Integrating Multi–Omics Data for Gene-Environment Interactions

. 2021 Jan 29;10(1):3. doi: 10.3390/biotech10010003

Algorithm 1 The Integrative analysis for G×E Interaction

Step 1: Estimate the loading matrices of LRMs U and V: construct LRMs.
(a) For $j = 1, \dots, p_{g}$ , obtain ${\hat{θ}}_{j}$ by minimizing Equation (4). Then the estimate $\hat{Θ} = ({\hat{θ}}_{1}, \dots, {\hat{θ}}_{p_{g}})$ .
Initialize $l = 1$ .
for $l = 1, \dots, L$ do
(b) Apply rank-1 sparse SVD on $\hat{Θ}$ to obtain the singular vectors corresponding to largest singular values $(u_{l}, v_{l}, d_{l})$ .
(c) Update ${\hat{Θ}}^{(l + 1)} = {\hat{Θ}}^{(l)} - u_{l} d_{l} v_{l}^{⊤}$ .
(d) $l = l + 1$ .
end for
Step 2: Estimate regression coefficients $α, b_{1}, b_{2}, γ$ : construct the penalized G×E interaction model.
(a) Calculate $G V$ , $R U$ , $\tilde{G}$ and $\tilde{R}$ .
Initialize ${\hat{b}}_{1}^{(0)} = {\hat{b}}_{2}^{(0)} = {\hat{γ}}^{(0)} = 0$ .
At the $(k + 1)$ th iteration.
repeat
(b) Compute ${\hat{α}}^{(k + 1)} = {(E^{{(k)}^{⊤}} E^{(k)})}^{- 1} E^{{(k)}^{⊤}} (Y - X_{1} {\hat{b}}_{1}^{(k)} - X_{2} {\hat{b}}_{2}^{(k)} - \tilde{R} {\hat{γ}}^{(k)})$ .
(c) Obtain $({\hat{b}}_{1}^{(k + 1)}, {\hat{b}}_{2}^{(k + 1)}, {\hat{γ}}^{(k + 1)})$ by minimizing Equation (9) through bi-level selection.
until convergence