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parameters than it is really possible to estimate separately. An inappropriate
parameterization may therefore not be obvious just from the results of the
fitting process.
8.6.2 Possible Candidate Models
The type of approach that is used for analyzing data involves defining a set
of candidate models from which one will be chosen. For example, if the sex
is recorded for each individual that is marked, then the following models
might be entertained for capture probabilities: (1) sex*time, for which the
capture probability varies with the sample time and differs for males and
females; (2) sex + time, for which the capture probability varies with the
sample time and the constant term in the logistic model also varies for males
and females; (3) sex, for which the capture probability is constant over time
but is different for males and females; (4) time, for which the capture prob-
ability varies with time but is the same for males and females; (5) trend, for
which there is a trend in capture probabilities; and (6) constant, for which
capture probabilities are constant.
These models can be expressed using the logistic function of Equation
(8.20) through judicious choice of covariates. Therefore, for example, the
model sex*time is set up by making the argument
u
i
of this function equal
to a sum of suitable indicator variables for the effects of time, sex, and the
interaction between these factors, as discussed by Lebreton et al. (1992, p. 77).
On the other hand, the model sex + time only includes variables for the main
effects of sex and time in the sum
u
i
.
The models time and trend both allow the survival probability to vary
with time. The difference is that with the time model the value
u
i
of Equation
(8.20) includes a component that is different for every sample time, whereas
with the trend model the sample time is included as a quantitative variable
with a coefficient to be estimated. This means that with the trend model
there is only one parameter to be estimated to account for time changes, but
the time model requires several parameters. Furthermore, the trend model is
a special case of the time model.
The six models for survival probabilities would probably be the same as
those used for capture probabilities. By considering all different combina-
tions of the six capture probability models and the six survival probability
models, there are then 6 × 6 = 36 possible models to be considered for the
data. It must be stressed that these 36 models may or may not be suitable for
a particular mark-recapture study.
8.6.3 Maximum Likelihood Estimation
Most models for capture-recapture data require the numerical maximization
of the likelihood function. Many different algorithms can be used for this pur-
pose, but variable metric methods have the advantage of not requiring the
