Exact Size and Power Properties of Five Tests for Multinomial Proportions

ABSTRACT

In this article, exact properties of the chi-square test, likelihood ratio test (LRT), Hoel's test (1938), Nas' test (1959), 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.

Keywords:

• Binomial distribution
• Chi-square approximation
• Confidence interval
• Coverage probability

Mathematics Subject Classification:

• Primary 62H15
• Secondary 62H17

Notes

1 = usual chi-square test; 2 = Nass′ approximate test; 3 = exact test.

1 = usual chi-square test; 2 = Nass′ approximate test; 3 = exact test.

*These chambers contained one host animal.

The numbers in parentheses are Monte Carlo estimates of the p-values.