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On a Comparison of Tests of Homogeneity of Binomial Proportions

Martin Klein and Peter Linton


Chi-squared approximation, Edgeworth series, exact test, moment-matching approximation, nuisance parameter, power study, simulation.


There are multiple tests of homogeneity of binomial proportions in the statistics literature. However, when working with sparse data, most test procedures may fail to perform well. In this article we review nine classical and recent testing procedures, including the standard Pearson and likelihood ratio tests; exact conditional and unconditional tests; tests based on moment matching chi-squared approximations; a recently proposed test based on a normal approximation in an asymptotic framework for sparse data; and a recent test based on higher order moment corrections using an Edgeworth approximation. For each test we review its theoretical underpinning, and show how to calculate the P-value. Most of the P-values can be readily calculated in a statistical computing software package such as R. We compare type I error probability and power via simulation. As expected, none of the procedures uniformly outperforms the others in terms of type I error probability and power, but we can make some recommendations based on our empirical results. In particular, we indicate scenarios in which certain otherwise reasonable test procedures can perform inadequately.


Martin Klein and Peter Linton. (2013). On a Comparison of Tests of Homogeneity of Binomial Proportions. Center for Statistical Research & Methodology Research Report Series (Statistics #2013-03). U.S. Census Bureau. Available online at <>.

Source: U.S. Census Bureau, Center for Statistical Research & Methodology, Research and Methodology Directorate

Published online: April 8, 2013
Last revised: April 8, 2013


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Source: U.S. Census Bureau | Research and Methodology Directorate | Center for Statistical Research & Methodology | (301) 763-3215 (or |   Last Revised: September 11, 2013