 Likelihood-ratio test of independence in a contingency table

For observed counts $y_{ij}$ and fitted counts ${\hat{\mu }}_{ij}=y_{i+}{y}_{+j}/y_{++}$ under independence, the likelihood-ratio statistic isUnder independence and standard large-sample regularity conditions, $G^2$ converges in distribution to ${\chi }_{(r-1)(c-1)}^2$.