Web - based Supplementary Materials for “ More powerful genetic association testing via a new statistical framework for integrative genomics ”

In Section 3 we showed that our method can have more power to detect o-eSNPs. Here we discuss its power to detect SNPs whose functional mechanisms have non-regulatory components. For simplicity we again consider only continuous Yi, Gi, and a single SNP Si in the ordinary linear model, where the variables have all been centered. We now consider the outcome model Yi = G T i αG + αSSi + i1, where the direct effect αS is nonzero. The transcript model remains Gi αG = βSSi + i2. We again compare to the usual approach of fitting Yi = β ∗ SSi +N(0, σ ∗2). We are interested in comparing the power of tests based on β̂S and β̂ ∗ S under this direct effect model, so we study βS/var (β̂S) and β ∗ S/var (β̂ ∗ S). First, we still have var (β̂ ∗ S) = (σ 2 1 + σ 2)/var (Si). To calculate var (β̂S) note that Step 1 of our integrative procedure is equivalent to marginalizing over Si in the true outcome model, which gives E(Yi | Gi) = Gi αG + αSE(Si | Gi) and var (Yi | Gi) = α Svar (Si | Gi) + σ 1. Without knowing more about the distribution of Si given Gi it is hard to draw further conclusions. But when Si is weakly correlated with Gi, the outcome model is approximately correctly specified and var (β̂S) ≈ σ 2/var (Si) + {σ 1 + α Svar (Si | Gi)}ΣSGΣ GGΣGS/var (Si). (1) Second, β∗ S = αS + βS. Thus when αS is large relative to βS with the same sign, standard analysis will be more powerful. Otherwise, denoting c = ΣSGΣ −1 GGΣGS/var (Si), var (β̂S) < var (β̂∗ S) if α 2 S < σ 2 1(1 − c)/var (Si | Gi)c. Since c is small when Si and Gi are weakly correlated and σ 1 tends to be large, our approach will still have more power unless αS is very large, though when it is so large that αSE(Si | Gi) is not close to zero the outcome model will not be approximately correctly specified.