Exact Size and Power Properties of Five Tests for Multinomial Proportions

Abstract

ABSTRACT In this article, exact properties of the chi-square test, likelihood ratio test (LRT), Hoel's test (1938 Hoel , P. G. ( 1938 ). On the chi-square distribution for small samples . Ann. Mathemat. Statist. 9 : 158 – 165 . [CSA] [Crossref] , [Google Scholar]), Nas' test (1959 Nass , C. A. G. ( 1959 ). The χ2-test for small expectations in contingency tables, with special reference to accidents and absenteeism . Biometrika 46 : 365 – 385 . [CSA] [Web of Science ®] , [Google Scholar]), and an exact test, for testing the multinomial proportions, are investigated numerically. Numerical studies show that the size behavior of the chi-square test for the multinomial distribution with three or more cells is erratic only when the probabilities are at the boundaries. Otherwise the chi-square test performs very satisfactorily. Hoel's test offers a little improvement over the χ2-test. The χ2-test is, in general, superior to the LRT, and inferior to the exact test with respect to size properties. The exact test exhibits very good size properties. Nas' test offers improvement over the χ2-test. Power studies indicate that the tests are equally powerful provided the sizes of the tests are close to each other for a given parameter configuration. Some recommendations regarding the choice of a test for practical applications are given, and the tests are illustrated using a biological example.