  In this paper we consider the effect of high dimensional Principal Component
(PC) adjustments while inferring the effects of variables on outcomes. This
problem is particularly motivated by applications in genetic association
studies where one performs PC adjustment to account for population
stratification. We consider simple statistical models to obtain asymptotically
precise understanding of when such PC adjustments are supposed to work in terms
of providing valid tests with controlled Type I errors. We also verify these
results through a class of numerical experiments.
