BhapkarTest: Bhapkar Marginal Homogeneity Test

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BhapkarTestR Documentation

Bhapkar Marginal Homogeneity Test


Bhapkar (1966) tested marginal homogeneity by exploiting the asymptotic normality of marginal proportion, and so this test is also called Bhapkar's test. The idea of constructing test statistic is similar to the one of generalized McNemar's test statistic used in StuartMaxwellTest, and the major difference lies in the calculation of elements in variance-covariance matrix.


BhapkarTest(x, y = NULL)



either a 2-way k \times k contingency table in matrix form, or a factor.


a factor with the same levels as x; ignored if x is a matrix.


Although the Bhapkar and Stuart-Maxwell tests are asymptotically equivalent (Keefe, 1982). Generally, the Bhapkar (1966) test is a more powerful alternative to the Stuart-Maxwell test. With a large N, both will produce the same Chi-square value. As the Bhapkar test is more powerful, it is preferred.


Andri Signorell <>


Bhapkar V.P. (1966) A note on the equivalence of two test criteria for hypotheses in categorical data. Journal of the American Statistical Association, 61: 228-235.

Ireland C.T., Ku H.H., and Kullback S. (1969) Symmetry and marginal homogeneity of an r x r contingency table. Journal of the American Statistical Association, 64: 1323-1341.

Keefe T.J. (1982) On the relationship between two tests for homogeneity of the marginal distributions in a two-way classification. Biometrika, 69: 683-684.

Sun X., Yang Z. (2008) Generalized McNemar's Test for Homogeneity of the Marginal Distributions. SAS Global Forum 2008: Statistics and Data Analysis, Paper 382-208.

See Also

StuartMaxwellTest, mcnemar.test, chisq.test, MHChisqTest, BreslowDayTest


# Source:
mc <- as.table(matrix(c(20,3,0,10,30,5,5,15,40), nrow=3))


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