Homogeneity Test for Multinomial Distribution

Definition

Let ${\mathbf{X}}_1\sim \mathrm{Mult}(n_1,p_{11},\dots ,p_{c1}),{\mathbf{X}}_2\sim \mathrm{Mult}(n_2,p_{12},\dots ,p_{c2}),\dots ,{\mathbf{X}}_r\sim \mathrm{Mult}(n_r,p_{1r},\dots ,p_{cr})$ and consider a null hypothesis $H_0:p_{11}=p_{12}=\cdots =p_{1r},p_{21}=p_{22}=\cdots =p_{2r},p_{c1}=p_{c2}=\cdots =p_{cr}$. Then, the test Statistic, which follows Chi-squared Distribution, is defined as $Q=\sum _{j=1}^r\sum _{i=1}^c\frac{(X_{ij}-n_j{\hat{p}}_{ij}{)}^2}{n_j{\hat{p}}_{ij}}\stackrel{D}{\to }{\chi }^2((r-1)(c-1))$ where ${\hat{p}}_{ij}=\frac{{\sum }_{j=1}^r{X}_{ij}}{{\sum }_{j=1}^r{n}_j}$