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 ${\chi }_{(r-1)(c-1)}^2$.

Combining levels of a categorical variable changes both the null hypothesis and the degrees of freedom of an independence test. Association that is coherent across the combined levels may become easier to detect because the grouped table has fewer degrees of freedom, while associations that cancel within groups may disappear. Conclusions from the original and aggregated tables therefore need not agree.