Overview of the considered deconfounding methods.

Large-scale studies of gene expression are commonly influenced by biological and technical sources of expression variation, including batch effects, sample characteristics, and environmental impacts. Learning the causal relationships between observable variables may be challenging in the presence of unobserved confounders. Furthermore, many high-dimensional regression techniques may perform worse. In fact, controlling for unobserved confounding variables is essential, and many deconfounding methods have been suggested for application in a variety of situations. The main contribution of this article is the development of a two-stage deconfounding procedure based on Bow-free Acyclic Paths (BAP) search developed into the framework of Structural Equation Models (SEM), called SEMbap(). In the first stage, an exhaustive search of missing edges with significant covariance is performed via Shipley d-separation tests; then, in the second stage, a Constrained Gaussian Graphical Model (CGGM) is fitted or a low dimensional representation of bow-free edges structure is obtained via Graph Laplacian Principal Component Analysis (gLPCA). We compare four popular deconfounding methods to BAP search approach with applications on simulated and observed expression data. In the former, different structures of the hidden covariance matrix have been replicated. Compared to existing methods, BAP search algorithm is able to correctly identify hidden confounding whilst controlling false positive rate and achieving good fitting and perturbation metrics.