The binomial scan statistic evaluate the statistic which compares the node attribute within
the subgraph with that outside the subgraph while the node attribute follows the binomial distribution.
Usage
binom.stat(obs, pop, zloc)
Arguments
obs
Numeric vector of observation values.
pop
Numeric vector of population values.
zloc
Numeric vector of selected nodes.
Value
Three values will be returned. The first value is test statistic.
The second is the estimated means which estimated outside the selected nodes.
The third is estimated means estimated within the selected nodes.
Details
A network with interested attributes is denoted as $G=(V,E,X)$,
where $X=(x_1,\ldots,x_{|V|})$ follows a defined distribution. Suppose a subgraph, $Z$, is selected.
$$\lambda_A(Z) =n_z \ln\big(\frac{p_{11}}{p_0}\big)+ (N_z-n_z) \ln\big(\frac{1-p_{11}}{1-p_0}\big)+(n_G-n_z) \ln\big(\frac{p_{10}}{p_0}\big) +[(N_G-N_z)-(n_G-n_z)] \ln\big(\frac{1-p_{10}}{1-p_0}\big),$$
where $p_0=n_G/N_G$, $p_{10}=(n_G-n_z)/(N_G-N_z)$, and $p_{11}=n_z/N_z$.
In addition, $N_G$ and $n_G$ are population sizes and number of successful cases of whole graph $G$.
$N_Z$ and $n_Z$ are expressed in the same way in the selected subgraph $Z$.
References
Kulldorff, M. (1997). A spatial scan statistic.
Communications in Statistics-Theory and methods, 26(6), 1481--1496.