Introduction to Mark-Recapture Sampling and Closed-Population Models - Introduction to Ecological Sampling - page 93

M b , trap response: Every unmarked individual has the same capture
probability p , which changes to c after the first capture ( P ij = p until
first capture, P ij = c for any recapture);
M bh , heterogeneity and trap response: The i ith individual has its own
unique capture probability p i before it is captured, which changes to
c i after the first capture ( P ij = p i until first capture and P ij = c i for any
recaptures, known as the generalized removal model);
M th , time variation and heterogeneity: The i ith individual has its own
unique capture probability, which varies from sample to sample
( P ij = p i e j );
M tb , time variation and trap response: The probability of capture is e j
for uncaptured individuals and c j for captured individuals in the
j th occasion ( P ij = e j until first capture, P ij = c j for any recapture); and
M tbh , time variation, trap response, and heterogeneity: The i ith animal
has its own unique capture probability, which varies with time and
changes after the first capture ( P ij = p ij until first capture, P ij = c ij for
any recapture).
It is desirable to estimate the parameters of these models using the prin-
ciple of maximum likelihood (ML); that is, parameters should be estimated
by the values that make the probability of obtaining the observed data as
large as possible. However, the usual type of mark-recapture data does not
provide enough information to estimate models M th , M tb , and M tbh by ML.
This was the main problem faced by the implementation of ML estimation in
CAPTURE. Nevertheless, CAPTURE still can choose between the other five
models on the basis of goodness-of-fit tests. Hence, after giving a full account
of the most recent estimation methods available (e.g., ML, log-linear models,
moment estimators, estimating equations, jackknife, etc.), Chao and Huggins
(2005b) warned that “there is no objective method to select a model from the
various heterogeneous models” (page 72).
The main interest for the last 30 years has been the development of alter-
native estimation procedures. As an example, to reduce the bias of estima-
tors, it is a standard practice to invoke jackknife methods (Manly, 2006).
This is particularly effective for fitting model M h . The jackknife approach is
already implemented in the CAPTURE program. Jackknifing is also a suit-
able estimation method for model M bh (Pollock and Otto, 1983); for models
M h and M th , the sample coverage approach (Lee and Chao, 1994) has practical
advantages as it makes it possible to summarize the heterogeneity effects in
terms of the coefficient of variation (CV) of the capture probabilities, where
the larger the CV the greater the degree of heterogeneity among animals
(Chao and Huggins, 2005b). The program CARE-2 (Chao and Yang, 2003)
implements this sample coverage approach for closed-population models.
For the most complex model, M tbh , CARE-2 uses a general estimation equa-
tion approach (Mukhopadhyay, 2004) by which the probability of recapture