Miettinen and Nurminen score statistics revisited

It is commonly necessary to perform inferences on the difference, ratio, and odds ratio of two proportions p1 and p2 based on two independent samples. For this purpose, the most common asymptotic statistics are based on the score statistics (S-type statistics). As these do not correct the bias of the estimator of the product pi (1–pi), Miettinen and Nurminen proposed the MN-type statistics, which consist of multiplying the statistics S by (N–1)/N, where N is the sum of the two sample sizes. This paper demonstrates that the factor (N–1)/N is only correct in the case of the test of equality of two proportions, providing the estimation of the correct factor (AU-type statistics) and the minimum value of the same (AUM-type statistics). Moreover, this paper assesses the performance of the four-type statistics mentioned (S, MN, AU and AUM) in one and two-tailed tests, and for each of the three parameters cited (d, R and OR). We found that the AUM-type statistics are the best, followed by the MN type (whose performance was most similar to that of AU-type). Finally, this paper also provides the correct factors when the data are from a multinomial distribution, with the novelty that the MN and AU statistics are similar in the case of the test for the odds ratio.