Asymptotic Efficiency of the Chi-Square Test for Heterogeneity of Proportions in Some Sparse-Data Cases

We consider the problem of testing for heterogeneity of K proportions when K is not small and the binomial sample sizes may not be large. We assume that the binomial proportions are normally distributed with variance σ². The asymptotic relative efficiency (ARE) of the usual chi-square test is found relative to the likelihood-based tests for σ²=0. The chi-square test is found to have ARE = 1 when the binomial sample sizes are all equal and high relative efficiency for other cases. The efficiency is low only in cases where there is insufficient data to use the chi-square test.