  We consider large-scale studies in which thousands of significance tests are
performed simultaneously. In some of these studies, the multiple testing
procedure can be severely biased by latent confounding factors such as batch
effects and unmeasured covariates that correlate with both primary variable(s)
of interest (e.g. treatment variable, phenotype) and the outcome. Over the past
decade, many statistical methods have been proposed to adjust for the
confounders in hypothesis testing. We unify these methods in the same
framework, generalize them to include multiple primary variables and multiple
nuisance variables, and analyze their statistical properties. In particular, we
provide theoretical guarantees for RUV-4 and LEAPP, which correspond to two
different identification conditions in the framework: the first requires a set
of "negative controls" that are known a priori to follow the null distribution;
the second requires the true non-nulls to be sparse. Two different estimators
which are based on RUV-4 and LEAPP are then applied to these two scenarios. We
show that if the confounding factors are strong, the resulting estimators can
be asymptotically as powerful as the oracle estimator which observes the latent
confounding factors. For hypothesis testing, we show the asymptotic z-tests
based on the estimators can control the type I error. Numerical experiments
show that the false discovery rate is also controlled by the Benjamini-Hochberg
procedure when the sample size is reasonably large.
