3. Identifying the Ecological Correlates of Extinction-Prone Species
29
If comprehensive surveys, detailed ecological studies, and the development of
explicit population models are not feasible, then agencies may consider other
methods to generate extinction risks or estimates of vulnerability to assist in
setting management priorities. An alternative is to develop simple models with the
data at hand. Although population data are difficult to come by, morphological
data are often easier to obtain or infer. Insofar as morphological traits are correlated to ecological factors, morphology may provide proxy measures of unobserved ecological factors.
Data on the distribution of the threatened species frequently are coarse and
unreliable. For instance, data may be composed of a mixture of presence records
and presence-absence records from a small sample of locations (see Elith, this
volume). The purpose of this work is to describe approaches to estimating extinction risks that use taxonomic, morphological, and spatial information to develop
models that explain extinction risks, using maximum likelihood estimation. The
methods are intended to make better use of information that is commonly available to biologists and to enhance the ability of managers to make decisions that are
sensitive to the relative risks faced by different populations and species.
Maximum Likelihood Estimation
Generalized linear models (GLM) provide an effective means of developing
models for categorical response variables. A distribution is assumed for the response variable, commonly binomial for binary data and Poisson for count data.
When using nonnormal response data, the expectations of the response have to be
constrained (e.g., when modeling a binary event such as extinct-extant, estimates
have to be constrained to be between zero and one). A link function, with a
corresponding probability density function, achieves this. For binary response
data (e.g., extinct-extant or present-absent), logit and probit links are common. In
the probit case, the normal distribution is used as the probability density function.
In the logit case, the logistic function is used as the probability density function.
The resulting expectations are expressed as a probability (of success, if 1 =
success).
In GLMs, likelihood functions are used to estimate parameters for the explanatory variables (Amemiya 1981). The values given to the parameters are those that
maximize the probability of obtaining the observed set of data. The equations that
determine the maximum likelihood estimations are nonlinear, and with GLMs,
iterative algorithms are used for solving them. For mathematical reasons, it is
actually the log-likelihood that is maximized. The use of a probability criterion for
categorical response variables, generally nonlinear in form, differs from leastsquares techniques. Least-squares models apply a linear distance criterion that can
be represented in the Cartesian space formed by the explanatory and dependent
variables. This forms the basis of goodness-of-fit measures in least squares.
Classical regression models deal with response variables that have normally
29
If comprehensive surveys, detailed ecological studies, and the development of
explicit population models are not feasible, then agencies may consider other
methods to generate extinction risks or estimates of vulnerability to assist in
setting management priorities. An alternative is to develop simple models with the
data at hand. Although population data are difficult to come by, morphological
data are often easier to obtain or infer. Insofar as morphological traits are correlated to ecological factors, morphology may provide proxy measures of unobserved ecological factors.
Data on the distribution of the threatened species frequently are coarse and
unreliable. For instance, data may be composed of a mixture of presence records
and presence-absence records from a small sample of locations (see Elith, this
volume). The purpose of this work is to describe approaches to estimating extinction risks that use taxonomic, morphological, and spatial information to develop
models that explain extinction risks, using maximum likelihood estimation. The
methods are intended to make better use of information that is commonly available to biologists and to enhance the ability of managers to make decisions that are
sensitive to the relative risks faced by different populations and species.
Maximum Likelihood Estimation
Generalized linear models (GLM) provide an effective means of developing
models for categorical response variables. A distribution is assumed for the response variable, commonly binomial for binary data and Poisson for count data.
When using nonnormal response data, the expectations of the response have to be
constrained (e.g., when modeling a binary event such as extinct-extant, estimates
have to be constrained to be between zero and one). A link function, with a
corresponding probability density function, achieves this. For binary response
data (e.g., extinct-extant or present-absent), logit and probit links are common. In
the probit case, the normal distribution is used as the probability density function.
In the logit case, the logistic function is used as the probability density function.
The resulting expectations are expressed as a probability (of success, if 1 =
success).
In GLMs, likelihood functions are used to estimate parameters for the explanatory variables (Amemiya 1981). The values given to the parameters are those that
maximize the probability of obtaining the observed set of data. The equations that
determine the maximum likelihood estimations are nonlinear, and with GLMs,
iterative algorithms are used for solving them. For mathematical reasons, it is
actually the log-likelihood that is maximized. The use of a probability criterion for
categorical response variables, generally nonlinear in form, differs from leastsquares techniques. Least-squares models apply a linear distance criterion that can
be represented in the Cartesian space formed by the explanatory and dependent
variables. This forms the basis of goodness-of-fit measures in least squares.
Classical regression models deal with response variables that have normally