Closed Form Methods to Compare Two Independent Proportions for Clustered Data

Objective: Many medical researches involve the collection of multiple observations for each subject studied. The statistical literature typically refers to such data as "clustered data". In order to compare two proportions estimated from independent samples, chi-squared test can be used. However, this test is not appropriate for clustered data since it doesn't take the dependencies among observations within the same cluster into account. According to simulation studies, when the within-correlation coefficient is positive, ignoring the correlation results in inflation to Type I error rate. So, methods taking the within-cluster correlation into account should be used to perform chi-square test in clustered data. Material and Methods: In this study, closed form methods for chi-square test to compare the proportion of decayed tooth estimated from independent samples (public and private school) and Mantel-Haenzel chi-square test supposing that age can be a confounder for the proportion of decayed tooth, will be introduced and applied on an ophthalmology data. Results: According to the results, although there were significant differences between the proportion of decayed tooth among the selected public and private schools at standard chi-squared test (p=0.047), this difference was not significant at methods which was adjusted to the clustered binary data (p>0.05). When the Mantel-Haenszel chi-square test adjusted for age was applied, the p value associated with standard Mantel-Haenszel test was 0.06 and those for the adjusted methods were 0.497 and 0.351. Conclusion: When comparing two independent proportions for clustered binary data, methods taking the within-cluster correlation into account should be used.