  Large-scale genome-wide association studies (GWAS) have offered an exciting
opportunity to discover putative causal genes or risk factors associated with
diseases by using SNPs as instrumental variables (IVs). However, conventional
approaches assume linear causal relations partly for simplicity and partly for
the availability of GWAS summary data. In this work, we propose a novel model
{for transcriptome-wide association studies (TWAS)} to incorporate nonlinear
relationships across IVs, an exposure/gene, and an outcome, which is robust
against violations of the valid IV assumptions, permits the use of GWAS summary
data, and covers two-stage least squares as a special case. We decouple the
estimation of a marginal causal effect and a nonlinear transformation, where
the former is estimated via sliced inverse regression and a sparse instrumental
variable regression, and the latter is estimated by a ratio-adjusted inverse
regression. On this ground, we propose an inferential procedure. An application
of the proposed method to the ADNI gene expression data and the IGAP GWAS
summary data identifies 18 causal genes associated with Alzheimer's disease,
including APOE and TOMM40, in addition to 7 other genes missed by two-stage
least squares considering only linear relationships. Our findings suggest that
nonlinear modeling is required to unleash the power of IV regression for
identifying potentially nonlinear gene-trait associations. Accompanying this
paper is our Python library \texttt{nl-causal}
(\url{https://nonlinear-causal.readthedocs.io/}) that implements the proposed
method.
