1/n - Proportional Correction to Tests of Independence

It is known that both the Pearson and G-test of independence have the same asymptotic distribution, namely ${\chi }^2$ with $\left(K？1\right)\left(M？1\right)$ degrees of freedom, where $K$ and $M$ are the number of levels, respectively, of two attributes. However, when extending the accuracy ot this approximation by $\frac{1}{n}$ -proportional terms, the resulting corrections differ quite dramatically. The purpose of this article is to derive each of these corrections and demonstrate their use in practical application.