  An important phenomenon in high dimensional biological data is the presence
of unobserved covariates that can have a significant impact on the measured
response. When these factors are also correlated with the covariate(s) of
interest (i.e. disease status), ignoring them can lead to increased type I
error and spurious false discovery rate estimates. We show that depending on
the strength of this correlation and the informativeness of the observed data
for the latent factors, previously proposed estimators for the effect of the
covariate of interest that attempt to account for unobserved covariates are
asymptotically biased, which corroborates previous practitioners' observations
that these estimators tend to produce inflated test statistics. We then provide
an estimator that corrects the bias and prove it has the same asymptotic
distribution as the ordinary least squares estimator when every covariate is
observed. Lastly, we use previously published DNA methylation data to show our
method can more accurately estimate the direct effect of asthma on methylation
than previously published methods, which underestimate the correlation between
asthma and latent cell type heterogeneity. Our re-analysis shows that the
majority of the variability in methylation due to asthma in those data is
actually mediated through cell composition.
