Limiting conditional distribution for tests of independence in the two way table
Seung-Ho Kang, Jerome H. Klotz · Communication in Statistics- Theory and Methods · 1998
For testing the hypothesis of independence in the rxc two way contingency table, we consider the likelihood ratio statistic –2In(λ) and the χ2statistic conditionally given the sufficient statistic of row and column margin totals. We use Stirling’s formula in the conditional characteristic function of normalized cell counts to give a direct proof that the limiting null conditional distribution is multivariate normal Z : Nrc (0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to Z T λ- Z, which has a central distribution with degrees of freedom ν=(r-l)(c-1) =rank(Λ).