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calculation of second derivatives of the likelihood function. Furthermore, if the
function maximized is the log-likelihood rather than the likelihood itself, then
these algorithms output an approximation to the Hessian matrix, which in
our application is the covariance matrix for the estimated parameters. SURGE
(Pradel and Lebreton, 1991) includes an algorithm of this type, and for the pro-
gram MRA-LGE (Supplementary Material for this chapter available at https://
sites.google.com/a/west-inc.com/introduction-to-ecological-sampling-
supplementary-materials/) the implementation of this algorithm in the
FORTRAN subroutine DFMIN provided by Press et al
.
(1992) has been used.
The 'mra' package (http://cran.r-project.org/web/packages/mra/index.html)
associated the R computer language, and MARK (http://www.phidot.org/
software/mark/ downloads) also implement this type of algorithm.
8.6.4 Likelihood Ratio Tests to Compare Two Models
One of the useful tools that is available when maximum likelihood estima-
tion is used is a test for whether one model is a significantly better fit than an
alternative simpler model. To use this test, it is necessary that the first model
has more parameters than the second model and includes the second model
as a special case. That is, by constraining one or more of the parameters of
the first model in some way, the second model is obtained.
Let
L
1
be the value of the maximum of the likelihood function for the first
model, which has
p
1
estimated parameters, and
L
2
be the maximum of the
likelihood function for model 2, which has
p
2
estimated parameters. Then,
p
1
>
p
2
, and it can be anticipated that
L
1
>
L
2
because of the extra parameters
in model 1. A standard result, then, is that if the simpler model 2 is in fact
correct, the statistic
D
= 2{log
e
(
L
1
) - log
e
(
L
2
)}
(8.24)
will approximately have a chi-squared distribution with
p
1
-
p
2
degrees of
freedom (df).
8.6.5 Model Selection Using the Akaike's Information Criterion
In recent years the Akaike's (1973) information criterion (AIC) has become
popular for selecting models for mark-recapture data (Anderson et al., 1994;
Burnham and Anderson, 1992, 2002; Burnham et al., 1994; Huggins, 1991;
Lebreton et al. 1992; Lebreton and North, 1993). Burnham et al. (1995) have
shown, through a simulation study, that this criterion gives a good balance
between bias caused by using models with too few parameters and high
variance caused by using models with too many parameters.
The criterion is calculated as
AIC = -2 log
e
(
L
max
) + 2
K
,
(8.25)
