Geoscience Reference
In-Depth Information
questions could be relevant in identifying potential costs of reproduction or
source-sink dynamics.
9.5 Including Covariates
Any of the probabilities in both the single- and multiseason models could be
modeled as a function of covariates or predictor variables. Indeed, often it
will be these relationships that are of primary interest to explore which land-
scape features are important for the distribution of the species. However,
while the covariate may potentially take any positive or negative value, a
probability must be on the scale of 0-1. Therefore, a transformation, or
link
function
, must be used to ensure that the probabilities and covariates are on a
comparable scale. There are a range of options available for doing so, includ-
ing the probit link, the log-log link, and the complementary log-log link,
although the one discussed here is the logit link, which is the basis of logistic
regression and is a commonly used technique for analyzing binary data. The
logit link function is defined as
=β +β ⋅
θ
−θ
(
θ=
logit
ln
i
x
β⋅
x
++β⋅
...
x
i
0
1
1, 2 ,
i
i
r
r i
,
1
i
where θ
i
is the probability of interest for unit
i
,
x
1,
i
to
x
r, i
are the values of the
covariates of interest measured for unit
i
, and β
0
to β
r
are the regression coef-
ficients to be estimated. Note that the ratio θ
i i
is the odds of the event
occurring; hence, the logit link is also referred to as the log-odds link. For the
probabilities used in the multinomial parameterizations, the multinomial-
logit link should be used.
Note that the logit link is a form of a generalized linear model (GLM);
hence, users need to specify the functional form of the relationship between
the probability and the covariate (e.g., linear, quadratic, etc.). This is not to
say that alternative approaches such as generalized additive models (GAMs)
cannot be used with occupancy models for more flexible curve-fitting rela-
tionships with covariate.
There are two broad types of covariates that could be considered with
occupancy models. First, unit-specific covariates such as vegetation type,
distance from water, elevation, and so on can be used as covariates for all the
probabilities discussed previously, both occupancy related and those asso-
ciated with the detection process. These covariates are essentially charac-
teristics of each unit and are assumed to be constant for a particular season,
but they may change between seasons. Second, covariates with survey-spe-
cific values (e.g., the time of survey, wind conditions, air temperature, etc.)
(1
θ
)
