  Gene expression and phenotype association can be affected by potential
unmeasured confounders from multiple sources, leading to biased estimates of
the associations. Since genetic variants largely explain gene expression
variations, they can be used as instruments in studying the association between
gene expressions and phenotype in the framework of high dimensional
instrumental variable (IV) regression. However, because the dimensions of both
genetic variants and gene expressions are often larger than the sample size,
statistical inferences such as hypothesis testing for such high dimensional IV
models are not trivial and have not been investigated in literature. The
problem is more challenging since the instrumental variables (e.g., genetic
variants) have to be selected among a large set of genetic variants. This paper
considers the problem of hypothesis testing for sparse IV regression models and
presents methods for testing single regression coefficient and multiple testing
of multiple coefficients, where the test statistic for each single coefficient
is constructed based on an inverse regression. A multiple testing procedure is
developed for selecting variables and is shown to control the false discovery
rate. Simulations are conducted to evaluate the performance of our proposed
methods. These methods are illustrated by an analysis of a yeast dataset in
order to identify genes that are associated with growth in the presence of
hydrogen peroxide.
