Upcoming Workshop Announcements:
- None scheduled at this time.
The Program MARK hypertext-based online discussion forum, Analysis of Data from Marked Individuals, is found at: http://www.phidot.org/forum/index.php.
Program MARK, a Windows Vista or XP program, provides parameter estimates from marked animals when they are re-encountered at a later time. Re-encounters can be from dead recoveries (e.g., the animal is harvested), live recaptures(e.g., the animal is re-trapped or re-sighted), radio tracking, or from some combination of these sources of re-encounters. The time intervals between re-encounters do not have to be equal, but are assumed to be 1 time unit if not specified. More than one attribute group of animals can be modeled, e.g., treatment and control animals, and covariates specific to the group or the individual animal can be used. The basic input to program MARK is the encounter history for each animal. MARK can also provide estimates of population size for closed populations. Capture (p) and re-capture (c) probabilities for closed models can be modeled by attribute groups, and as a function of time, but not as a function of individual-specific covariates.
Parameters can be constrained to be the same across re-encounteroccasions, or by age, or by group, using the parameter index matrix (PIM). A set of common models for screening data initially are provided, with time effects, group effects, time*group effects, and a null model of none of the above provided for each parameter. Besides the logit function to link the design matrix to the parameters of the model, other link functions include the log-log, complimentary log-log, sine, log, and identity.
Program MARK computes the estimates of model parameters via numerical maximum likelihood techniques. The FORTRAN program that does this computation also determines numerically the number of parameters that are estimable in the model, and reports its guess of one parameter that is not estimable if one or more parameters are not estimable. The number of estimable parameters is used to compute the quasi-likelihood AIC value (QAICc) for the model.
Outputs for various models that the user has built (fit) are stored in a database, known as the Results Database. The input data are also stored in thisdatabase, making it a complete description of the model building process. The database is viewed and manipulated in a Results Browser window.
Summaries available from the Results Browser window include viewing and printing model output (estimates, standard errors, and goodness-of-fit tests),deviance residuals from the model (including graphics and point and click capability to view the encounter history responsible for a particular residual), likelihood ratio and analysis of deviance (ANODEV) between models, and adjustments for over dispersion. Models can also be retrieved and modified to create additional models.
These capabilities are implemented in a Microsoft Windows interface. Context-sensitive help screens are available with Help click buttons and the F1 key. The Shift-F1 key can also be used to investigate the function of a particular control or menu item. Help screens include hypertext links to other help screens, with the intent to provide all the necessary program documentation on-line with the Help System.
The theory and methods used in Program MARK are described in more detail in an “electronic book“.
Sixteen different parameterizations of encounter data are providedin Program MARK.
Live recaptures are the basis of the standard Cormack-Jolly-Seber. Marked animals are released into the population, often by trapping them from the populations. Then, marked animals are encountered by catching them alive and re-releasing them. If marked animals are released into the population on occasion 1, then each succeeding capture occasion is one encounter occasion. Consider the following scenario:
Release —-S(1)—–> Encounter 1——-S(2)——> Encounter 2
Animals survive from initial release to the first re-encounter with probability S(1), andfrom the first encounter occasion to the second encounter occasion with probability S(2). The recapture probability at encounter occasion 1 is p(2), and p(3) is the recapture probability at encounter occasion 2. At least 2 encounter occasions are required to estimate the survival rate between the first release occasion and the first encounter occasion, i.e., S(1). The survival rate between the last two encounter occasions is not estimable because only the product of survival and recapture probability for this occasion is identifiable.
Generally, the survival rates of the CJS model are labeled as phi(1), phi(2), etc., because the quantity estimated is the probability of remaining available for recapture. Thus, animals that emigrate from the study area are not available forrecapture, so appear to have died in this model. Thus, phi(i) = S(i)(1 – E(i)),where E(i) is the probability of emigrating from the study area.
Lebreton et al. (1992) develop this model, and use SURGE (Pradel and Lebreton 1993)to provide parameter estimates. MARK provides the same capabilities as SURGE, plus additional types of models. Another program applicable to live recaptures is POPAN,which provides for estimation of population size and recruitment with the Jolly-Sebermodel. A third program is SURPH, which issimilar in its capability to MARK for live recapture and known fate data. None of the above 3 programs will handle the band recovery models, the joint live recapture and dead recovery models, robust design model, or the multi-state model.
Band Recovery Model
With dead recoveries, marked animals are released into the population, and re-encountered as dead animals, typically harvested. This theory has been developed by Brownie et al. (1985). Parameters estimated are survival rate, S(i),and band reporting rate, r(i), following Seber (1970). The primary model used by MARK differs somewhat from the parameterization of Brownie et al. (1985) because the f(i) of Brownie et al. are reparameterized as (1 – S(i))r(i). The primary parameterization of MARK results in better numerical estimation properties, plus, makes the band recovery models consistent with the parameterization of the CJS models. In particular, the use of covariates with the S(i) and r(i) is reasonable, because each parameter represents a particular process in the the overall band recovery process (unlike the f(i) parameter of the Brownie et al. model). However, the last S(i) and r(i) are confounded. In addition, with the S(i) and r(i) parameterization, S(i) is always estimated between zero and one. However, when the estimate of S(i) is at the boundary, i.e., close to or equal to one, the standard error is not estimated correctly. An equivalent situation occurs with the binomial distribution when either no successes occur in the data, or all successes occur in the data, and the standard error is estimated as zero. Both the S(i), r(i) and S(i), f(i) parmeterizations of the band recovery model are included in MARK.
Joint Live and Dead Encounters
The joint live and dead model is based on theory developed byBurnham (1993). The parameter space consists of survival rates [S(i)], recapture rates[p(i)], reporting rates [r(i)], and fidelity [F(i)]. An extension developed by Barker (1997) that allows live resightings during the interval between live recaptures is also available. Barker’s model extends the capability of Burnham’s model, plus allows for the option of no dead recoveries and live recaptures and live resightings.
Known Fate Model
Known fate data assumes that there are no nuisance parameters involved with animal captures or resightings. The data derive from radio-tracking studies,although some radio-tracking studies fail to follow all the marked animals and so would not meet the assumptions of this model. A diagram illustrating this scenario is
Release —–S(1)—-> Encounter 2 —–S(2)—-> Encounter 3 —–S(3)—->Encounter 4 …
where the probability of encounter on each occasion is 1 if the animal is alive or dead.
Closed Captures Models
The closed captures models allow the modeling of the initial captureprobability (p) and the recapture probability (c) to estimate populationsize (N). This data type is the same as is analyzed with Program CAPTURE(White et al. 1982). All the likelihood models in CAPTURE can be duplicated in MARK. However, MARK allows additional models not available in CAPTURE, plus comparisonsbetween groups and the incorporation of time-specific and/or group-specific covariatesinto the model.
Individual Covariatescannot be used with the closed captures data type because animals that were never captured(and hence, whose individual covariates could never be measured) are incorporated into thelikelihood as part of the estimate of population size (N). Models that canincorporate individual covariates existing in the literature (Huggins 1989, 1991) havebeen implemented in MARK. Estimates of population size are given for the Huggins’models, but these estimates are not quite as efficient as the closed captures data typewhere the statistical models are equivalent to those in Program CAPTURE. However,the ability to incorporate individual covariates makes the Huggins’ models moreappropriate if individual heterogeneity exists in the data. Further, the Huggins models seem to provide more reasonable estimates of N when nearly all the population has been captured. The Huggins models provide the population size as a derived parameter, and MARK allows these derived parameters to be used in model averaging and variance components analyses.
In addition, the Pledger(2000) models using mixtures of p values to model individual heterogeneity have been incorporated into all the closed capture models available in MARK. Thus, there are a total of 6 different different data types that can be used to estimate population size.
Robust Design Models
Robust Design Models are a combination of the CJS live recapturemodel and the closed capture models, and are described in detail by Kendall et al. (1997,1995) and Kendall and Nichols (1995). Instead of just 1 capture occasion betweensurvival intervals, multiple (>1) capture occasions are used that are close together intime. These closely-spaced encounter occasions are termed “sessions”.
For each trapping session (j), the probability of first capture(p(ji)) and the probability of recapture (c(ji)) are estimated (where i indexes the number of trapping occasions within the session), along with the number of animals in thepopulation (N(j)). For the intervals between sessions, the probability of survival(S(j)), the probability of emigration from the study area or more precisely, the probability of the animal not being available for capture on the jth occasion given that it was available on the j-1st occasion (gamma” (j)), and the probability of staying away from the study area or the probability of an animal not being available forcapture on the jth occasion given that it was not available for capture on the j-1stoccasion (gamma’ (j)) are estimated. Indexing of these parameters follows thenotation of Kendall et al. (1997). Thus, gamma”(2) applies to the second trapping session, and gamma’ (2) is not estimated because there are no marked animals outside the study area at that time. To provide identifiability of the parameters for the Markovian emigration model, Kendall et al. (1997) suggest setting gamma” (k-1) = gamma”(k) and gamma'(k-1) = gamma'(k), where k is the number of trapping sessions. To obtainthe “No Emigration” model, set all the gamma parameters to zero. To obtain the “Random Emigration” model, set gamma'(i) = gamma”(i).
Individual Covariates can be used to model the parameters S, gamma”, and gamma’ in the Robust Design data type. Individual Covariates cannot be used with the Robust Design data type for the p’s, c’s, and N’s with the closed capture models that include N because animals that were never captured (and hence, whose individual covariates could never be measured) are incorporated into the likelihood as part of the estimate of population size (N). Models that can incorporate individual covariates existing in the literature (Huggins 1989, 1991) are implemented in MARK, and individual covariates can be used to model the p’s and c’s. Estimates of population size are given for the Huggins’ models, but these estimates are not quite as efficient as the closed captures data type where the statistical models for M0, Mt, and Mb are equivalent to those in Program CAPTURE. However, the ability to incorporate individual covariates makes the Huggins’ models more appropriate if individual heterogeneity exists in the data. The Pledger (2001) models are also available to model individual heterogeneity in capture probabilities.
The multi-state model of Brownie et al. (1993) and Hestbeck et al.(1991) allows animals to move between states with transition probabilities. At this time, only the movement model without memory is implemented. An extension to the multi-state model to include dead recoveries is also implemented, as well as the robust-design multi-strata data types.
Additional extensions to the multi-state models include the open robust design multi-state model (Kendall and Bjorkland 2001), and multi-state models with misclassification (Kendall ).
Jolly-Seber Models (Jolly 1965; Seber 1965, 1982, 1986, 1992;Pollock et al. 1990, Schwarz and Arnason 1996) extend the CJS live recaptures models toinclude recruitment into the populations. In addition to the apparent survival and recapture probabilities of the Cormack-Jolly-Seber model (recaptures only model), the Jolly-Seber model allows estimation of the population size (N) at the start ofthe study, plus the rate of population change (lambda) for each interval. Also included in MARK are the 3 models developed by Pradel (1996) where only recruitment is estimated, both recruitment and apparent survival are estimated, and apparent survival and rate of population change are estimated. The POPAN model is also available in MARK for the Jolly-Seber situation.
Nest Survival Model
Estimation of nest survival has been a problem of interest since the Mayfield estimator. The nest survival model implemented into MARK allows estimation of daily nest survival rates as a function of both time of season and age of nest (Dinsmore et al. 2002). The nest survival model is also useful for “ragged” radio-tracking datasets, where all animals in the radioed population are not checked simultaneously, as required for the known fate model.
Estimation of the proportion of sites occupied is a common problem in ecology. MacKenzie et al. (2002) have formalized the model to incorporate the probability of detection of a species at a site. MacKenzie et al.’s model, plus a robust-design extension, (MacKenzie et al. 2003) have both been implemented into MARK. In addition, the single-season occupancy model of Royle and Nichols (2003), plus some extensions, have been implemented. Other occupancy models include the multi-site occupancy model (Nichols et al. 2008), and single-season and multi-season occupancy models with multiple states and state uncertainty (Nichols et al. 2007, MacKenzie et al. 2009).
Estimation of population size when marks are only applied once can be performed with the models in the NOREMARK software. However, Brett McClintock has developed likelihood-based models that provide improvements over the NOREMARK models, plus with being implemented in MARK, allow model selection with AICc, model averaging of population estimates, and variance components analysis.
The Encounter Histories File is the file that contains the encounterhistories, i.e., the raw data needed by Program MARK. Format of the file depends on the data type and examples are given in the help file. The convention of Program Mark is that this file name must end in the INP suffix. The root part of the file name dictates the name of the dBASE file used to hold model results. For example, the input file MULEDEER.INP would produce a Results File with the name MULEDEER.DBF and 2 additional files (MULEDEER.FPT and MULEDEER.CDX) that would contain the memo fields and index orderings, respectively. MULEDEER.CDX will be erased upon exit from MARK.
Encounter Histories Files do not contain any PROC statements, but only encounter histories or other special formats such as recovery matrices. You can have group label statements and comment statements in the input file, just to help you remember what the file contains. The interactive interface adds the necessary program statements to produce parameter estimates with the numerical algorithm based on the model specified.
Once the encounter histories file is created with an ASCII text editor, the next step is to execute the program and select File, New. You then enter the number of Encounter Occasions, number of Groups, and the Data Type. After this input is provided,the Parameter Matrices are created, one for each parameter and group. These matrices default to Time matrices, which you can then modify to other possibilities using menu options. If you don’t need any additional constraints, which can be specified via the Design Matrix, then choose the Run menu option to produce the numerical estimates. The Run Window has additional requests for input, including the Run Title, Model Name, Time Intervals, and Encounter Histories File Name. When you click the OK button to run compute the numerical estimates, you must wait for this process to complete before proceeding. At that time, a Results data base will be created (if you request it), and the output stored in the data base for comparison with other models you may provide.
The input file for the example data from American Fisheries Monograph No. 5 (Burnham et al. 1987) is provided as AFSMONGR.INP. This Cormack-Jolly-Seber data set has 5 re-encounter occasions, 2 groups, and is live recapture data. Specify these values when you start the program from the File | New menu choices. In the File Name Dialog Window, select the AFSMONGR.INP file as the Encounter Histories Input File. Alternatively, the results database for this example is also included with the program in the Examples subdirectory. Use the File | Open menu choices to open this file, and review the model results provided.
No paper documentation is available for MARK. Electronic documentation is provided in the Windows help file that accompanies the program and available here as HTML files. Open up the Help document with the program, and read some of the documentation, or check out the HTML version. You can print any of this material if you really want hard copy.
A reasonably complete description of Program MARK was developed for the Euring 97 conference, available as a PDF file. I consider this paper as the primary citation for Program MARK:
White, G.C. and K. P. Burnham. 1999. Program MARK: Survival estimation from populations of marked animals. Bird Study 46 Supplement, 120-138.
An electronic book, Program MARK A Gentle Introduction, is being developed by Evan Cooch at Cornell University. For the complete novice, this is the place to start to learn how to run MARK. This guide is a work in progress, so is not complete, nor ever will be as long as MARK continutes to be developed.
Notes concerning the theory and use of MARK from the graduate course taught at Colorado State University: FW663, Analysis of Vertebrate Populations, are available. This is the same material provided as “Technical Background” from Evan’s site referenced in the preceding paragraph.
A set of slides that illustrate the concepts of MARK is available for viewing. These slides give a general overview, and portions of them are used in the slide talks listed below.
A one-day workshop on Program MARK was given at the Second International Wildlife Management Congress in G�d�ll�, Hungary, July 2, 1999. The following are the slide talks given:
|Introduction to Program MARK — Gary C. White|
|Exploring Ecological Relationships in Survival and Estimating Rates of Population Change Using Program MARK — Alan B. Franklin|
|The Robust Design for Capture-Recapture Studies: Analysis using Program MARK — William L. Kendall|
|Jointly Analyzing Live and Dead Encounters using MARK — Richard J. Barker|
|Advanced Features of Program MARK — Gary C. White|
In addition, the following papers were published from this workshop.
|First Steps with Program MARK: Linear Models — Evan Cooch|
|Exploring Ecological Relationships in Survival and Estimating Rates of Population Change Using Program MARK — Alan B. Franklin|
|The Robust Design for Capture-Recapture Studies: Analysis using Program MARK — William L. Kendall|
|Jointly Analyzing Live and Dead Encounters using MARK — Richard J. Barker and Gary C. White|
|Advanced Features of Program MARK — Gary C. White, Kenneth P. Burnham, and David R. Anderson|
One of the problems with obtaining software from the Web is that hard copy documentation is not available, such is the case for Program MARK. The following site provides information on how to cite electronic documents: MLA-Style Citations.
Standard Version of MARK Generated Using Visual Objects 2.8
Copy the single setup.exe file to your hard disk, and execute it to install MARK. This setup file should place a MARK icon on your desktop, register the necessary DLL files, and put the examples distributed with the program in an Examples subdirectory under the MARK directory.
Windows XP Setup Problems
Some folks are having difficulties downloading MARK onto Windows XP operating systems. The problem concerns the setup.exe program wanting to create a file entitled TGETUP9 when XP already has one. Here’s the work around from Jon Runge:
1. Through Window Explorer go to Tools: Folder Options: View. Check the “show hidden files and folder” box and uncheck the “hide protected operating system files” box.
2. Go to the folder C:\WINDOWS:\TEMP. Rename TGSETUP9.TMP to something like TGSETUP~9.TMP.
3. Run Setup.exe for MARK.
4. When done, go back and restore TGSETP9’s original name.
Mac and Linux Machines
To run MARK on a Mac (from Evan Cooch):
Equipment Tested: Macintosh PowerBook G3 (Lombard) 333 MHz with 192 MB of ram (note that Mac clock speed numbers are NOT the same as Windows/Intel clockspeeds, i.e., a 333 MHz Mac is faster than a comparable WinTel machine).
Software: Virtual PC version 3.0.3 with Windows 98. Able to use MARK under Virtual PC with Windows 98. Also able to use Microsoft Access under Virtual PC.
Recommendations: The more ram you have the better. Set your Virtual PC program’s memory to as much ram as you can afford. The emulator program (Virtual PC)actually runs Windows using the amount of ram that you set aside for the emulator. I set the Virtual PC to use 69MB of memory and find that this allows Windows/Dos software to run as fast as a real contemporary WinTel machine. Also, I’ve had best results running the Mac OS with an abbreviated set of Extensions. You can easily do this by creating a reduced Extension set with the Extension Manager (this is a Control Panel).
Update (2/3/06) from Martin Renner:
Equipment tested: 800 Mhz G4 Dual Processor, Mac OX 10.3.9, Virtual PC 6 running Windows 98 and MARK version 4.10.
While not really fast, this configuration is perfectly usable. Allocating more RAM helps.
When preparing .inp files on the Macintosh it seems to be important to convert the end-of-line character from mac <CR> or unix <LF> to dos/windows <CR/LF>. This can be easily done in BBedit, a number of free utilities, or by opening and saving the file in WordPad.
To run MARK on a Linux machine (from Len Thomas):
Software: VMWare — a BIOS emulator for both Linux and WinNT that effectively lets you run one or more “virtual computers” inside your current operating system. So, for example, you can open a Win95 window from your linux box, and everything within that window thinks its in Windows 95. Of course you do need a Win95 license for this, but at least it gets around the problem of wanting to run linux for most things, but having some legacy software in windows.
Many people use VMWare because they do most things in linux (SPlus, C++,F90), but then some people want or have to use MS Office for their word processing, for example. I use it the other way around: I do most things in WinNT (Visual Basic, etc), but need to be able to test my programs in “vanilla” Windows NT, 98, 95, 2000 systems, so I can run these inside my main machine.
Communication between virtual computers is via virtual networking.
At this time MARK has never been tested under VMWare in linux, but MSOffice works, so MARK is expected to work.
Older changes are stored here. Recent changes include the following:
190. The robust design multi-state data type with open primary sessions and mis-classification of states is now working correctly. More details are provided in the help file.
191. Data cloning is implemented as an option in the Results Browser under the Output | Specific Model Output menu choice. Data cloning is useful for determining estimability of parameters. Output from the analysis is presented in an Excel spreadsheet.
192. A model name is now displayed in the caption heading of the design matrix, along with a menu choice (included in the right click button pop-up menu) for the user to change the model name.
193. A bug in the robust design Pradel models that included N was fixed. The first c parameter of the last primary session was getting set to a log link instead of the value specified for the PIM (i.e., the first c parameter was treated as an N parameter which gets the log link by default). This bug only appeared in models that included N in the likelihood, not the Huggins parameterizations that do not include N.
194. An option from the Results Browser | File menu was added to replace the encounter histories file and rerun all of the existing models. If you replace the input data with a different data set, you have to rerun all of the models because chaning the data means that none of the results in the Results Browser are now correct.
195. An option to view the encounter histories file in the editor was provided under the Results Browser | Output menu choice. Note that the input data summary procedure is also available in the same menu.
196. The odds ratio estimator of lambda for multi-season occupancy models (labeled lambda’ on page 200 of the MacKenzie et al. occupancy book) was added as a derived parameter for parameterizations of the multi-season occupancy models.
197. The numerical output from the random effects model that is placed in the Results Browser when an AICc value is calculated is now stored in the Model Notes field of the Results Browser.
198. The data bootstrap estimator was modified to fix 2 issues. First, encounter histories files with aggregated frequency counts are now de-aggregated so that individual encounter histories are sampled, although specifying a covariate to cluster the encounter histories still works correctly. Second, the number of encounter histories in the original data for each group is used to determine the number of bootstrap samples to include, rather than the number of clusters as was what previously was done. Third, specification of a c (over-dispersion) parameter >1 in the simulation input window means that this value will be applied during the resampling. As an example a value of c = 1.5 means that approximately 1/2 of the encounter histories sampled will get a frequency count of 1 and the other 1/2 a value of 2. However, the total number of encounter histories will remain approximately the same as the original data for each group.
199. The psiB (occupancy of species B) and psiAB (joint occupancy of both species) parameters were added as derived parameters for the 2-species conditional occupancy model of Richmond, O. M. W., J. E. Hines, and S. R. Beissinger. 2010. Two-species occupancy models: a new parameterization applied to co-occurrence of secretive rails. Ecological Applications 20:2036-2046.
200. A bug in the robust design occupancy models with heterogeneity (mixtures for p) was fixed in all three parameterizations.
201. Simulators were added for the CJS Pleder, CJS random effects, single-season multiple state occupancy, and multiple-season multiple state occupancy data types.
202. An option was added to the Help menu choice to list out all of the data types available in MARK.
203. Code was added to check the true model when specified in the simlation module to see if a simulator is actually available for the specifiedd data type. Plus, you can list all of the data types that can be simulated with an option under the Simulation menu choice.
204. The Barker robust design model was updated to a proper definition of the temporary emigration parameters: the gamma’s were changed to a‘s (availability) to properly reflect their meaning. Also, this data type now properly handles unequal time intervals (L) between primary sessions. S and F are corrected as S^L and F^L, and the R and R‘ parameters are corrected as 1 – (1 – R)^L and 1 – (1 – R‘)^L. The a” and a‘ parameters cannot be corrected for unequal time intervals, so must remain time-specific. No correction is needed for r because no matter how long the time interval, an animal can only die once.
205. The Barker model was updated to correctly handle unequal time intervals (L). S is corrected as S^L, and the R and R‘ parameters are corrected as 1 – (1 – R)^L and 1 – (1 – R‘)^L. The F and F‘ parameters cannot be corrected for unequal time intervals, so must remain time-specific. No correction is needed for r because no matter how long the time interval, an animal can only die once.
206. The regular robust design model was updated to change the effect of unequal time intervals (L) between primary sessions. S is still corrected as S^L. However, because the gamma” and gamma’ parameters cannot be corrected for unequal time intervals, they must remain time-specific to accommodate unequal intervals. For the case where time intervals are multiples, e.g., L = 1 and L = 2, a dummy primary session can be used with all values equal to dots (.). However, you better understand which parameters remain estimable and which will become unidentifiable when doing this.
207. A bug with dots in the encounter history was fixed in the Huggins robust design data types, so that the estimate of N is now correctly computed. In addition, the robust design data types with N in in the likelihood were changed to not allow dots in the encounter history because N cannot be correctly estimated in these data types when dots are in the encounter histories.
208. The multi-season occupancy models with gamma (colonization) and epsilon (extinction) were updated to not correct for unequal time intervals using L as a power. This change was made because the previous correction did not work correctly.
May, 2012 (Version 6.2)
209. Two versions of the FORTRAN numerical estimation code are now supplied with MARK in the setup.exe file, with both now generated with the gfortran compiler. Depending on whether you are running a 32-bit or 64-bit version of the operating system, either the 32-bit or 64-bit version of the mark.exe file is used for numerical estimation. Both include parallel processing using multiple threads You can specify the number of threads to use for parallel processing in the File | Preferences menu choice. The number of threads used and the maximum available are reported at the top of the full output text file.
210. The Pledger and Schwarz (2002) mixture model for the Seber (1970) band recovery model was added, available from the “Change Data Type” menu from either the Seber or Brownie dead recoveries data type.
211. The individual heterogeneity random effects model for the Seber (1970) band recovery model was added, available from the “Change Data Type” menu from either the Seber or Brownie dead recoveries data type. Although both sigmaS and sigmar are included in the model, the sigmar parameter is not identifiable.
February, 2013 Presto (Piping) Plover Version (Version 7.1)
212. The Richmond et al. (2010) 2-species occupancy model was extended to a multi-season model using the transition matrix described in Miller et al.(2012). The help file is titled “Occupancy Estimation Robust Design 2 Species”.
213. Simmulation capability for the single-season Richmond et al. (2010) 2-species occupancy model was added.
214. Simmulation capability for the multi-season Richmond et al. (2010) 2-species occupancy model was added.
215. Two bugs with the specification of threads were fixed, so that multiple threads now run as specified.
216. The data type names for the closed captures data types were changed to be more informative. This also changes the names of all robust design data types.
217. The dead recoveries data types were consolidated into a single entry on the new data analysis screen. These data types were the Seber, Brownie et al. and the BTO dead recoveries.
218. A bug that was apparently introduced in December, 2012, concerning retrieval of PIMs that were fully specified was fixed.
219. Added the derived parameter of survival over all occasions to the data type Lukacs survival of young with a marked adult.
220. The product of columns menu choice was modified to use the design matrix product function for columns containing individual covariates.
221. We have made the following change to the multistate robust design (open and closed) with state uncertainty (3 data types). We have reparameterized the mixture parameters for the first primary period, so that pi1^s = w1^s*p1^*s / sum[w1^s*p1^*s] (see Kendall et al. 2012 Ecology). Therefore pi1 no longer exists as a parameter in the likelihood, and there are now K-2 parameters in the pi PIMs, where K is the number of primary periods. The first parameter listed is for primary period 2, and the last pi is for primary period K-1. There is a pi estimate only for the first S – 1 states, where S is the number of states. The pi for the last state is obtained by subtraction. We made this change because for the common case where a given state is never known with certainty, pi1 and therefore the survival and transition probabilities for primary period 1 for that state were not estimable.
222. The ability to “lasso” blocks in the PIM Chart was extended so that once you have lassoed a set of blocks, as shown by changing to green instead of blue, you can right click and use the Constant, Time, Age, or All Different pop-up menu choices to make the selected change to the lassoed blocks. Note that to lasso a block, you only need to include the lower left corner inside the lasso rectangle. You lasso blocks by holding down the shift key and then the left mouse button and draging out the resulting rectangle to capture blocks.
223. The pent parameter of the data types (1) Open Robust Design Multi-state, and (2) Open Robust Design Multi-state with classification uncertainty, has been changed to obtain the last value by substraction, rather than the first as was originally programmed. This change makes it easier to fit linear and quadratic models to the probability of entry parameter in these models.
224. The Open Robust Design Multi-state with State Uncertainty data type was extended to create a new data type that allows seasonality. The idea is that the attribute that allows determination of the state may not be identifiable, so that an additional set of parameters, alpha (PIM for each primary session and each state) to allow the attribute to become identifiable has been added. In addition, the attribute may go away, so yet another set of parameters, c (again with a PIM for each primary occasion and state) was added to allow the attribute to cease. See the updated help file for more details on these models.
225. The ability to save the summary statistics from the MCMC procedure into a CSV (comma sep arated variable) file that can be read by Excel was added. If the file name is set to blank, then no CSV file will be created. The addition of this option to the MCMC dialog window forced a reformatting of the window.
226. An option was added to File | Preferences dialog window to make the first row of the time effect in a design matrix the reference row, instead of the last row as was previously the default. This option affects the Full Design Matrix and Pre-defined Models that build a design matrix.
227. The MCMC output now includes 80%, 90%, and 95% highest posterior density (HPD) credible intervals (CI) for each parameter posterior distribution. In addition, these values are also saved to the CSV file.
228. The multi-scale occupancy model (data type number 123) was changed to be easier to understand and the notation standardized with the original paper. I changed the name of the p PIMs, to be ‘Primary’ instead of ‘Sampling Occasion’ as previously. Thus a case with L = 16 devices and K = 3 visits still results in 16 p PIMs, each with 3 entries. However, p PIMs are now labeled as ‘Primary 1’, ‘Primary 2’, etc. Existing DBF and FPT files will not work properly with the new version just installed on the web, in that this name change means that you can retrieve a model, but not run it again because of the name change. The encounter histories file does not need to be changed as it is still organized the same way. The help file has also been expanded and more fully explains the definition of parameters in the PIMs, how to input the parameters K and L, plus how to organize the encounter histories. The simulator still works for this data type.
229. The occupancy model with correlated detections (Hines, J. E., J. D. Nichols, J. A. Royle, D. I. MacKenzie, A. M. Gopalaswamy, N. S. Kumar, and K. U. Karanth. 2010. Tigers on trails: occupancy modeling for cluster sampling. Ecological Applications 20:1456–1466.) was added to MARK (data type 143). The model was extended to handle multiple secondaries within each segment.
230. The occupancy model relaxing the closure assumption (Kendall, W. L., J. E. Hines, J. D. Nichols, and E. H. C. Grant. 2013. Relaxing the closure assumption in occupancy models: staggered arrival and departure times. Ecology 94:610–617.) was added to MARK (data type 144).
231. An option was addd to File | Preferences to use the 32-bit mark.exe file (for more speed) instead of the 64-bit version (for very large jobs needing additional memory).
232. A rather severe problem was uncovered with the gfortran compiler when a “large” problem is optimized in the mark.exe code. The problem occurred on all machines running the gfortran code: PC, Cray, or Unix. Specifically, when a design matrix of dimension 460 X 460 was used, the cpu time to allocate thread-specific copies for the multiple threads was ~100 times what would take a run without multiple threads. Several changes were made to circumvent this problem. For analyses without individual covariates, only the original design matrix is needed — not multiple copies — so a test to determine this condition was added and processing without multiple copies then proceeds. Further, an option was added to not use parallel processing with a single thread, and thus avoid the overhead of the OpenMP with multiple threads. If you find that a “large” design matrix with individual covariates is taking an exorbitant amount of time, try specifying threads=1.
233. Output from the variance components/random effects analysis was better labeled, and the design matrix is now listed at the bottom of the output. All of this output is stored as a model memo when the model is run to obtain weights.
234. The variance components/random effects analysis was re-written to provide more error messages during execution. I’ve had issues with running this moodification on 32-bit XP machines. If you have trouble, let me know.
June, 2014 Version 8.0 California Sea Lion
235. The N parameter in all of the closed captures data types and robust design derivatives has been changed to be labeled f0 to prevent users from mistakenly thinking setting the N parameters equal is evaluating this hypothesis. N is still provided as a derived parameter.
236. The random effects version of the Huggins estimator has been added for closed captures, robust designs, closed multi-state, and Pradel robust designs. The estimator uses Gaussian-Hermite quadrature to integrate out individual random effects on detection probability, p. Population estimates are provided as derived parameters based on the estimated mean detection probability.
237. The Fletcher chat estimator (Fletcher 2012) has been added to the full output file, and also for collection by the simulator. This estimator requires knowing the total number of possible encounter histories, which can be problematic when parameter estimates preclude some histories. Examples of this problem are p = 0 in the CJS data type, or transition probabilities (psi) fixed to 0 or 1 in multi-state models. Other similar problems are caused by dots in the encounter history, or losses on capture.
238. The random effects version of the occupancy estimator has been added for single-season occupancy, and multi-season robust designs. The estimator uses Gaussian-Hermite quadrature to integrate out individual random effects on detection probability, p.
239. The random effects version of the known fate estimator has been added, mainly for use as a way to simulate overdispersion in the form of individual heterogeneity or parameter heterogeneity. The estimator uses Gaussian-Hermite quadrature to integrate out individual random effects on survival, S. However, because the saturated model is one of the useful models for known fate data, the random effects estimator of sigma will be non-identifiable if the saturated model is used.
240. The ability to select a subset of the PIMs and view the subset in the PIM Chart has been added. Selection of the PIMs to view can be done from a menu choice under the PIM main menu, right-clicking on the PIM Chart, or using the lasso and right-clicking on the PIM Chart.
241. A menu choice under the Run menu in the Results Browser has been added to compute variable weights when a set of models has been constructed with the ‘Subset of DM Models’. Care in naming the variables when creating the models should be used to avoid unintended overlap of variable names. The user must be careful to not have additonal models in the Results Browser that will cause the computed weights to be invalid. Basically, the set of models should be a balanced set of models for the variables being considered.
242. Computation of the p* for the zPNE mark-resight model was changed to provide a better approximation when there is a lot of individual heterogeneity.
243. Previously the number of points to perform the numerical integration in the sigma individual heterogeneity models was 15. This value was changed to 101 to provide a better approximation of this integral. For a single integration, the increase in computation time is not too bad. But when you have double integrals, this change will noticely slow down the optimization process.
244. A bug in replacing the effective sample size value when using the replace data option was fixed. This bug affected the value of the AICc in the Results Browser.
245. Occasion-specific population estimates were added as a derived parameter to the immigration-emigration mark-resight estimator.
246. The ability to recompute real and derived parameter estimates in the Results Browser without re-optimizing the model was added. The capability is useful when you have not been consistent with the values of individual covariates used to compute the real and derived parameters, but now want to model average a set of them. This option is available under the Results Browser menu choices Run | Regenerate Real and Derived Estimates.
247. Model-averaging an individual covariate plot is now possible with the menu choices from the Results Browser of Output | Model Averaging | Indiviudal Covariate Plot.
248. The ability to click on the model name or number of parameters in the Results Browser and change their value has been fixed so that you now get the same results as if you had used the menu choices to make changes.
249. Derived parameters have been added to the Robust Design Multi-state Conditional Occupancy data type (= 124) for the psi estimates through time.
250. A bug was fixed that resulted in an error message when you tried to retrieve a model with the PIM Chart open.
251. The false-positive models of Miller et al., both single and multi-season, have been implemented.
252. The predefined models option was modified to provide the standard closed captures models when closed captures are being used.
253. Three new mark-resight models were added that appropriately handle marked animals that were not individually identified. An additional parameter, r, models the probability that a marked animal is identified to individual, and not just recognized as marked.
254. The Huggins random effects estimator was add to the list of models for use with density estimation using telemetry data.
255. The Huggins random effects closed captures estimator was add to the list of models for use with the Robust Design Barker live-dead data type.
256. The median c-hat procedure was extended to include the CJS random effects (135), Burnham live/dead random effects (138), and the multi-state live-dead (21) data types.
257. The median c-hat procedure was extended to include the Huggins closed captures models. To assess goodness-of-fit of these models, you have to condition on M(t+1), the number of animals captured one or more times. The simulation procedure was modified for this median chat situation, but still uses N for regular simulations.
258. The ability to record M(t+1) from simulations is now provided.
259. Data types for the robust design Pradel model were added where lambda in the Pradel portion of the likelihood is replaced by the rato of population estimates from the closed captures portion of the likelihood. These 7 new data types can be specified from the “Change Data Type” option under the PIM menu when any of the robust design Pradel data types are specified.
260. Floating point numerical issues (underflow, overflow, and divide by zero) are now reported as warnings in the MARK output. Overflows and divide by zeros should be considered serious errors, and output checked carefully for non-sensical values. Underflow occurs when a numerical value is small enough that it becomes zero. Underflows are not uncommon in MARK, particuarly with Gaussian-Hermite numerical integration for the individual random effects data types. Underflows will also occur during numerical optimization when starting values are far enough from the final estimates that some of the computed encounter histiory probabilities are basically zero. For these reasons, generally underflows are not terribly serious assuming the optimization worked through them.
261. The capability to specify more informative real parameter labels/names has been added. You can set the real parameter labels from the Run menu in the Results Browser, or from the Appearance menu from the design matrix window. However, this capability may be confusing in that if you change the PIMs, then the values you have specified are lost and the default labels return. Hence, this capability is most useful for when you set up your PIMs and then never change them while you build your models in the design matrix. The dipper example distributed with the program shows an example of the use of user-specified labels. To reset the real parameter labels from a previous model, just retrieve the model.
262. The ability to simulate Barker robust design data types has been added.
263. The multi-season occupancy model with relaxed closure (data type 170, Chambert et al. 2015 Methods in Ecology and Evolution 6:638-647) has been added.
264. Three additional derived parameters were added to the single-season occupancy model with relaxed closure (data type 144, Kendall et al. 2013 Ecology 94:610-617): probability of presence (alpha), mean arrival time, and mean departure time.
265. A long-term bug that showed itself when users opened MARK files with some models having large values of delta AICc was fixed. You should no longer have to hit “Ignore” to get through the list of models to get the Results Browser to open.
266. The capability to handle dots in the encounter history was added to the POPAN data type (data type 19).
267. The capability to specify the number of nodes to use in Gaussian-Hermite quadrature (used in random effects individual heterogenity models) was added to the File | Preferences dialog window. The default value is 101 nodes, with minimum of 15 and maximum of 505.
268. Processing of individual covariates in the mark-resight data types (114, 115, 120, 158, 159, and 160) was updated based on changes provided by Brett McClintock. These data types now give identical answers when an individual covariate of all 1’s is used in a model compared to the same model with no individual covariate.
269. The Poisson mark-resight models were updated to add a data type with a zero-inflated Poisson distribution, useful for modeling animals that are marked but that may not be available in the next resighting occasion. The additional parameter (w) is the probability that a newly marked animal remains alive and on the study area during its first resighting occasion. In addition, all 3 of the Poisson models were extended to assume the same individual random effect across primary periods, rather than the current implementation of individual random effects changing between primary periods. There are now 6 Poisson mark-resight data types:
|Data Type #||Code||Description|
|115||PoissonMR||Poisson Mark Resight with Robust Design within primary periods|
|160||UnIdPoissonMR||Unidentified Marks Poisson Mark Resight with Robust Design within primary periods|
|171||ZiUnIdPoissonMRwithin||Zero-inflated Unidentified Marks Poisson Mark Resight with Robust Design within primary periods|
|172||PoissonMRacross||Poisson Mark Resight with Robust Design across primary periods|
|173||UnIdPoissonMRacross||Unidentified Marks Poisson Mark Resight with Robust Design across primary periods|
|174||ZiUnIdPoissonMRacross||Zero-inflated Unidentified Marks Poisson Mark Resight with Robust Design across primary periods|
Data types 115, 160, 171, and 174 have simulators built. Generally, I would recommend using data types 171 and 174 for simulation because these models are more flexible.
March, 2017 Version 8.2 Hawaiian Goose Photo
270. Closed captures data types that use the full likelihood parameterization, including robust designs, now report population size (N), instead of f0 as a real parameter in the MCMC output. Previously, the parameter was labeled f0, but was actually N.
271. The robust design (multi-season) version of the multi-site occupancy model has been added (data type 175). Parameters for the robust design extention are psi for the first primary occasion, epsilon (extinction), and gamma (colonization). A simulator has also been built.
272. The option to export estimates for each encounter history to Excel from the individual covariate plot now includes SE and confidence intervals, plus the individual covariate values that generated the estimate.
273. The multi-state data types now show the matrix of transitions when the input data summary is requested.
274. Derived parameter estimates are now included in MCMC output. Derived parameter names are provided with the DLABEL statement. Four abbreviations are used to identify parameters: Grp for group, Str for strata or state, Ses for primary session, and Occ for occasion.
275. When derived parameter estimates are requested in an Excel file, parameter names are now provided.
276. The Watanabe-Akaike information criterion, WAIC, was added to MCMC output.
277. The input data summary command has been extended to provide summaries of the primary session encounters for robust designs, including multi-state robust designs.
278. A bug was fixed that allowed the user to compute profile likelihood confidence intervals for real parameters that were modeled with an individual covariate. Such a confidence interval is nonsense because each encounter history has its own real parameter estimate. Now a warning message is printed when a profile confidence interval is specified for a real parameter modeled with an individual covariate.
279. The Mark-Resight models have been modified so that individual covariates are now properly handled for the r, w, and g parameters.
280. The NLBETA function was added. All the beta parameters in MARK are linear parameters. The NLBETA function allows the user to have a non-linear parameter embedded in the design matrix. The NLBETA function adds parameters to the beta parameter list. As an example, to model an asymptotic threshold function, the design matrix entry
would estimate the threshold with the slope specified in the nlbeta(1) function. Details are provided in the MARK help file. The addition of this capability required major changes to MARK code, so if you encounter errors, let me know.
281. A Hidden Markov model (data type = 178) has been added. This model is NOT robust design, but models state uncertainty with observed events. See the MARK help file for more details.
April, 2018. Version 9.0 — Black-bellied Whistling Ducks
282. The numerical computation of the first and second derivatives has been tweaked to improve accuracy.
282. A major change in estimating the number of parameters that were estimated in a model has been implemented. Two methods are now used. First, a numerical threshold is estimated from the gradient (G) vector as 2 times the maximum absolute value in the gradient. This numerical threshold is then used to determine the number of values in the singular-value decomposition (S) vector that exceed the numerical threshold, with this value taken as the number of parameters estimated. Second, the S vector is searched for the largest ratio S(i)/S(i + 1) between 2 consecutive values, as well as the next largest ratio between 2 consecutive values. If the ratio is >50, the index of the numerator for the maximum ratio is taken as the number of parameters estimated. When both of these estimates agree, all is well. If the 2 estimates disagree, the maximum of the 2 is reported as the number of parameters estimated, and a warning is printed in the full output that the 2 estimates disagree. An option has been provided in the File | Preferences menu choice to make this warning very explicit. The model name has the phrase “Check Par. Cnt.” added to the front of the name, and the model name is shown in blue in the Results Browser. The user should then check the full output to see if the estimate reported is reasonable, or if the number of parameters estimated should be changed. Once an appropriate value is set, the blue coloring can be eliminated by clicking on the model name and deleting the phrase “Check Par. Cnt.”. Unfortunately, neither of the 2 methods can detect that a parameter estimated at its boundary should be counted, e.g., p-hat = 1 with a logit link, or pent-hat = 0 with a MLogit link. Improved numerical precision of the derivatives just made this problem worse. Users should use the sin link when possible to detect parameters estimated at the boundary.
283. All data types can now be used with the Bootstrap Data menu choice under Simulations. You won’t even be hounded to catch the moving OK button!
284. When the number of marked individuals is known on a given encounter occasion, the zero-inflated Poisson log-normal mark-resight models (data types 171 and 174) can now account for marked individuals that were known to be alive but were temporarily unobservable (e.g., off the study area). Previously these individuals would require a ‘..’ in their encounter histories when the number of marks was known, but these can now be accounted for by including a ‘*0’ in the encounter history to indicate when a marked individual was known to be alive but temporarily unobservable.
285. The Fletcher c-hat estimator was updated to include a small sample correction suggested by David Fletcher.
286. Fixed a bug in the Barker (data type = 8) and Barker robust design (data types = 18, 43, 44, 45, 46, 47, 162) models. This bug was caused by changes in March, 2018. So if you’ve used any of these data types since then, you need to re-run the models to get correct results.
287. Details of how the simulated annealing optimization algorithm is progressing have been added to the screen output. The last 4 values of the -2log L function are provided, plus the parameter estimates for the minimum -2log L value. These parameter values can be copied from the screen to start a regular optimization if simulated annealing is taking too long. A description of the output is provided in the help file under Optimization Method.
288. The Variance Components module was updated to allow multiple variance component model to be run simultaneously.
289. The Append and Append Subdirectory options to add model results files run on a different machine now do not leave you directory in the location of the files to add, but rather the directory which is associated with the Results DBF file.
290. The individual random effects sigma models were added to the multi-state data type for the state-specific S and p parameters. The new data type is 181. You can access this new data type from either the File | New creation process, or by changing data type from the traditional multi-state data type (=9). Individual random effects were not added to the psi parameters because each psi would require its own sigma parameter, creating a model that would be too complex to estimate the parameters.
291. Data simulators were built for data types 178 (Hidden Markov with state uncertainty), 179 (Barker multi-state with state uncertainty and transitions at the end of intervals) and 180 (Barker multi-state with state uncertainty and transitions at the beginning of intervals).
292. The multi-state Jolly-Seber data type = 183 was added. This data type is a combination of the multi-state and POPAN data types. However, this model is conditioned on the number of captures, so population abundances of the super population (N*) and each of the states at each occasion are available as derived parameters. The ability to specify which of the probability of entry (pent) parameters (default is time 0 before the first capture occasion, or before the last occasion) and which of the transition (psi) parameters for each state is available through the Change PIM Definition option.
293. A bug in the Link-Barker data types (31, 134, 136) when unequal time intervals were used was fixed.
294. I added a feature to the simulated annealing code to save the current best estimates in a temporary file in the same directory as the output file is being created. The temporary file has the same name as the output file, except that the suffix is changed to “SIM”, or if no suffix, “.SIM” is added. So if the run is killed by your or other reasons, the temporary file will provide estimates to start the usual optimization algorithm closer to the optimum. But if the run completes normally, the temporary file disappears.
August, 2022. Version 10.0.
295. I have made major changes to the code to use FORTRAN 2008 features to change how the PIMs were addressed. These changes make the code much more readable and easier to modify and add data types, and should not be noticed by users.
296. Four additional derived parameters were added to data type 183, multi-state Jolly Seber: expected ingress time, expected egress time, and expected residence times 1 and 2.
For questions or to let me know about problems you have encountered, send email. Please try to provide as much documentation as possible to help me duplicate your problem. In particular, I would like to have the input file that caused the problem, and the values you entered for the number of occasions, the number of groups, and the data type. Further, if you have created a results file, please send these via a zipped attachment. Both the *.DBF and *.FPT files must be forwarded — both are needed to see the models you have built.
Email: Gary.White at ColoState.edu
An alternative to a week-long workshop is to take FW663, Analysis of Vertebrate Populations, a 5-credit graduate course taught by Larissa Bailey and William Kendall in alternate spring semesters at Colorado State. Out-of-state tuition for the course is approximately $2,700, and cheaper for Colorado residents. The class meets MWF from 8-12 from mid-January until the first of April. The class will next be taught spring semester, 2014, beginning mid-January and ending early April.
Another intermediate level workshop is scheduled for 1-6, 2014, in Fort Collins, Colorado.
Individuals desiring a comprehensive treatment of the background material of Program MARK, and gaining a familiarity with using the program, are encouraged to take the course FW663, Sampling and Analysis of Vertebrate Populations, co-taught by Larissa Bailey and William L. Kendall. The course meets from mid-January until the last week of March, MWF from 8-12. The class will next be taught spring semester, 2014. We strongly encourage students from outside Colorado State University to participate in this course.
Some known problems that you should be aware of:
- When plotting real parameters as functions of individual covariates, you must be cautious about what link function is being used. As an example, suppose lambda (population rate of change) is being plotted as a function of an individual covariate. The model may have been run with the default link function of logit or sine, but the numerical processor automatically switches the link function to a log because lambda estimates must be >0. However, the individual covariate plot does not know this subtlety and assumes a logit link as was specified when the model was run. To properly plot lambda with a log link, you must re-run the model with the parameter-specific link option, specifying log for the lambda parameter.
- The deviance for the closed captures model divided by its degrees of freedom is not a valid estimate of c-hat. This problem carries over to the robust design model. I believe the problem is because these models are not in the exponential family, and this type of estimate of c-hat is only valid for models in the exponential family. This problem also extends to the robust design model.
- The deviance is not a good test of the goodness-of-fit of the model for sparse data. Sparse data may result from few releases, but even with lots of releases and a large number of encounter occasions, the data will be sparse. Sparse data result in small expected values for lots of the capture histories. To have observed a capture history, at least 1 animals must have been observed with this history. For a small expected value, the contribution to the deviance is large, i.e., (1 – Expected)^2 / Expected is large. A pooling algorithm does not seem possible that will fix this problem. To obtain a valid test of the goodness-of-fit of the model, and an appropriate estimate of c-hat, a parametric bootstrap procedure has been implemented in the Tests menu choice.
- When the design matrix is opened with exactly 66 rows, only 65 are shown. One workaround to get the browser to show the last row is to specify an intercept that writes a “1” to the 66th row.
- Some Windows 7 systems generate uniform 0,1 random numbers that are negative. I’ve inserted a test when MARK starts to see whether your system generates invalidd random numbers. If your system fails the test, please send me the details, particularly a copy of your System screen from the Control Panel.
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Barker, R. J., and G. C. White. 2001. Joint analysis of live and dead encounters of marked animals. Pages 361-367 in R. Field, R. J. Warren, H. Okarma, and P. R. Sievert, editors. Wildlife, land, and people: priorities for the 21st century. Proceedings of the Second International Wildlife Management Congress. The Wildlife Society, Bethesda, Maryland, USA.
Barker, R. J., G. C. White, and M. McDougal. 2005. Movement of paradise shelduck between molt sites: a joint multistate-dead recovery mark recapture model. Journal of Wildlife Management 69:1194-1201.
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Kendall, W. L., and J. D. Nichols. 1995. On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions. Journal of Applied Statistics 22:751-762.
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MacKenzie, D. I., L. L. Bailey, and J. D. Nichols. 2004. Investigating species co-occurrence patterns when species are detected imperfectly. Journal of Animal Ecology 73:546.
MacKenzie, D. I., J. D. Nichols, M. E. Seamans, and R. J. Gutierrez. 2009. Modeling species occurrence dynamics with multiple states and imperfect detection. Ecology 90:823-835.
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McClintock, B. T., G. C. White, M. F. Antolin, and D. W. Tripp. 2009a. Estimating abundance using mark-resight when sampling is with replacement or the number of marked individuals is unknown. Biometrics 65:237-246.
McClintock, B. T., G. C. White, K. P. Burnham, and M. A. Pryde. 2009b. A generalized mixed effects model of abundance for mark-resight data when sampling is without replacement. Pages 271-289 in D. L. Thomson, E. G. Cooch, and M. J. Conroy, editors. Modeling Demographic Processes in Marked Individuals. Springer, New York, USA.
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