The authors [J. Am. Stat. Assoc. 76, 221-230 (1981; Zbl 0473.62010
), and Current topics in survey sampling, Proc. int. Symp., Ottawa/Can. 1980, 247-265 (1981; Zbl 0488.62009
)] have shown that the Pearson chi-squared test statistic $X\sp 2$ for testing goodness of fit and independence in a two-way table and homogeneity of proportions across populations are asymptotically distributed as weighted sums of independent chi-squared random variables $X\sp 2\sb 1$ of 1 degree of freedom.
In this paper, the authors propose generalized results to multiway tables, and obtain the asymptotic distribution of $X\sp 2$ as a weighted sum of independent $X\sp 2\sb 1$ random variables under nested loglinear models, and further obtain a simple correction to $X\sp 2$ which requires only the cell design effects (deffs) and the deffs of collapsed tables, whenever the likelihood equation under multinomial sampling admits an explicit solution.
Finally, they take an empirical example on the relative performance of $X\sp 2$ and some corrected $X\sp 2$ statistics in a three-way table from the Canada Health Survey.