phi 1 514 0 324 The estimates of Mtb model converged 0 ye
10. ion of GRTM As the computer restarts in Window interface doubly click the icon GSRUNSO on the desktop of your computer to initialize Gauss Run Time Module and then the interface is shown below Figure The GAUSS Working Environment NOTERTE Commanl Inpai Ontpni Fie Eit Yer Goute Bw Debug Dob Wini Help D k bA S E R CARIE 2 pgm CAGSALINED E Command Input Ontpat All the necessary background and estimation methodologies are provided in the following paper Hwang W H and Chao A 2002 Continuous time capture recapture models with covariates Statistica Sinica 12 1115 1131 For models without covariates more details are given in the following paper Hwang W H Chao A and Yip P S F 2002 Continuous time capture recapture models with time variation and behavioral response Australian and New Zealand Journal of Statistics 44 41 54 When you download the GAUSS source program from the website please note that we have also distributed four illustrative data sets the two data files simeg dat and coul dat contain covariates whereas simeg2 dat and coul2 dat do not include covariates that are used to demonstrate the data input format and the running procedure in this guide As long as you will not distribute CARE 3 in any commercial form you are welcome to use CARE 3 for your own research and applications If you publish your work based on the results from CARE 3 please use the following reference
11. o the other records When covariates are not recorded or considered the data input format is similar except that there are no columns for covariates In the distributed data files simeg2 dat and coul2 dat contain identical capture records as those in simeg dat and coul dat but the covariates are dropped Remark The data file must be saved as an askii file If your data were originally processed by EXCEL please re save it as prn file using blank as separation of numbers CARE 3 will not work if you re save it as txt using tab as separation file 3 MODELS AND ESTIMATES FEATURED IN CARE 3 Assume there are N individuals indexed by 1 2 N Also assume that the experiment period is relatively short so that the population size remains fixed in the study period Suppose that the experiment terminates at the time 7 and N t denotes the number of times the ith animal has been caught in 0 t Each N t 0 lt t lt 7 is a continuous time counting process with intensity 2 7 The intensity for the ith animal 4 t is A t dt P dN t 1 F_ where F is the capture history generated by N u N u N u O lt u lt t Let the associated covariates for the ith individual be Z Z Z Z where p denotes the number of covariates In program CARE 3 we only handle the estimation procedure for time independent covariates because an experiment s duration is usually short for a closed model as a matter
12. s 0 000 N and se N 183 657 12 535 phi and se phi 0 960 0 236 The estimates of Mth model Note The first reg coefficient is an intercept beta_O In landa_tau converged 0 yes 0 000 N and se N 197 201 9 463 reg coefficients 1 690 0 643 0 072 standard error 0 689 0 144 0 034 Note The first reg coefficient is an intercept beta_0 is the estimate of the intercept and the last coefficient is the behavioral response effect theta_ p 2 In phi converged 0 yes 0 000 N and se N 239 685 30 749 reg coefficients 0 564 0 693 0 073 0 495 standard error 0 745 0 151 0 036 0 223 Note The first reg coefficient is an intercept theta_O In landa_tau and the last coefficient is the behavioral response effect theta_ p 2 In phi converged 0 yes 0 000 N and se N 205 824 31 970 reg coefficients 1 582 0 658 0 072 0 127 standard error 0 694 0 154 0 033 0 323 To help the user interpret the results we discussed the numerical outputs for two models Min and Ma Let Z denote the covariate gender and let Zz denote the covariate weight The intensity function for the most general model Min was assumed to be A t A t exp Z Z 2 Z aX t where Z J the ith individual is a male Z weight and X t Z the ith individual has been captured in 0 The parameter estimates under model M n are B 0 658 s e 0 15 B 9 0 072 s e 0 032 and 0 127 s e 0 316
13. tory 0 1 00 101 1 0 1 for which the individual was caught on occasions 2 5 7 8 and 10 corresponds to capture times 2 5 7 8 and 10 2 DATA INPUT FORMAT Corresponding to the two types of experiments there are two types of data input formats for CARE 3 1 Continuous Type Data Input Data are collected from a continuous time experiment as described above The exact capture times for each individual are recorded along with some relevant covariates In the distributed data file simeg dat we provide a simple simulated data with two covariates for illustrating the necessary input The termination time is 2 units and there are two covariates gender and weight A total of 161 individuals were captured and the maximum frequency is 7 In the input covariates are first given and followed by capture times The first five records in the data file are displayed as follows O 20 585 0 005 1 098 1 551 0 0 0 0 O 21 876 0 007 1 604 1 741 0 0 0 0 1 18 579 0 018 0 256 0 365 0 495 0 526 0 624 1 636 1 20 969 0 041 0 171 0 174 0471 1 860 1 869 0 1 18 538 0 044 0 488 0 539 0 679 1 407 1 733 0 The first column shows the gender 1 male O female the second column denotes the weight in gram and the other columns record the exact capture times For example the first record indicates that a female individual with weight 20 585 grams was caught in time units of 0 005 1 098 and 1 551 Note that four extra 0 s must be added to fit our required inp
14. ut The third record shows that a male individual with 18 579 grams was caught seven times no additional 0 s are added because seven times is the maximum frequency The fifth record means that a male with weight 18 538 grams was caught six times in time units of 0 044 0 488 0 539 0 679 1 407 and 1 733 A zero is added in the last column to fit the required input 2 Discrete Type Data Input Data are collected from a discrete time experiment and are arranged in the usual individual capture history as described in CARE 2 for discrete time data analysis In the distributed file coul dat we store the data for a ten occasion house mouse collected by Coulombe and discussed in Section 4 of Hwang and Chao 2002 The reader is referred to the above paper for further details There were 171 animals caught with two covariates gender and age adult and non adult For each record the covariates must precede the capture history For example the first five records in the file coul dat are given below 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 1 0 1 1 1 1 1 0 0 0 1 0 0 1 0 0 The first column shows the gender 1 male O female the second column denotes the age 1 adult 0 non adult and the other columns record presence 1 or absence 0 in each occasion For example the first record means a female adult was caught on occasions 1 2 3 4 and 10 and not in the others Similar interpretation pertains t
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