The Distribution of the Chi-Squared Statistic under Clustered Sampling from Contingency Tables

Abstract When the entries in a contingency table arise from clustered sampling, the chi-squared statistic conventionally calculated to testsimple and complex hypotheses about the parameters of the table may not have the distribution of a χ2 variate. Assuming a model for positively associated clustering, this article finds the distribution of the conventional chi-squared statistic and shows how to correct it to an asymptotically χ2 variate. A numerical example from the epidemiology of mental illness is given.