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An important phenomenon in high dimensional biological data is the presenceof unobserved covariates that can have a significant impact on the measuredresponse. When these factors are also correlated with the covariate(s) ofinterest (i.e. disease status), ignoring them can lead to increased type Ierror and spurious false discovery rate estimates. We show that depending onthe strength of this correlation and the informativeness of the observed datafor the latent factors, previously proposed estimators for the effect of thecovariate of interest that attempt to account for unobserved covariates areasymptotically biased, which corroborates previous practitioners' observationsthat these estimators tend to produce inflated test statistics. We then providean estimator that corrects the bias and prove it has the same asymptoticdistribution as the ordinary least squares estimator when every covariate isobserved. Lastly, we use previously published DNA methylation data to show ourmethod can more accurately estimate the direct effect of asthma on methylationthan previously published methods, which underestimate the correlation betweenasthma and latent cell type heterogeneity. Our re-analysis shows that themajority of the variability in methylation due to asthma in those data isactually mediated through cell composition.

Accounting for unobserved covariates with varying degrees of estimability in high dimensional biological data