On Chi-Squared Tests for Multiway Contingency Tables with Cell Proportions Estimated from Survey Data

The impact of survey design on standard multinomial-based methods for a multiway contingency table is studied, under nested loglinear models. The asymptotic null distribution of the Pearson chi-squared test statistic, X² (or the likelihood ratio test statistic, G²) is obtained as a weighted sum of independent ²₁ random variables, and the weights are then related to the familiar design effects (deffs) used by survey samplers. A simple correction to X² (or G²) is also obtained which requires the knowledge of only the cell deffs and the deffs for collapsed tables (marginals), whenever the model admits a direct solution of likelihood equations under multinomial sampling. Finally, an example on the relative performance of X² and some corrected X² statistics in a three-way table is given, using some data from the Canada Health Survey, 1978-1979.