On Power Approximations and Comparison of Several Asymptotic Tests to Detect a Specified Difference between Two Proportions

Power evaluation of asymptotic tests for comparing two binomial proportions to detect direct and indirect association in large-scale studies

Asymptotic tests are commonly used for comparing two binomial proportions when the sample size is sufficiently large. However, there is no consensus on the most powerful test. In this paper, we clarify this issue by comparing the power functions of three popular asymptotic tests: the Pearson's χ² test, the likelihood-ratio test and the odds-ratio based test. Considering Taylor decompositions under local alternatives, the comparisons lead to recommendations on which test to use in view of both the experimental design and the nature of the investigated signal. We show that when the design is balanced between the two binomials, the three tests are equivalent in terms of power. However, when the design is unbalanced, differences in power can be substantial and the choice of the most powerful test also depends on the value of the parameters of the two compared binomials. We further investigated situations where the two binomials are not compared directly but through tag binomials. In these cases of indirect association, we show that the differences in power between the three tests are enhanced with decreasing values of the parameters of the tag binomials. Our results are illustrated in the context of genetic epidemiology where the analysis of genome-wide association studies provides insights regarding the low power for detecting rare variants.

Applications of asymptotic confidence intervals with continuity corrections for asymmetric comparisons in noninferiority trials

A large-sample problem of illustrating noninferiority of an experimental treatment over a referent treatment for binary outcomes is considered. The methods of illustrating noninferiority involve constructing the lower two-sided confidence bound for the difference between binomial proportions corresponding to the experimental and referent treatments and comparing it with the negative value of the noninferiority margin. The three considered methods, Anbar, Falk-Koch, and Reduced Falk-Koch, handle the comparison in an asymmetric way, that is, only the referent proportion out of the two, experimental and referent, is directly involved in the expression for the variance of the difference between two sample proportions. Five continuity corrections (including zero) are considered with respect to each approach. The key properties of the corresponding methods are evaluated via simulations. First, the uncorrected two-sided confidence intervals can, potentially, have smaller coverage probability than the nominal level even for moderately large sample sizes, for example, 150 per group. Next, the 15 testing methods are discussed in terms of their Type I error rate and power. In the settings with a relatively small referent proportion (about 0.4 or smaller), the Anbar approach with Yates' continuity correction is recommended for balanced designs and the Falk-Koch method with Yates' correction is recommended for unbalanced designs. For relatively moderate (about 0.6) and large (about 0.8 or greater) referent proportion, the uncorrected Reduced Falk-Koch method is recommended, although in this case, all methods tend to be over-conservative. These results are expected to be used in the design stage of a noninferiority study when asymmetric comparisons are envisioned.