deletions | additions       diff --git a/Correcting for unknown covariates using principal components.tex b/Correcting for unknown covariates using principal components.tex  index fde8ae4..c45e6a9 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 Both known and unknown covariates such  as time of experiment, age of sample, etc.) and unknown etc. can  affect molecular trait measurements and confound QTL studies. Principal component analysis (PCA) is sometimes  used to capture and remove these effects from QTL studies \cite{Stephens_Gilad_Pritchard_2010}\cite{Pickrell,XXXXX}. In order to \cite{Stephens_Gilad_Pritchard_2010}\cite{Pickrell_2010, Degner_2012}. To  leverage PCA while maintaining the discrete nature of the count data, the CHT directly models the covariate effects. To do this we include a user defined number of PCA loadings $u_{i\bullet}$ and fit coefficients $c_{h\bullet}$ when calculating $\lambda_{hi}$. \[  \lambda_{hi} = \left\{