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.

Two-tailed asymptotic inferences for the odds ratio in cross-sectional studies: evaluation of fifteen old and new methods of inference

Various asymptotic methods of obtaining a confidence interval (CI) for the odds ratio (OR) have been proposed. Surprisingly, insofar as we know, the behavior of these methods has not been evaluated for data proceeding from a cross-sectional study (multinomial sampling), but only for data that originate in a prospective or retrospective study (two independent binomials sampling). The paper evaluates 15 different methods (10 classic ones and 5 new ones). Because the CI is obtained by inversion in θ of the two-tailed test for H_0(θ): OR =[Formula: see text] (null hypothesis), this paper evaluates the tests for various values of θ, more than the CIs that are obtained. The following statements are valid only for the two-tailed inferences based on 20 ≤ n ≤ 200 and 0.05≤ OR≤20, since these are the limitations of the study. The two best methods are the classic Cornfield chi-squared method for 0.2≤ OR≤5 and, in other cases, the new method of Sterne for chi-squared; but the adjusted likelihood ratio method is a good alternative to the two previous methods, especially to the first when the sample size is large. The three methods require iterative calculations to obtain the CI. If one is looking for methods that are simple to apply (that is, ones that admit a simple, explicit solution), the best option is the Gart logit method for 1/3≤ OR≤3 and, if in other cases, the Agresti logit method. The Cornfield chi-squared and Gart logit methods should not be used outside the specified interval OR. The paper also selects the best methods for realizing the classic independence test (θ = 1).