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Asymptotic Efficiency of the Chi‐Square Test for Heterogeneity of Proportions in Some Sparse‐Data Cases
Author(s) -
Anderson S.,
Hauck W. W.
Publication year - 1989
Publication title -
biometrical journal
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.108
H-Index - 63
eISSN - 1521-4036
pISSN - 0323-3847
DOI - 10.1002/bimj.4710310514
Subject(s) - mathematics , statistics , efficiency , chi square test , binomial (polynomial) , square (algebra) , sample size determination , binomial proportion confidence interval , binomial distribution , variance (accounting) , negative binomial distribution , statistical hypothesis testing , poisson distribution , geometry , accounting , estimator , business
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 σ 2 . The asymptotic relative efficiency (ARE) of the usual chi‐square test is found relative to the likelihood‐based tests for σ 2 =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.