Miettinen and Nurminen score statistics revisited.

It is commonly necessary to perform inferences on the difference, ratio, and odds ratio of two proportions <italic>p</italic><italic>1</italic> and <italic>p</italic><italic>2</italic> based on two independent samples. For this purpose, the most common asymptotic statistics are based on the score statistics (<italic>S</italic>-type statistics). As these do not correct the bias of the estimator of the product <italic>p</italic><italic>i</italic> (1–<italic>p</italic><italic>i</italic>), Miettinen and Nurminen proposed the <italic>MN</italic>-type statistics, which consist of multiplying the statistics <italic>S</italic> by (<italic>N</italic>–1)/<italic>N</italic>, where <italic>N</italic> is the sum of the two sample sizes. This paper demonstrates that the factor (<italic>N</italic>–1)/<italic>N</italic> is only correct in the case of the test of equality of two proportions, providing the estimation of the correct factor (<italic>AU</italic>-type statistics) and the minimum value of the same (<italic>AUM-</italic>type statistics). Moreover, this paper assesses the performance of the four-type statistics mentioned (<italic>S</italic>, <italic>MN</italic>, <italic>AU</italic> and <italic>AUM</italic>) in one and two-tailed tests, and for each of the three parameters cited (<italic>d</italic>, <italic>R</italic> and <italic>OR</italic>). We found that the <italic>AUM-</italic>type statistics are the best, followed by the <italic>MN</italic> type (whose performance was most similar to that of <italic>AU-</italic>type). Finally, this paper also provides the correct factors when the data are from a multinomial distribution, with the novelty that the <italic>MN</italic> and AU statistics are similar in the case of the test for the odds ratio. [ABSTRACT FROM AUTHOR]