In ophthalmologic or dental studies, observations are frequently taken from multiple sites (called units), such as eyes or teeth, of each subject. In this case, observations within each subject (called clusters) may be dependent, although those from different subjects are independent. When a categorical observation is made from each site, application of the usual Pearson chi-square tests is invalid since sites within the same subject tend to be dependent. We propose a modified χ 2 statistic for testing no treatment effect in these cases. The proposed methods do not require correct specification of the dependence structure within cluster. Simulation studies are conducted to show the finite-sample performance of the new methods. The proposed methods are applied to real-life data.