deletions | additions       diff --git a/Correcting for unknown covariates using principal components.tex b/Correcting for unknown covariates using principal components.tex  index 59c711e..7824623 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 both measurable (such as time of experiment, age of sample, etc.) and unknown can affect molecular trait measurements and confound QTL studies. Principal components component analysis (PCA)  can be used to capture and remove these effects from QTL studies \cite{Pickrell,XXXXX} \cite{Pickrell,XXXXX}. In order to leverage this while maintaining the discrete nature of the count data, the CHT directly models PCA effects. To do these, we include PCA effects in the calculation of $\lambda_{hi}$.  \[  \lambda_{hi} = \left\{  \begin{array}{ll}  2 \alpha (c_{h1} u_{i1} + c_{h2} u_{i2} + \ldots) T_i & \textrm{if } G_{im} = 0 \textrm{ (homozygous allele 1)} \\  \\  \left( \alpha + \beta \right) T_i & \textrm{if } G_{im} = 1 \textrm{ (heterozygous)} \\  \\  2 \beta T_i & \textrm{if } G_{im} = 2 \textrm{ (homozygous allele 2)}  \end{array} \right.  \]