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