Biology Reference
In-Depth Information
need to assume any underlying distribution of the independent variables,
which is, again, convenient as detailed mechanisms regarding disease trans-
mission is often unknown.
Logistic regression models the independent variables to be linear with
respect to the logarith m of the odds of the outcome (
logit
). If
p
denotes the prob-
ability of a disease occurrence, and
xi
is the environmental variable
i
,
then
p
=++ + +++
log
ββ β
x
x
……
β
x
β
x
0
1
1
2
2
ii
NN
1
−
p
(3.1)
and
1
p
=
+
−+ +
1
e
(
ββ β
x
x
+ + + +
……
β
x
β
x
)
011
2
2
ii
NN
(3.2)
The methods in logistic regression will estimate the parameters β
i
while
minimizing the error between data and predicted value. One can then
take the exponent of the parameter (
e
β
i
) to determine the amount of change
in the outcome for a unit increase in the environmental variable
i
. The
predictive accuracy of the model can be evaluated using receiver-operator
characteristic (ROC) curve analysis (Brooker et al. 2002). Briefly, predic-
tions on disease risk are calculated based on the probability threshold,
and, as ROC varies across this threshold, the specificity and sensitivity
of the model can be obtained. The Akaike Information Criterion (AIC) is
often used to eliminate variables that do not contribute significantly to the
model (Miller 2002).
3.5.1.2 Regression Trees
Another statistical tool for modeling infectious diseases with remote sens-
ing data is called Classification and Regression Trees (CART) (Breiman et al.
1984). CART is a nonparametric, decision tree-based method that is widely
applicable given that it does not assume any underlying distribution for both
the predictor and response variables. Typically, a regression tree is gener-
ated when the outcome variable is continuous, whereas a classification tree
is used for categorical outcome variable. Since our interest is in predicting
disease occurrence rates, we will briefly discuss regression tree method in
the following text.
CART uses binary recursive partitioning in building the tree. Starting
with a root of the tree, the method splits the data into two child nodes that
maximizes the homogeneity within these nodes. The partitioning continues
iteratively with each of the resulting child nodes. For each iteration, CART
will find the most suitable (optimized) partitioning given previous actions
