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 p_i (1-p_i), 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.