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\subsection {Analysis and Approaches}  %\begin{itemize}  % \item Summary paragraph giving an overview of the different types of methods and the ideas motivating them.  %\end{itemize}  \begin{enumerate}  \item Single-variant approach  \begin{itemize} 
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\begin{itemize}  \item VT \cite{Price_2010}:   \begin{itemize}  \item The variable threshold (VT) approach of \citeNP{Price_2010} is based on the regression of phenotypes onto the counts of variants whose minor allele frequencies (MAFs) are below some threshold (e.g. $1 \%$ or $5\%$) CHECK THIS. $5\%$).  \item The idea is that, variants with MAF below some threshold are more likely to be functional than the variants with higher MAF.   \item For each possible MAF threshold, VT computes a z-score and uses the maximum of z-score over all allele frequency thresholds. The statistical significance is then assessed by permutation testing on phenotypes.  %\item For each possible MAF threshold, a genotype score is computed based on a given collapsing theme(CHECK THIS). The chosen MAF threshold maximizes the association signal and permutation testing is used to adjust for the multiple thresholds.  \item In their simulations, \citeNP{Price_2010} found that the VT approach had high power to detect the association between rare variants and disease traits when effects are in one direction.    %\item We used VTWOD function in RVtests R package \cite{Xu_2012}.  \end{itemize}  \item C-alpha \cite{Neale_2011}:%Test the variance of the effect size for variants in a specific genomic window (No effect, increase or decrease risk).  \begin{itemize}  \item C-alpha test of \cite{Neale_2011} is a variance components approach that assumes the effects of variants are random with mean zero.   \item C-alpha procedure tests the variance of genetic effects under the assumption that variants observed in cases and controls are a mixture of risk, protective or neutral variants.  \item \citeNP{Neale_2011} found that C-alpha showed greater power than burden test when protective and risk variant exist.  \end{itemize}  \item We applied VTWOD function in R package RVtests \cite{Xu_2012} for VT-test and SKAT function in R package SKAT SKAT\cite{Wu_2011}  for C-alpha test across the simulated region by using sliding windows of 20 SNVs overlapping by 5 SNVs. \end{itemize}    \item Joint-modeling methods