For testing the hypothesis of independence in the r x c two way contingency table, we consider the likelihood ratio statistic -2 ln(λ) and the χ2 statistic 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 : script Nrc(0, Λ). We then establish the two statistics have a limiting conditional distribution equivalent to ZT Λ-Z, which has a central χv2, distribution with degrees of freedom v = (r - 1)(c - 1) = rank(Λ).
All Science Journal Classification (ASJC) codes
- Statistics and Probability