Tag (or "band" if you're a North American bird bander; "ring" if you're a European) recovery models are in a sense a special case of CJS models, in which encounters happen in a single way: dead recovery, either shot and reported by hunter (the common method for game birds and fish and waterfowl), or found dead and reported by the general public. In common with CJS these models are all conditional on the animals having been previously marked and released, so unmarked animals do not appear in the likelihood; this, it is not possible to estimate abundance. Because the only information possible for an animal is the time of the initial releases, and the subsequent dead recovery (if any), the data structure is much simpler than for CJS models, where an individual may be encountered multiple times. We will get into the specific of how data can be read and analyzed using MARK and RMark below, but first some basic concepts and terminology.
Data structure
LDLD format
The "standard" format for dead recovery data (as well as for joint live-dead and known fates as we'll see) is an augmented encounter history with 2 columns for each occasion, known as "LDLD" format. The first column is a 1 if the animal was observed alive at that occasion, and the second is a 1 if the animal is recovered dead during the interval between the occasion and the next. For dead recovery data, the first column can only be 1 at the release occasion, and the second column 1 at the recovery occasion; all other columns must be zero (since the animal is never observed as alive except upon release, and once dead can never be re-encountered). For example
100100
indicates "released at 1 and recovered at 2".
110000 is "released at 1 and recovered at 1" (e.g., the hunting season immediately after release).
100000 is "released at 1 and never recovered".
Either MARK or RMark can read data in this format. If individual covariate information is used, this format (and not the summarized format below) must be used.
Summarized format
If there is no need to retain information on individual animals (e.g., we are not modeling individual covariates), then it is possible to summarize dead recovery data in a compact, tabular form, rather than the LDLD format above. However, the summarized format can be used as input only for MARK (not RMark).
The basic summarized data structure for a dead recovery problem involves a vector of animals that have been marked and released, and a matrix of recoveries from each banding occasion (i) at each recovery occasion (j). For example the data for a i = 3 banding occasions (e.g., years) and j= 4 recovery occasions (e.g., hunting seasons) would be
Model parameterizations and parameters
Parameters estimated
There are 2 types of parameter that can be estimated in a band recovery model: survival rates (denoted S) and recovery or "reporting" rates (denoted f or r depending on the specific model parameterization used; more on this below. The idea is that animals survive over an interval (usually corresponding to the interval between marking periods, and often 1 calendar year) with probability S. They then are recovered with probability f. Exactly how we express this depends on the situation. In North American game bird band recovery, most birds are banded and released immediately before the hunting season, so the opportunity for other mortality sources to kick in is low. Therefore, the probability that a banded bird is recovered in the first recovery period (hunting season) after banding is f, that it survives to the second season and is recovered is S x f, to the third and is recovered is S2 x f, etc. This is under a simple model where both parameters are assumed constant through time, below I generalize to allow time variation, group covariates, etc.
"Real" vs. "apparent" survival
You will hear these terms tossed about in the CMR and recovery literature. You might also have notice that earlier we referred to survival in CJS models by the notation φ , but now are labeling survival S. This notational differences arises because there is in fact a very real difference in the way that the "survival" parameter can be interpreted in CMR vs. recovery studies.
In recovery, we ordinarily have a situation where tags can be potentially reported from animals wherever they go, and thus the encounter process is independent of the original marking process. Thus, in recovery studies there is no notion of permanent emigration from the study area: animal are always subject to encounter. By contrast, in CMR ordinarily the marking and encountering (recapture or re-sighting) happen in the same location, often at the same time. In subsequent encounter occasions, animals can and do move (either temporarily or permanently) to locations where there is no sampling effort, and where thus there is no chance they will be caught. They have effectively emigrated from the study.
If we have overlapping data of both recoveries and recaptures, we potentially can take advantage of these different types of survival parameter. Simply put, CMR models estimate φ = S(1-E)., where E is the probability of emigration. By contrast, recovery models estimate "pure" survival, S. Putting these structures together allows us to estimate E or 1-E = site fidelity as a derived parameter. Both MARK and RMark implement various types of analyses of combined recovery and CMR data (see Barker and Burnham models).
Parameter estimation
Given the parameters as defined above, and the data, a multinomial model is constructed to estimated survival (Si) and recovery (fi) rates from the data, via maximum likelihood methods. The standard Brownie model for the above data structure would provide expected values for the data (i.e, given MLEs of the parameters) of :
EXPECTED NUMBER
In this example, 3 recovery rates (f1 , f2, f3 ) and 2 survival rates (S1 , S2) can be estimated uniquely; the remaining 2 can only be estimated as a product S3f4 (i.e, a single parameter), which is the case anytime the number of recovery occasions (l) > the number of banding occasions (k).
The above representation is fairly general but is tied to a specified parameterization known as the Brownie parameterization, most used in North American game tag recoveries where f is interpretable as closely related to annual harvest rates. E.g., in US waterfowl management, f is adjusted by estimates of band reporting rate (rate at which found bands are turned in, not to be confused with r below). In Britain and Europe, most "ring recoveries" are reported on non-harvested birds found dead and reported to the various ringing schemes coordinated by EURING (by the way, I hosted the last EURING Analytical Meeting in Athens in 2013). Under this system, recovery the first year following ringing occurs because (obviously) the bird first died, and then is found and reported. This is expressed under the Seber parameterization as (1-S)r. This leads to an equivalence between the Brownie f and Seber r as f =(1-S)r or equivalently r = f/(1-S). Programs MARK and RMark provide either the Seber or Brownie parameterization. If you happen to obtain estimates under a different parameterization than desired, you can either re-run the analysis with the alternate parameterization, or transform estimates of recovery by the above expressions.
Next: Implementation in MARK