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\end{itemize}  \end{itemize}  \end{enumerate}   \item Tree-Based method  \begin{itemize}  \item Reconstructed genealogical trees at each SNV (Blossoc, \citeNP{Mailund_2006}): A fast method to localize the disease-causing variants.  \begin{itemize}  \item Approximates perfect phylogenies for each site, assuming infinite site model of mutation and scores according to the non-random clustering of affected individuals.  \item \citeNP{Mailund_2006} have found Blossoc to be a fast and accurate method to localize {\bf common} disease-causing variants but how well does it work with rare variants?   \item Can use either phased or unphased genotype data. However, it is impractical to apply it to unphased data with more than a few SNPs due to the computational burden associated with phasing. We will thereform assume the SNV data are phased, as might be done in advance with a fast-phasing algorithm such as fastPHASE \cite{Scheet_2006}, BEAGLE \cite{Browning_2011}, IMPUTE2 \cite{Howie_2009} or MACH \cite{Li_2010,Li_2009}.  \end{itemize}  \item True trees (MT-rank of the coalescent events, \citeNP{Burkett_2013}): Detect co-clustering of the disease trait and variants on genealogical trees.   \begin{itemize}  \item In practice, the true trees are unknown. However, the cluster statistics based on true trees represent a best case insofar as tree uncertainty is eliminated. A previous simulation study \cite{Burkett_2013} established the optimality of these tests for detecting association. We therefore include two versions of Mantel test as a benchmark for comparison.  \begin{itemize}  \item Version 1: Naive-Mantel test, phenotype is scored according to whether or not haplotype comes from a case.  \item Version 2: Informed-Mantel test, phenotype is scored according to whether or not haplotype comes from a case and carries a risk variant.  \end{itemize}  \item Upweight the short branches at the tip of the tree. %(DESCRIBE BRIEFLY HOW WE ACHIEVE UPWEIGHTING OF THE SHORT BRANCHES AT THE TIPS).  We assign a branch-length of one to all branches, even the relatively longer branches that are close to the time to the most recent common ancestor.  %[NOW CAN REMOVE: Expected number of time it takes for the final two of k lineages to coalesce is $ E(T_{2}) = 0.5 \times E(TMRCA) $. So, if we rank the coalescence events(i.e. intercoalescence times are 1 time unit), $ T_{2} $ becomes 1, as well as $T_{k}$ is one. So, this has the effect of upweighting the branch.]  \item Success in localization was declared if the strongest signal was in the risk region.  \end{itemize}   \end{itemize}