User Guide for Program CARE-2
Contents
1. Input 2 7 Please input the number of continuous individual covariates Input 0 8 8 Please input the filename containing the capture history and individual covariates continuous type covariates must follow by the categorical type covariates Input c program files CARE 2 data exampl61 dat 9 Please input the number of categorical occasional covariates Input 1 10 Please input the number of continuous occasional covariates Input 0 11 Please input the filename containing the occasional covariates continuous type covariates must follow by the categorical type covariates Input c program files CARE 2 data exampl62 dat 11 Do you want to include the unknown time effects y or n We input n 12 Please input the filename to save the output Input for example c program files CARE 2 output out Please wait a moment and the results will be shown in the GAUSS window Moreover the output is also saved in c program files CARE 2 output out The standard output is shown in Table 9 Table 9 The output of covariate analysis for rodent data CARE 2 for capture recapture anal ysis wth covari ates Authors Anne Chao and Hsi n Chou Yang THHE 3HHF Version 1 5 April 2006 THHE 22 Sunmary Statistics Total nunter of distinct aninals 171 Nunber of capture sanples 10 E 68 0 68 0 2 68 2 3 27 6 6 6 amp 2 74 3 2 36 6 10 40 7 4 2 40 5 172. 39 09 55 48 38 73 49 29 M JK 45 5 3 58 3 71 41 09 56 22 4133 49 67 M JK2 48 3 5 78 5 68 41 69 66 72 39 73 57 83 MYIntJK 45 5 8 35 3 71 39 29 8158 4133 70 24 Mx EE 40 2 2 14 0 50 38 44 48 89 38 00 43 76 Mb CME 48 0 12 78 11 98 2 95 39 46 106 76 38 78 85 55 Mb UME 43 6 11 12 6 90 2 34 38 47 104 74 38 07 80 31 Mb EE 47 1 8 51 10 78 2 82 39 91 81 09 3800 68 25 Mh SCl 43 6 3 97 3 77 0 51 39 62 57 57 39 70 51 76 Mh SC 42 5 3 4 3 45 0 49 39 18 5485 3890 48 89 Mh EE 40 3 2 20 0 51 38 48 49 14 38 00 44 26 Mh SO 50 5 23 43 0 60 39 13 176 57 38 89 125 72 MhJK 53 0 9 43 42 84 84 47 38 00 73 00 Mh EE 43 5 4 44 1 68 0 40 39 36 60 04 3800 51 33 Mbh EE 44 2 4 58 1 89 0 36 39 72 60 60 3810 53 58 The first part of the output shows basic information including the data filename c program files CARE 2 data example1 dat for this example the number of distinct animals caught in the experiment 38 in this case the number of trapping occasions 6 in this case and the number of bootstrap replications 1000 in this case The summary statistics are listed in the second part of the output We use these data to introduce some notation The numbers of captures for the six occasions are n4 No Ne 15 20 16 19 25 25 Out of the n animals there are u firs
3. 2001 yields an estimate of 123 with an estimated bootstrap s e of 11 75 A 95 confidence interval associated with this estimate under model Mn is 113 169 or 114 156 based on two methods Example 3 Mouse data aggregated categorical data In Example 2 we used the mouse data with individual capture history Example3 dat files the data in a format of aggregated categorical data The user can view Example3 dat for the required format for CARE 2 All running procedures are similar to those in Examples 1 and 2 except that aggregated categorical data is selected in step 6 The output is exactly the same as that in Example 2 except for the bootstrap s e and confidence intervals Example 4 Cottontail rabbit data individual capture history Edwards and Eberhardt 1967 conducted an 18 trapping occasion capture recapture experiment on a confined population of known size In their study 135 wild cottontail rabbits were penned in a 4 acre rabbit proof enclosure Out of 142 captures there were 76 distinct rabbits An advantage of this data set is the true population size is known The basic data information and the summary statistics are shown in Table 5 Otis et al 1978 pp 84 87 found that for these data there was significant time variation and heterogeneity but little behavioral response Hence we select all models with time and or heterogeneity models Mi M and Min along with the most general 12 model M n This dat
4. k a ge j i ae Noy Y UG Df 206 90 95 Van Fhe Max r Rd T D m ujl J a u 1 0 gt where Nop is a simple estimator valid under model Mpn Here 5 e6 e denotes the unknown relative time effect of sample k A convenient estimator of Pxj4 l6 4 Is afunction of and can be presented as Byja Papal uy M 19 uj 419 233
5. Mfg o LIU Df j2 maxi A 1 0 2yYn j k J No 2 M C Estimating Equation Mth EE 2001 t N M Equation for N 2n ja i3 1 C pt 14 7 Equationfora a n N j 1 2 t where a pe j 1 2 C 1 u n M M fua 2T f max and N Ma 1 fir K gh sgj No Y iU Wf 29 n n j k J 7 Model Mp Pollock and Otto 1983 Lee and Chao 1994 Chao et al 2001 Jackknife Mbh JK Noy M t u Sample Coverage Mbh SC wv M j u a Ngo ud Ex y where j max k u lt u e k 21 t 1 C 1 u uU u u pi max ct L Lao U Non Mia Cis e Estimating Equation Mbh EE toy Equation for N gt t Equation for gt m Mj p 0 i amp jJuj N M P m Mi p E EP Sacra Scri a j l C C 1 u u m M M j 1 u P3 nt Tf 1 j 1f Dale 29 00 M ol Pa t 1 PNE 4n 4 uf T fa 0 max i is a simple estimator valid under model Mh that is Mi h PRI fe I f sa 8 Model Mn Chao et al 2001 Estimating Equation Mtbh EE Equation for N y oe i j l 1 C 1 6 1 C 1 75 6 t Nm Equation for 2 Di Q A Aj 4AN n 1 tn I 2NO 1 Pon a N where Aj amp N 2 N n 1473 A INC _ 1 72 m X i ia x z C C _ 1 u u m M M gt
6. USA Yip P S F 1991 A martingale estimating equation for a capture recapture experiment in discrete time Biometrics 47 1081 88 Zippin C 1956 An evaluation of the removal method of estimating animal populations Biometrics 12 163 89 26 Appendix In this Appendix we give formulas for the estimators featured in CARE 2 under various models Refer to Tables 1 and 2 for definitions and references 1 Model M Otis et al 1978 Darroch 1958 Yip 1991 Unconditional MLE MO UMLE Mia logt L IN j 1 4 tlog 1 p 0 jal PIDIE UL NES ahere n gt p p tp Conditional MLE MO CMLE Equation for N Equation for p janr Equation for N 1 M a t 2T SEE ee 1 N 1 p logL n Nt n op p 1 p Estimating Equation MO EE Equation for p 0 where n gt jar Equation for N Y IN M p u N M p 0 j 1 Equation for p n Np 0 where n x jan 2 Model M Otis et al 1978 Darroch 1958 Yip 1991 e Unconditional MLE Mt UMLE OlogL Meat Equation for N ET Y N j i y fat 1 ClogL n N n Equation for ej Le j 1 2 t oe e 1 6 J Conditional MLE Mt CMLE M t Equation for N 1 1 e j n L j 1 2 t Equation for e e P 2g Estimating Equation Mt EE t Equation for N YN M 1 e u N M e 0 ja Equationfore n Ne 0 j 212 t 3 Mo
7. by the Horvitz Thompson estimator which is Nur gt IL where P is the estimated capture probability evaluated i 1 ij at the conditional MLE The variance of the resulting estimator can be estimated by an asymptotic variance formula derived in Huggins 1989 1991 Below two examples are used for CARE 2 to illustrate the estimation and model selection Table 6 Models with covariates in CARE 2 The effect c is optional Model Assumption Restriction in model M ton M toh logit P a c v Y B W r R M logit P a v Y B W setcj 0 r 0 M e logit P a c v Y r R set B 0 Mth logit P a c B W r R set v 0 M n logit P a B W set cj 0 r 0 v 0 M logit P a v Y set B 0 c 0 r 0 M logit P a c r R set B 0 v 0 Mo logit P a set B 0 c 0 r 0 v 0 Running Procedures by Examples In the following we provide two examples to demonstrate the procedure of CARE 2 16 for covariate analysis They are Example 5 Same capture data as in Example 1 but three individual covariates are included data in file example5 dat Refer to Huggins 1991 and Chao and Huggins 2003 for detailed analysis Example 6 Rodent data with two individual covariates and one occasional covariate capture data and individual covariates are in file exampl61 dat occasional data are in file exampl62 dat Refer to Huggins 1989 for detailed analysis Exampl
8. captures and 1 s captures possibly followed by some individual covariates The maximum size for capture history matrix input in CARE 2 is 2000 individuals and 80 occasions 2 Aggregated Categorical Data In some studies with many captured individuals the individual capture history matrix becomes very large Itis more convenient to represent the raw data in a categorical data by a tally of the frequencies of each capture history The two types of data input will be illustrated by examples in the following sections 4 Analysis Without Covariates Models Estimators Featured The models considered in CARE 2 are originally proposed in Otis et al 1978 and White et al 1982 and are tabulated in Table 1 Assume that there are N animals in the study area and capture recapture experiments are conducted over f occasions The purpose is to estimate the unknown parameter N Under each model there are many available estimators in the literature The estimators featured in CARE 2 and their abbreviations in output see later sample output for four examples are shown in Table 2 All the estimators are shown in the Appendix Table 1 Models without covariates in CARE 2 Pj denotes the capture probability of the ith animal on the jth occasion pi heterogeneity effect of the ith individual i 21 2 N ej time or occasional effect of the jth occasion j 1 2 behavioral response effect Model Assumption Rest
9. moment and the results will be shown in the GAUSS window Moreover the output is also saved in c program files CARE 2 output out The standard output for CARE 2 with this example with the above input is shown in Table 8 Remark If you have abundant data it may take a long time to get your output due to complicated iterative estimation in GAUSS program operating on a large array or high dimensional matrix 18 Table 8 The output of covariate analysis for deer mice data 3HHF CARE 2 for capture recapture anal ysis wth covariates Authors Anne Chao and Hbi n Chou Yang HHH 3HHF Version 1 5 April 2006 HE Total nunber of distinct aninals 38 Nunber of capture sanples 6 1 15 0 15 0 9 15 2 8 12 20 15 6 11 3 6 10 16 23 7 14 4 3 16 19 29 6 11 5 3 22 25 32 6 8 6 3 22 25 35 4 9 7 38 m The Fit amp Esti nati on of all nodas Model Estinte SE MN LL AC 95 d Stat us MO 38 47 0 72 15727 316 54 38 06 42 04 Converge Mt 38 40 0 66 15242 3168 38 04 41 80 Converge Mrb 42 25 3 76 150 43 304 87 38 96 56 86 Converge Mh 39 85 1 72 144 87 297 75 38 39 46 67 Converge Mtb 46 48 12 65 148 18 310 36 39 02 108 74 Converge Mth 39 66 1 61 13955 297 10 38 34 46 20 Converge Mbh 47 15 7 17 139 54 289 09 40 35 73 52 Converge Mtbh 47 13 10 08 137 33 294 66 39 59 90 50 Converge M l Descri pti on The general logistic no
10. night daily for five days Two individual covariates are recorded gender male and female and age young semi adult and adult The summary statistics for capture history are shown in Table 9 below Otis et al 1978 concluded there is no behavior response effect but time variations and individual heterogeneity are strong No suitable estimators were available at the time and thus they suggested the use of the number of the distinct animals caught in the experiment There are two types of covariates individual covariates and occasional covariates in this example The individual capture history and individual covariates gender and e 2 age are stored in c program files CARE 2 data exampl61 dat The experiment time morning or night is treated as an occasional covariate The data format for filing an occasional covariate is shown in c program files CARE 2 data exampl62 dat where 1 denotes for morning and 2 denotes night There are two rodents with missing covariates hence we exclude these two records in the following analysis It leads to somewhat different results from those in Huggins 1989 The running steps 1 to 3 are similar to those for Example 5 so we begin with step 4 4 Please input the number of distinct individuals n this example we input 171 5 Please input the number of sampling occasions Input 10 4 5 6 Please input the number of categorical individual covariates
11. set to be 0 as the reference group Suppose there are k categories for the first covariate then in the output we have k 1 coefficients beta1 1 beta1 2 beta1 k 1 where betan j denotes the effect of the jth group relative to the reference group for the nth covariate From Table 7 male is coded as 0 and female is coded as 1 in data entry thus group 1 the larger numerical value is set to be the reference group Therefore in Table 8 the coefficient beta1 1 0 92 is the effect for male the female is set to be 0 so males have larger probabilities Also young is coded as y and adult is coded as a in data entry thus in an alphabetical order the group y is used for reference group The second coefficient beta2 1 1 88 is the effect for adult the young effect is set to be 0 so young have larger capture probabilities The last coefficient in the output beta3 0 16 is the effect for a unit change of body weight This implies the heavier the weight the larger the capture probability Then from the summary of model fitting the estimated population size under the selected model M pn is 47 2 s e 7 17 with a 95 confidence interval of 40 4 73 5 Example 6 Rodents data two individual covariates and one occasional covariate The data of salt marsh rodents were originally collected by Coulombe and analyzed by Otis et al 1978 pp 62 67 and Huggins 1989 The experiment was carried out in the morning and
12. 0 otherwise The general logistic model incorporating covariates considered in CARE 2 is logit P a c v Y B W r R where a denotes the baseline intercept c1 co cz1 represents the unknown occasional or time effect and c 0 is used for the reference group These time effects may or may not be included in the model You can specify whether these effects are needed for each data analysis Table 6 summarizes all sub models The interpretation of the coefficient of any 8 is based on the fact that when gt 0 the larger the covariate is the larger the capture probability is while if 8 lt O then the larger the covariate is the smaller the capture probability is Similar interpretation pertains to the coefficient of any r for occasional covariate The parameter v represents the effect of a recapture which implies that v gt O corresponds to a case of trap happy and v 0 corresponds to a case of trap shy 215 The parameters in the logistic models are estimated by a conditional ML method based on the captured individuals Huggins 1989 1991 The default of maximum number of iterations in CARE 2 is 500 Model selection can be performed using Akaike information criterion AIC which is defined as 2logL 2m where L denotes the likelihood computed at the conditional MLE and m denotes the number of parameters in the model A model is selected if AIC is the smallest among all models considered The population size is estimated
13. 19 28 221 72 122 92 206 70 9 Mb EE 1395 9 21 84 ESS 37 2 36 118 32 217 71 120 80 195 35 Mtb CMLE T3 46 20 55 69 3 63 127 83 337 77 123 85 293 48 Mtb UMLE I6l l 42 71 41 72 3 19 122 255 322 92 121 45 285 52 Mtb EE 152 0 28 68 32 87 2 87 122 46 251 21 118 99 224 05 Mtbh EE 123 2 LT 75 m 1 03 052 112 95 169 00 113 51 156 30 As in Example 1 estimation results for the selected models follow the basic data information and summary statistics The model selection procedure in Otis et al 1978 pp 92 96 shows that the most likely model is model Miy and model M is the next most likely model In the following discussion we interpret the results for these two models based on the above output 1 The unconditional MLE for model M Mb UMLE in Table 4 yields an estimate of 142 2 with an asymptotic s e of 16 42 and a bootstrap s e of 22 68 A 9596 confidence interval constructed by a log transformation is in the range of 119 222 the bootstrap percentile method gives an interval range of 123 207 The ratio of recapture and first capture probabilities is estimated to be 2 42 Phi 2 42 in the output which shows a trap happy situation The conditional MLE estimate is 145 5 and the estimate based on an optimal estimating equation is 139 9 Their associated variance and confidence intervals are shown in the above output If model Min is assumed an estimating equation approach Chao et al
14. 2 ME 0 38 0 28 0 02 0 46 S E 0 08 0 11 0 13 0 11 Model Mtb a V r1 1 ME 0 31 0 01 0 67 S E 0 16 0 00 0 10 Model Mth a beal 1 beta21 beta2 2 r1 1 ME 0 63 0 28 0 02 0 47 0 68 S E 0 17 0 11 0 14 0 12 0 11 Model M bh a v betal 1 beta2 1 beta22 ME 0 24 0 18 0 28 0 02 0 46 S E 0 12 0 13 0 11 0 14 0 11 Model Mitbh a v beal1 betaX1 beta2 2 r1 1 ME 0 65 0 03 0 28 0 02 0 47 0 68 S E 0 16 0 05 0 11 0 15 0 12 0 11 From the results of AIC listed in Table 9 model Min is selected The conclusion is consistent with that in Otis et al 1978 pp 62 64 For gender data entry is 1 for male and 2 for female the female is served as the reference group The negative regression coefficient beta1 1 0 28 demonstrates that the females have larger capture probabilities than the males For age data entry is 1 for young 2 for semi adult and 3 for adult thus the adult group with the largest numerical value is regarded as a reference group The regression coefficient beta2 1 0 02 is not significant hence there is no significantly difference of capture probabilities between the young and adult However the regression coefficient beta2 2 0 47 is significantly different from 0 which implies that adults have higher capture probabilities than the semi adult For the occasional covariate data entry is 1 for morning and 2 for night the coeff
15. 2 11 h SC2 132 8 22 05 20 62 0 65 103 26 194 39 103 47 181 51 h JK1 116 6 8 54 8 89 103 201 137 0 I07T17 125 11 h JK2 141 4 14 25 14 87 C118 92 175 79 3 E 120 76 162 13 h IntJK 142 3 38 07 15 18 99 27 264 74 107 17 252 1 h EE 125 3 16 41 Saga 0 67 102 15 169 10 100 39 154 64 th SC1 138 9 24 35 224 05 0 70 106 23 206 84 108 82 194 47 th SC2 134 6 22 56 21 22 0 68 104 29 197 46 105 93 183 40 th EE noc a SSS me llt EPSBES tbh EE aaa CERTE SE IEEE N E EEUU EE E OE RE ERI ee en eee ee ee ee ee ee eee ES kkk iterative steps do not converge Edwards and Eberhardt 1967 reported that the usual estimators based on equal catchability considerably underestimated the true number 135 Itis readily seen from the output that all estimates based on model M Mt CMLE Mt UMLE and Mt EE in the output are about 95 or 96 Burnham and Overton 1978 suggested modeling 243 these data by model My and adopted an interpolated jackknife estimator In the output the first order Mh JK1 and the second order jackknife Mh JK2 are also shown the interpolated jackknife Mh IntJK yields an estimate of 142 with an asymptotic s e of 15 18 The confidence interval proposed by Burnham and Overton 1978 was 112 172 based on the asymptotic s e This interval is different from ours in Table 5 because we use a bootstrap s e The asymptotic s e is
16. 7 The first part of the output shows all summary statistics The second part shows the fitting and estimation results for the logistic model and all sub models followed by model description For each model the corresponding estimated population size number under the heading Estimate in Table 8 its s e under the heading S E negative value of the minimum log likelihood under the heading MIN LL the Akaike information criterion AIC and 95 confidence interval Chao 1987 are calculated From the values of AIC we select model M y because AIC of this model is the smallest among all models There are slight differences between our estimates and those in Huggins 1991 because different numerical algorithms are used The last part of the output shows all fitted parameter estimates Under model M pn the fitted intercept is 2 91 the behavioral response effect is 1 18 for re capture the first capture effect is set to be 0 so recaptures have higher probabilities Then there are 20 several coefficients corresponding to the three individual s covariates according to the order of data entry Generally one coefficient is associated with a continuous covariate For a categorical covariate there are k 1 coefficients associated with a covariate with k categories When groups are in a numerical order or in an alphabetical order according to the data entry The category with the largest numerical value or the last alphabetical order is always
17. M e NU 1 e 1 9 e j zem Equation for e 29 e Model M Burnham and Overton 1978 Lee and Chao 1994 Chao et al 2001 The First order Jackknife Mh JK1 t 1 Rl Mass CL The Second order Jackknife Mh JK2 2t 3 t 2 Rs c Mc CO t ls Interpolated Jackknife Mh IntJK N SN g 1 N EN y CN 4 1 lt g lt 5 N 55 gz5 N sgh C Te ze g min P gt a P isthe P value and a is the significant level Coefficients a can be referred to Burnham amp Overton 1978 e Sample Coverage1 Mh SC1 e Ma fen Na cn ye where C 1 538 adfg 1 1 NS JUD p P yy max di 1 0 and N MIO N jf e Sample Coverage2 Mh SC2 M hs Nsco per P where C 1 fi 2f t 9 2 fi C G6 Noob Dy jG f p2 max lt 2 1 0 and N M C W ada Estimating Equation Mh EE alin E t u Equation for N j l 1 C4 Equation for p p 5 n tN j 1 i j a Ca 1 fy 5 n Mp M thy fs t gt jj f f max s 1 0p where Mu IT fi Kf t 9 9 jf 30 6 Model Mi Lee and Chao 1994 Chao et al e Sample COverage1 Mth SC1 M NM 1 47 where C x 1 f 0 Jf jt Nor uU fs f max 1 0 Yn j k sci Nos M C e Sample Coverage2 Mth SC2 Back to Table2 Nooo Mas t Is 7 where 2 2 C 2 1 Fy 2f t 1 y
18. Version 1 5 April 2006 User Guide for Program CARE 2 Anne Chao and Hisn Chou Yang Institute of Statistics National Tsing Hua University Hsin Chu Taiwan Table of Contents 1 Introduction 2 Download and Setup 3 Data Input Format 4 Analysis without Covariates Example 1 Deer mice data individual capture history data Example 2 Mouse data individual capture history Example 3 Mouse data aggregated categorical data Example 4 Cottontail rabbit data individual capture history 5 Analysis with Covariates Example 5 Deer mice data with three individual covariates Example 6 Rodents data two individual covariates and one occasional covariate Appendix 1 Introduction Program CARE 2 calculates population size estimates for various closed capture recapture models The program consists of two parts one part written in C Language deals with models without covariates and the other part written in GAUSS language deals with models with covariates In this manual we outline the downloading and setup procedures Section 2 data input formats Section 3 Operation procedures models and estimators featured in CARE 2 are described in Section 4 for models without covariates and Section 5 for models with covariates Examples are provided and sample outputs are shown Results for each example are also discussed to help the user interpret the numerical output Before using CARE 2 the user is suggested to read two introductor
19. a was analyzed in the literature e g Burnham and Overton 1978 Chao et al 1992 This data set with individual capture history is filed in example4 dat The output for models Mi Mn and Mn is given in Table 5 Table 5 The output of cottontail rabbit data analysis 1 Basic Data Information Data filename c program files CARE 2 data example4 dat Total distinct animals 76 Number of capture occasions 18 Bootstrap replications 1000 2 Summary Statistics T u i m i n i M i Tt 35 T as SACAN E Doe eee ee e T ee S S 1 9 0 9 0 43 9 2 6 2 8 9 16 13 3 3 6 9 LS 8 12 4 EIE 3 14 18 6 22 5 4 4 8 29 0 24 6 1 4 5 33 2 23 7 10 8 18 34 1 29 8 7 4 LI 44 0 35 9 1 3 4 51 0 35 10 1 2 3 52 0 35 11 9 7 16 53 0 43 12 0 5 5 62 0 41 13 T 1 2 62 0 41 14 5 2 7 63 0 46 15 6 3 9 68 0 50 16 0 0 0 74 0 50 17 0 4 4 74 0 47 18 2 8 10 74 0 43 19 76 ft is of individuals that were captured exactly i times on occasions 1 2 cor fX of individuals that were captured exactly once on occasions 1 2 p ou 3 Estimation Results odel Est Boot s e Asy s e Phi CV 95 CI log transf 95 CI percentile Pte x Ep Set bos e NE La MAS DP LADO MORE OCUPA C c EUN cL he ee eC Dd RENDUM OR GN ORDRE Oe t CMLE 96 0 8 13 6 70 5 27 119 04 3 86 63 112 19 t UMLE 95 8 36 6 58 84439 1119 31 85 98 110 34 t EE 95 0 8 81 64 51 823 97 121 L0 85 46 112 43 h SC1 137 0 21 50 21 44 0 67 107 20 195 31 106 43 18
20. a3 65 5 15 58 73 139 16 5 6 3 38 4 14 B 45 7 2 6 76 15 5 Al 8 0 35 35 16 1 2 9 2 74 76 169 0 9 10 0 38 38 11 1 2 1 171 Model Esti nte S E MN LL AC 9596 Status MO 173 99 1 83 1093 07 2188 14 171 99 180 02 Converge Mt 173 79 176 1071 43 2146 86 171 90 179 68 Converge Mrb 172 99 1 60 1092 39 2188 78 171 50 178 96 Converge Mh 175 38 2 33 1080 36 2168 72 172 65 182 64 Converge Mtb 173 74 1 74 1071 43 2148 86 171 87 179 57 Converge Mth 175 14 2 26 1058 44 2126 89 172 52 182 26 Converge M bh 173 86 2 05 1079 44 2168 87 171 81 181 09 QGonverge Mstbh 174 86 2 21 1058 42 2128 84 172 36 181 95 QGonverge M l Descri pti on The general logistic nodel Mtbh is logt Pij za cj v Yij beta Wi r Rj where i refers to the ith i ndi vi dual j refers to the j th sanpl e or jth capture occasi on a baseline intercept cj the unknown tine or occasional effects of the jth capture occasi on set c t 0 where t the nunber of capture occasi ons V behavi oral response the effect wr t the past capture history indi cator Y ij beta the effect of individual covari ates Wi r the effect of occasional covariate Rj The MEs of Regression Coefficients Model MKO a ME 0 69 S E 0 05 224 eK Model Mrt a r1 1 ME 0 31 0 68 S E 0 16 0 10 Model Mrb a V ME 0 58 0 15 S E 0 10 0 11 Model Mth a beal 1 beta21 beta2
21. acteristics age sex body weight or wing length and occasional covariates could be environmental variables temperature on each occasion or known catch effort expended in trapping method e g number of traps on each capture occasion Occasional covariates should be stored in another file as will be shown in Example 6 below 4 Suppose for each animal there are s individual covariates Let the individual covariates for the ith animal be denoted as W W W W and B 6 B 8 denotes the effects of these covariates It is necessary to assume that the individual covariates are constant across the f capture occasions in the experiment as they cannot be measured on an occasion if the individual is not captured If heterogeneity is fully explained by individuals covariates then the heterogeneity effect can be expressed conveniently as B W B W B W B W Assume that there are b occasional covariates Ri Ri Rid Rei Ree Rot sig Rp Rip Ro For example Ri Rip xdg Rij may represent the temperature on each occasion and Ry Roe Rei may represent the capture effort on each occasion Let r n r r denote the effects of the occasional covariates Define Rj Ri Ra Rej then the occasional effect for the jth occasion can be expressed as rR riRij ro Ro amp Rp Define Y 1 if the ith animal has been captured at least once before the jth occasion and Y
22. also tabulated so that user can compute relevant intervals If model Min is assumed the coefficient of variation CV of the capture probabilities for all estimation methods is estimated to be about 0 70 as shown in the output This relatively large value of the CV gives strong evidence of heterogeneity because the CV 0 corresponds to no heterogeneity The two estimators using the sample coverage methods Mth SC1 and Mth SC2 proposed by Chao et al 1992 and Lee and Chao 1994 are respectively 138 9 s e 24 35 and 134 6 s e 22 56 The latter gives a 9596 confidence interval 104 197 using a log transformation and 106 183 using a percentile method The estimating equation approach does not yield an estimate due to insufficient capture and recapture information which causes failure of convergence in the numerical iterations If we adopt the most general model Mipn similar difficulty arises Therefore capture and recapture information is not sufficient for fitting a complicated model with three sources of variations We caution that in some cases estimates can still be obtained in the case of insufficient information but the standard error generally becomes so large that the model is useless 5 Analysis With Covariates Models Estimators Featured In program CARE 2 we distinguish covariates as two types individual covariates and occasional covariates as in Huggins 1989 1991 Individual covariates include individual s char
23. ation Table 3 The output of deer mice data analysis 1 Basi c Data I nf ornati on Data filename c program files CARE 2 data examplel dat Total of distinct aninals 38 Nunber of capture occasions 6 Bootstrap replications 1000 ui mhi ni Mi ft i fiui 4 1 15 0 15 0 9 15 2 8 R 20 15 6 11 3 6 10 16 23 7 14 4 3 6 19 29 6 11 5 3 2 25 2 6 8 6 3 22 25 3 4 9 7 4 38 ft i of indivi duals that were captured exactly i tines on occasions 1 2 t ffi of indivi duals that were captured exactl y once on occasions 1 2 i 3 Esti nati on Resul ts Model Estinte Boot_s e Asys e Phi CV 95 log transf 95960 percentile i iai ie d i Je a ee a MXOMEB 38 5 0 36 0 72 38 12 39 81 3813 39 55 MXUME 38 0 0 24 0 67 38 00 38 00 38 00 38 83 MX EE 38 0 0 36 0 68 38 00 38 00 3800 39 22 M QME 38 4 0 31 0 66 38 11 39 51 38 08 39 27 M UME 38 0 0 14 0 62 38 00 38 00 38 00 38 53 M EE 38 0 0 21 0 62 38 00 38 00 38 00 38 73 M COMEB 42 3 7 30 3 75 1 92 38 43 80 28 3877 57 41 M UME 40 8 6 91 3 05 179 38 18 81 43 3800 51 98 Mx EE 41 9 5 29 3 58 1 89 38 53 66 84 3800 53 28 M SC 43 5 3 81 3 72 0 50 39 64 56 78 39 65 50 94 M SC 42 4 3 52 3 40 0 48
24. clude eight models for comparisons The model description is listed in Table 1 4 Bootstrap Selection select whether you like to do the bootstrap for obtaining standard error estimates and confidence intervals or not If yes then select the number of replications 1000 is suggested 5 Confidence Interval Selection select whether you like to have a 95 confidence interval or not If your selection is yes you must also check yes in step 4 for the bootstrap selection and specify the number of replications 6 Data Structure Selection select the format of your data set Two types of data formats are described in Section 3 7 Click Load Data to input the filename of your data file e g c program files CARE 2 data example1 dat 8 Click Compute to get the results Wait a while for executing the program The execution time depends on the size of data and the number of bootstrap replications 9 Click Output from the top menu to view the results You can click Save Output to save all the output results to a designated file click Print to print the output from your printer or click Clear to remove all results and to proceed another run Examples Four examples are used to demonstrate the use of CARE 2 for analyzing animal capture recapture data without covariates All data sets used in this guide are distributed with CARE 2 and stored by default in the directory c program files CARE 2
25. covariates The covariate analysis is not embedded in the interface of Figure 1 A working environment of Gauss is provided by the following procedure first doubly click the GRTM exe to unzip all files of the Gauss Run Time Module GRTM in the previously specified folder Then doubly click the executable file setup exe to install the Gauss Run Time Module which is GUASS free ware for non commercial redistribution 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 doubly click the icon GSRUNSO on the desktop of your computer to initialize the Gauss Run Time Module and then the interface is shown below Figure 2 The interface of CARE 2 for analysis with covariates N GSRUN Command Input Output File Edit View Configure Run Debug Tools Window Help Deh B OM R CARE 2 pgm zi gt C AGSRUN5O sie P Command Input Output Ln 1 Col 1 3 Data Input Format Data must be read from an ascii file There are two types of data input formats 1 Individual Capture History Data are arranged in a matrix called individual capture history matrix with the rows representing the capture histories of each captured individual and the columns representing the captures on each occasion The capture history of each captured individual is expressed as a series of 0 s non
26. data The output will be shown and briefly described The four examples used in this section are Example 1 Deer mice data in a format of individual capture history data in file example1 dat Refer to Chao and Huggins 2003 for detailed analysis Example 2 Mouse data in a format of individual capture history data in file example2 dat Refer to Chao et al 2001 for detailed analysis Example 3 Same data set as in Example 2 but in a format of aggregated categorical form data in file example3 dat Example 4 Cottontail rabbit data in a format of individual capture history form data in file example4 dat Refer to Chao et al 1992 for detailed analysis Example 1 Deer mice data individual capture history data These data were collected by V Reid and are distributed with program CAPTURE Otis et al 1978 White et al 1982 Rexstad and Burnham 1991 The data arose from a live trapping experiment that was conducted for six consecutive nights with a total of 38 mice captured over these six capture occasions In data file example1 dat a matrix of 38 x 6 is recorded Analyses of these data include Otis et al 1978 p 32 Huggins 1991 and Chao and Huggins 2003 Using the procedure as described in the above and selecting all models in step 3 the following output is shown in the Output window after execution The output contains three parts 1 basic data information 2 summary statistics and 3 results of estim
27. del M Otis et al 1978 Zippin 1956 Lloyd 1994 Unconditional MLE Mb UMLE Mis Equation for N Ter Sin j t y Dont p j 1 Equation for logL 0 M m p 0 Og 9 1 p Nt M M _ Equation for p OEM MiM AM mg 0 where ep P 1 p 1 p t t n gt any m yam and M QM Conditional MLE Mb CMLE Equation for N N M 1 1 p Equation for ZIOgE Hh Mame og 9 1 p Nt M M Equation for p IONE n EMA ALLEY L 0 where ep P 1 p 1 6p n Man pm 37 m and M M Estimating Equation Mb EE t Equation for N S N M 1 p u N M p 0 1 Equation for 4 Xip p m M p gt p 1 p lu N M p 0 EM DN Equation for p lt I En 28 4 Model My Chao et al 2000 Lloyd 1994 Unconditional MLE Mtb UMLE Meas Equation for N dlogt Ni j ty Dloalt e j 1 oN t Equation for d rz 2 yy n i 0 where m m j 2 ej i n N M M m Equation fore PDDE uc jt Mj 09 6 j 1 2 t J J 1 i ey ej ge e Conditional MLE Mtb CMLE M Equation for N int 1 e jal t Equation for g Pogi Gar ae 5 eR 0 where m Ws j 2 j n N M M m Equation fore ilo NT jM D 9 j 1 2 t e e 1 e 1 4e e Estimating Equation Mtb EE Back to Table2 t u N M e Equation for N gt l i L amp N M 1 e Equation for J nde Ee N
28. del Mtbh is logit Pij za ccj v Yij beta Wi r Rj where i refers to the ith i ndi vi dual j refers to the jth sanple or jth capture occasi on a baseline intercept cj the unknown tine or occasional effect of the jth capture occasi on set c t 0 where t the nunber of capture occasi ons V behavi oral response the effect wr t the past capture history indi cator Y ij beta the effect of individual covari ates Wi r the effect of occasional covariate Rj The MEs of Regression Coefficients Model MKO a 19 ME 0 08 S E 0 13 eK Model Mrt a c4 c2 c3 c4 c5 ME 062 1 07 0 54 0 96 0 64 0 00 S E 0 24 0 42 0 42 0 42 0 42 0 17 Model Mrb a V ME 0 76 1 22 S E 0 34 0 38 eK Model Mth a betal 1 beta2 1 beta3 ME 1 95 0 81 1 90 0 16 S E 0 71 0 31 0 57 0 06 Model Mtb a V c1 c2 c3 c4 c5 ME 1 16 1 72 0 42 031 0 45 0 37 0 12 S E 1 09 0 98 0 80 0 57 0 49 0 45 0 42 Model Mth a betal 1 beta2 1 beta3 c 1l c2 c3 c4 c 5 ME 143 084 198 016 1 18 0 59 106 0 70 0 00 S E 0 74 0 32 0 58 0 06 0 44 0 43 0 44 043 019 Model Mkbh a v be amp al 1 beta2 1 beta3 M E 2 91 1 18 0 92 1 88 0 16 S E 0 87 0 40 0 35 0 63 0 06 Model Mtbh a v betal 1 beta2 1 beta3 c1 c2 c3 c4 c5 ME 2 76 1 21 094 1 92 0 16 0 11 0 02 0 71 0 50 0 08 SE 1 30 0 74 0 36 0 64 0 06 0 87 0 80 0 60 0 56 0 5
29. e 5 Deer mice data with three individual covariates For the data set discussed in Example 1 there were actually three covariates gender male or female age young semi adult or adult and weight collected for each individual in the deer mouse data Only three semi adult mice were caught so they were re classified as adults The user can view example5 dat for the complete data Part of the complete data is shown in Table 7 Table 7 Individual capture history of deer mice with three covariates Gender 0 male 1 female Age y young a adult and Weight in grams Occasion 1 Occasion 2 Occasion 3 Occasion 4 Occasion 5 Occasion6 Gender Age Weight 1 1 1 1 1 1 0 y 12 1 0 0 1 1 1 1 y 19 1 1 0 0 1 1 0 y 15 0 0 0 0 0 1 0 16 0 0 0 0 0 1 1 19 There are three individual covariates and there is no occasional covariate Since every covariate can be treated as either categorical or continuous the user has to specify the numbers of each For example there are two categorical gender and age and one continuous weight for individual covariates of this data In the data format the order of data entry should be capture history categorical covariates followed by the continuous covariates Occasional covariates are stored in a separate file with the 2472 same order of categorical variables first and then continuous variables We describe the procedures for analyzing deer mice data with covariates The following procedure must b
30. e executed in a GAUSS environment 1 Provoke GAUSS environment either by doubly clicking GSRUNSO on your desktop as described in Download and Setup or by clicking the executable file GSRUN exe stored in the directory GSRUNBO 2 Click File on the top menu of GAUSS and subsequently click Run Program and select the program CARE 2 gcg which is stored in a pre specified working directory The default is c program files CARE 2 It prompts you subsequently the following input steps 3 Please input the number of distinct individuals In this example we input 38 4 Please input the number of sampling occasions Input 6 5 Please input the number of categorical individual covariates Input 2 6 Please input the number of continuous individual covariates Input 1 7 Please input the filename containing the capture history and individual covariates continuous type covariates must follow by the categorical type covariates Input c program files CARE 2 data example5 dat 8 Please input the number of categorical occasional covariates Input 0 9 Please input the number of continuous occasional covariates Input 0 10 Do you want to include the unknown time effects y or n This means that whether the effects c1 C2 ci 1 are needed in the logistic model We input y 11 Please input the filename to save the output Input for example c program files CARE 2 output out Please wait a
31. icient r1 1 0 68 denotes the effect of morning time Thus the capture probabilities are higher in the night The population size estimate under model Min is 175 1 with an estimated s e of 2 3 and a 95 confidence interval of 172 5 182 3 These results here are slightly different from those obtained in Huggins 1989 due to the different ways of treating missing covariates 24 Reference Burnham K P and Overton W S 1978 Estimation of the size of a closed population when capture probabilities vary among animals Biometrika 65 625 33 Chao A 1987 Estimating the population size for capture recapture data with unequal catchability Biometrics 43 783 91 Chao A Chu W and Hsu C H 2000 Capture recapture when time and behavioral response affect capture probabilities Biometrics 56 427 33 Chao A and Huggins R M 2003 Closed population models To appear as a chapter in The Handbook of Capture Recapture Methods Edited by Manly B McDonald T and Amstrup S Princeton University Press Chao A Lee S M and Jeng S L 1992 Estimating population size for capture recapture data when capture probabilities vary by time and individual animal Biometrics 48 201 16 Chao A Yip P S F Lee S M and Chu W 2001 Population size estimation based on estimating functions for closed capture recapture models Journal of Statistical Planning and Inference 92 213 32 Darroch J N 1958 The multiple recap
32. ir first capture Chao and Huggins 2003 suggested considering further general models Mp and Mio by use of estimating equation EE approach The two models produce close estimates Mbh EE and Mtbh EE in Table 3 So it is reasonable to adopt the most general model Mipn and conclude that the population size is about 44 standard error 4 6 The data based on model Mi show strong trap happy behavior Phi 1 89 in Table 3 a low degree of heterogeneity the CV estimate is 0 36 where CV denotes the coefficient of variation of p1 P2 pu and slight time varying effects as the relative time effects are estimated to be pe pe pe 0 34 0 32 0 26 0 26 0 33 0 33 where p denotes the average of p s Time effects are not shown in the output Refer to Chao et al 2001 for calculation formula The 95 confidence interval using a log transformation under model Mph is 40 to 61 This interval is unavoidably wider than that for model M because more parameters are involved Usually a simpler model has smaller variance but larger bias whereas a general model has lower bias but larger variance For interval estimation a simpler model produces narrow confidence interval with possibly poor coverage probability whereas a more general model produces wide interval with more satisfactory coverage probability A trade off clearly occurs with this example Example 2 Mouse data individual capture history 10 The mouse data
33. riction in model Mion Mion pe until first capture l p e for any recapture Mon _ p until first capture set e 1 p for any recapture Generalized removal model Mi e until first capture set pj 1 ge for any recapture Min P p ej set d 1 M P p set e 1 1 Mp p P until first capture set p p ej 1 p for any recapture Removal model M P e set pj 1 d 1 Mo Pj p set p p ej 1 d 1 Table 2 Estimators and their abbreviations in program CARE 2 Model Estimators Approaches Estimators in Software CARE 2 Mo Unconditional MLE UMLE Otis et al 1978 Conditional MLE CMLE Darroch 1958 Estimating equations EE Yip 1991 M Unconditional MLE UMLE Otis et al 1978 Conditional MLE CMLE Darroch 1958 Estimating equations EE Yip 1991 Mp Unconditional MLE UMLE Otis et al 1978 Conditional MLE CMLE Zippin 1956 Estimating equations EE Lloyd 1994 Mi Unconditional MLE UMLE Chao et al 2000 Conditional MLE CMLE Chao et al 2000 Estimating equations EE Lloyd 1994 Chao et al 2000 Mi Jackknife JK1 JK2 IntJK Burnham and Overton 1978 Sample coverage SC1 amp SC2 Lee and Chao 1994 Estimating equations EE Chao et al 2001 Mn Sample coverage SC1 amp SC2 Lee and Chao 1994 Estimating equations EE Chao et al 2001 M n Jackknife JK Pollock and Otto 1983 Sample coverage SC Lee and Chao 1994 Es
34. t captures and mj recaptures so that uj m nj With U4 U2 ue 15 8 6 3 3 3 and mi ms me 0 12 10 16 22 22 The statistic M denotes the number of marked animals just before the jth occasion Thus M u1 ue uj1 and Mi Me M7 0 15 23 29 32 35 38 for these data That is the number of marked individuals in the population progressively increased from M 0 to M 38 Here M denotes the total number of distinct animals caught in the experiment The frequency counts for the six occasions are fie foe fee 9 6 7 6 6 4 where fj denotes the number of animals captured exactly j times on occasions 1 2 k Since singleton information is usually important we also list fii fia fig 15 11 14 11 8 9 The third part shows estimation results For these data Otis et al 1978 p 32 indicated that the most suitable model for these data was model My Based on the usual unconditional MLE approach Mb UMLE in Table 3 the estimated population size in model M is 41 with bootstrap s e of 6 9 and asymptotic s e of 3 1 The 95 confidence intervals are 38 2 81 4 and 38 0 52 0 for log transformation and percentile methods respectively based on the bootstrap procedure The proportion constant between the re capture probability and first recapture probability in Table 1 or Phi in Table 3 is estimated to be 1 79 suggesting animals became trap happy after the
35. timating equations EE Chao et al 2001 Min Estimating equations EE Chao et al 2001 Program CARE 2 calculates two standard error estimates One is the asymptotic s e Asy s e in output which is obtained by inverting a Fisher information matrix for models without heterogeneity or by a delta method for heterogeneous models For the estimating equation EE approach the asymptotic s e is not obtainable for models M Min Mon and Min because of complexity The other method is bootstrap s e Boot s e in output which is always obtainable for all estimators For interval estimation CARE 2 provides two 95 confidence intervals based on a log transformation method Chao 1987 and percentile method Efron and Tibshirani 1993 respectively Both intervals are constructed from the bootstrap s e We remark that the bootstrap standard error Boot s e and confidence intervals may vary from trial to trial because the bootstrap replication data vary with trials Running Procedures 1 Doubly click the executable file CARE 2 exe it prompts you the interface window as shown in Figure 1 2 Click Without Covariate from the top menu of CARE 2 There are four items to be specified before executing CARE 2 as shown in Figure1 They are Model Bootstrap Confidence Interval and Data Structure as explained in the following four steps 3 Model Selection select suitable model s for your data You can check all model boxes to in
36. ture census Estimation of a closed population Biometrika 45 343 59 Edwards W R and Eberhardt L L 1967 Estimating cottontail abundance from live trapping data Journal of Wildlife Management 31 87 96 Efron B and Tibshirani R J 1993 An Introduction to the Bootstrap Chapman and Hall New York Huggins R M 1989 On the statistical analysis of capture experiments Biometrika 76 133 40 Huggins R M 1991 Some practical aspects of a conditional likelihood approach to capture experiments Biometrics 47 725 32 Lee S M and Chao A 1994 Estimating population size via sample coverage for closed capture recapture models Biometrics 50 88 97 Lloyd C J 1994 Efficiency of martingale methods in recapture studies Biometrika 81 305 15 Otis D L Burnham K P White G C and Anderson D R 1978 Statistical inference from capture data on closed animal populations Wildlife Monographs 62 1 135 Pollock K H and Otto M C 1983 Robust estimation of population size in closed animal populations from capture recapture experiments Biometrics 39 1035 49 25 Rexstad E and Burnham K P 1991 User s Guide for Interactive Program CAPTURE Colorado Cooperative Fish and Wildlife Research Unit Fort Collins White GC Anderson D R Burnham K P and Otis D L 1982 Capture Recapture and Removal Methods for Sampling Closed Populations Los Alamos National Lab LA 8787 NERP Los Alamos New Mexico
37. were originally collected by S Hoffman and described and analyzed in Otis et al 1978 p 93 Trapping was conducted on five days and 110 distinct mice were caught We specifically select this example because a detailed analysis is given in Chao et al 2001 For this data set since Otis et al 1978 concluded that for these data behavior is the strongest factor affecting capture probabilities we select three models with behavioral response models My Mp and Men in step 3 of the procedures presented earlier The results are the following Table 4 The output of mouse data analysis nd Basic Data Information Data filename c program files NCARE 2NdataNexample2 dat Total distinct animals se ENO Number of capture occasions 5 Bootstrap replications 1000 2 Summary Statistics i u i m i n i M i ft i fl i MT AUR T Pe LOS Rue OR c bU OE MC ot i Oe eee E 1 37 0 37 0 34 37 2 31 23 54 37 20 45 3 9 49 58 68 28 27 4 21 44 65 77 15 38 5 12 53 69 98 13 34 6 110 ft i of individuals that were captured exactly i times on occasions 1 2 t fl i of individuals that were captured exactly once on occasions 1 2 i 3 Estimation Results Model Est Boot s e Asy s e Phi CV 95 CI log transf 95 CI percentile Du ete Ae I Joea ena MIA ne A2 furi iere Me tte Ser ATA Re a es ere ee int i Lad Mb CMLE 145 5 25 40 18 02 2251 120 09 235 16 124 23 214 34 Mb UMLE 142 2 22 68 16 42 2 42 1
38. y chapters in a Handbook of Capture Recapture Chao and Huggins 2003 where some backgrounds and historical development are provided You are welcome to use CARE 2 for your own research and applications as long as you will not distribute CARE 2 in any commercial form If you publish your work based on the results from CARE 2 please use the following reference for citing CARE 2 Chao A and Yang H C 2003 Program CARE 2 for Capture Recapture Part 2 Program and User s Guide published at http chao stat nthu edu tw The maximum input size in CARE 2 is 2000 individuals and 80 occasions f your data exceed these sizes please send a mail to us indicating your size we will send you a modified program that fits your data input 2 Download and Setup Program CARE 2 can be downloaded from Anne Chao s website at http chao stat nthu edu tw softwareCE html First doubly click the downloaded file care 2 exe to unzip all files to a specified folder Then doubly click the executable file setup exe to install the program The source files along with six illustrative data sets will be stored automatically in the specified folder in your computer Analysis without covariates After the setup doubly click the executable file CARE 2 exe to start the program with the interface shown in Figure 1 Figure 1 The interface of CARE 2 for analysis without covariates P CARE 2 for application to animal data Analysis with
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