deletions | additions       diff --git a/Correcting for unknown covariates using principal components.tex b/Correcting for unknown covariates using principal components.tex  index 829c785..a7311c6 100644  --- a/Correcting for unknown covariates using principal components.tex  +++ b/Correcting for unknown covariates using principal components.tex 
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\subsection{Correcting for unknown covariates using principal components}  Covariates that are both measurable (such as time of experiment, age of sample, etc.) and unknown affect molecular trait measurements and confound QTL studies. Principal component analysis (PCA) are used to capture and remove these effects from QTL studies \cite{Pickrell,XXXXX}. \cite{Stephens_Gilad_Pritchard_2010}\cite{Pickrell,XXXXX}.  In order to leverage PCA while maintaining the discrete nature of the count data, the CHT directly models the covariate effects. To do this we include PCA loadings $u_{i\bullet}$ and fit coefficients $c_{h\bullet}$ when calculating $\lambda_{hi}$. \[  \lambda_{hi} = \left\{