2.2.3. Omnibus Test

Let p_Ho, p_He, and p_BT denote the p values obtained by the HoK, HeK, and BT statistics. Based on the idea of the p value combination method through the Cauchy distribution [47–49], we propose the homogeneous omnibus test (HoO) and heterogeneous omnibus test (HeO).

(1) Homogeneous Omnibus Test. Combining the p_Ho with the p_BT, we construct the homogeneous omnibus test (HoO) as follows:

where F_C⁻¹ stands for the inverse cumulative distribution function of the standard Cauchy distribution.

(2) Heterogeneous Omnibus Test. Combining the p_He with the p_BT, we construct the heterogeneous omnibus test (HeO) as follows:

The null distributions of the O_Ho test and the O_He test asymptotically follow a standard Cauchy distribution [47–49]. The p values of the O_Ho test and the O_He test are calculated by the R package RNOmni [50].

The kernel statistic, the burden test, and the omnibus test are also applicable to the X chromosome. Additional technical information for extensions to the X chromosome is shown in Appendix S2.