  Epidemiological evidence suggests that simultaneous exposures to multiple
environmental risk factors (Es) can increase disease risk larger than the
additive effect of individual exposure acting alone. The interaction between a
gene and multiple Es on a disease risk is termed as synergistic
gene-environment interactions (synG$\times$E). Varying multi-index coefficients
models (VMICM) have been a promising tool to model synergistic G$\times$E
effect and to understand how multiple Es jointly influence genetic risks on a
disease outcome. In this work, we proposed a 3-step variable selection approach
for VMICM to estimate different effects of gene variables: varying, non-zero
constant and zero effects which respectively correspond to nonlinear
synG$\times$E, no synG$\times$E and no genetic effect. For multiple
environmental exposure variables, we also estimated and selected important
environmental variables that contribute to the synergistic interaction effect.
We theoretically evaluated the oracle property of the proposed variable
selection approach. Extensive simulation studies were conducted to evaluate the
finite sample performance of the method, considering both continuous and
discrete gene variables. Application to a real dataset further demonstrated the
utility of the method. Our method has broad applications in areas where the
purpose is to identify synergistic interaction effect.
