In closed CMR analysis we are operating under the assumption that the population is closed to changes in abundance (losses due to emigration or mortality; gains due to immigration or recruitment). See discussion of this under our earlier class on study design. It is important to note that lack of change in N is a necessary but not sufficient condition for closure; for instance, a population could remain at 100 with gain of 50 recruits and loss of 50 through mortality. This would most certainly be a serious violation of closure. As with any assumption, the assumption of closer will only be approximately met in most cases. If the investigator has reason to believe that serious violation of closure occurs, then he/she should either take steps to account for losses or gains through additional data collection and estimation, and probably should be using open cmr or the Robust Design instead, in order to avoid serious biases and other misleading inferences.
If we are operating under closed CMR with a single group (stratum), then there is a single parameter of interest, N (abundance), which is unchanging over time. A stratified (e.g., by sex or area) analysis will have N indexed by groups, so a separate parameter for each group in the most general model.
In addition to N, we will need to appropriately model and estimate the nuisance parameter capture probability, which potentially can vary over time (capture occasions), groups, individual animals, or in response to previous capture exposure. Conventionally, the probability that animal never previously captured is captured is denoted as p, and that a previously captured animal is recapture as c (recapture probability). Although p and c are technically different parameter types, they can "share" information under specific models as we will see.
The data used to estimate parameters are captures and recapture encounter histories of the type we have already seen under CJS and other models, e.g.,
10010
for an individual caught on occasions 1 and 4 of a 5 occasions study. Our data will consist of an array of such histories, one for each animal caught at least once on the study, e.g.,
10010
10001
00001
10000
In the above tiny example, we would know N was at least 4 (since we caught and released 4 unique individuals, and are assuming closure. What we don't know at the end of the study is how many animals were not caught
00000
since by definition we never observed this number. In fact, this is essentially what we are trying to estimate with our data and maximum likelihood models, since if we know how many we caught and can estimate the unmarked portion of the population, we have an estimate of N!
This gives us a clue that something is very different about closed (and open) CMR models, compared to CJS, tag recovery, known fates, and other previous applications of CMR. With CJS and the other methods, all statistical inference was based on animals that had been capture and released, and in fact to appear in the likelihood had to be recaptured at least once (conditional on previous capture and release). Thus in the earlier data array only
10010
10001
would appear in the likelihood. By contrast, closed abundance estimation requires us to make inference about animals that are never captured, so were unmarked just before the capture sample. In that case
00001
10000
do contribute to the estimation of N, and are part of the likelihood. Closed estimation models (except Huggins models) are therefore said to be unconditional models, meaning that the likelihood is not conditional on previous capture.
In the sections to follow, we will use CMR data and a variety of maximum likelihood approaches to estimate the parameters (N, p, and c) and evaluate alternative models of variation in p and c.
Next: Non-mixture models