Joint Location-Scale Testing Procedure for Single-SNP Analysis
Our proposed JLS testing framework, based on the working model of Equation 2, tests the following null hypothesis:H0joint:βG=0andσi=σjforalli≠j,i,j=0,1,2.The alternative hypothesis of interest isH1joint:βG≠0orσi≠σjforsomei≠j.For a SNP under study, different JLS test statistics can be considered. Let pL be the p value for the location test of choice (i.e., testing H0location:βG=0 using, for example, ordinary least-squares regression), and pS be the p value for the scale test of choice (i.e., testing H0scale:σi=σjforalli≠j using, for example, Levene’s test). We first consider Fisher’s method (JLS-Fisher) to combine the association evidence from the individual location and scale tests. The JLS-Fisher statistic is defined asWF=−2(log(pL)+log(pS)).
Large values of WF correspond to small values of pL and/or pS and provide evidence against the null H0joint. If pL and pS are independent under H0joint, WF is distributed as a χ42 random variable. Although Fisher’s method here is used to combine evidence from two tests applied to the same sample, the assumption of independence between pL and pS under H0joint holds theoretically for a normally distributed trait (Appendix A, Lemma 1), as well as empirically for approximately normally distributed traits in finite samples (Figures S1 and S2, Tables S1 and S2).
One can also consider the minimum p value (JLS-minP) approach, or various alternatives based on combining the individual test statistics themselves with or without weights.20–23 The JLS-minP statistic is defined asWM=min(pL,pS).If pL and pS are independent under H0joint, WM is distributed as a Beta random variable (with shape parameters 1 and 2) where small values of WM correspond to small values of pL and/or pS and evidence against the null.