A Sparse Additive Model for High-Dimensional Interactions with an Exposure Variable

A conceptual paradigm for onset of a new disease is often considered to be the result of changes in entire biological networks whose states are affected by a complex interaction of genetic and environmental factors. However, when modelling a relevant phenotype as a function of high dimensional measurements, power to estimate inter-actions is low, the number of possible interactions could be enormous and their effects may be non-linear. Existing approaches for high dimensional modelling such as the lasso might keep an interaction but remove a main effect, which is problematic for interpretation. In this work, we introduce a method called sail for detecting non-linear interactions with a key environmental or exposure variable in high-dimensional settings which respects either the strong or weak heredity constraints. We prove that asymptotically, our method possesses the oracle property, i.e., it performs as well as if the true model were known in advance. We develop a computationally effcient fitting algorithm with automatic tuning parameter selection, which scales to high-dimensional datasets. Through an extensive simulation study, we show that sail out-performs existing penalized regression methods in terms of prediction accuracy and support recovery when there are non-linear interactions with an exposure variable. We then apply sail to detect non-linear interactions between genes and a prenatal psychosocial intervention program on cognitive performance in children at 4 years of age. Results from our method show that individuals who are genetically predisposed to lower educational attainment are those who stand to benefit the most from the intervention. Our algorithms are implemented in an R package available on CRAN (https://cran.r-project.org/package=sail).