Pearson's Chi-Square Test of Independence

Pearsons's Chi-Square Test of Independence tests the independence between two categorical variables, $x$ and $y$ which contain $s$ and $r$ categories respectively. The test statistic is

$X^2=\sum _{k=1}^s\sum _{j=1}^r\frac{(o_{kj}-e_{kj}{)}^2}{e_{kj}}$

where:

$o_{kj}=\sum _{i=1}^n{w}_i{I}_{x=k,y=j}$,

$w_i$ is the Calibrated Weight of the $i$th of $n$ observations,

$e_{kj}=\frac{\sum _{k=1}^s{o}_{kj}\times \sum _{j=1}^r{o}_{kj}}{\sum _{i=1}^n{w}_i}$

$p\approx \mathrm{Pr}({\chi }_{(s-1)(g-1)}^2\ge X^2)$