Independence Test for Two Discrete Variables

Definition

Let category variables $A_1,\dots ,A_r$ and $B_1,\dots ,B_c$ and consider a null hypothesis $H_0:A,B$ are independent. Then, the test Statistic, which follows Chi-squared Distribution, is defined as $Q=\sum _{j=1}^b\sum _{i=1}^a\frac{(X_{ij}-n{\hat{p}}_{ij}{)}^2}{n{\hat{p}}_{ij}}\stackrel{D}{\to }{\chi }^2((r-1)(c-1))$ where $n=\sum _j\sum _i{X}_{ij}$, ${\hat{p}}_{ij}={\hat{p}}_{i.}{\hat{p}}_{.j}=\frac{X_{i.}}{n}\frac{X_{.j}}{n}$, $X_i=\sum _j{X}_{ij}$, $X_j=\sum _i{X}_{ij}$