deletions | additions       diff --git a/subsection_Several_popular_methods_begin__.tex b/subsection_Several_popular_methods_begin__.tex  index e41cde3..5290591 100644  --- a/subsection_Several_popular_methods_begin__.tex  +++ b/subsection_Several_popular_methods_begin__.tex 
...
\item The test statistic upweights the short branches at the tip of the tree by assigning a branch-length of one to all branches, even the relatively longer branches that are close to the time to the most common ancestor.  \item Pairwise distances between haplotypes on this re-scaled tree are then correlated to pairwise phenotypic distances.  \item We determined the distance measure matrix, $d_{ij}=1-s_{ij}$, where $s_{ij} = (y_i-\mu)(y_j-\mu)$ is the similarity score between haplotype $i$ and $j$, $y_i$ is the binary phenotype (coded as $0$ or $1$) and $\mu$ is the disease prevalence in the 1500 simulated individuals. We then used the Mantel statistic to compare the phenotype-based distance matrix, $d$, with the re-scaled tree-distance matrix.  \item Note that we defined a 'phenotype' for each haplotype within an individual. Therefore, an individual has two phenotypes rather than one.    \item Reconstructed genealogical trees at each SNV (Blossoc, \citeNP{Mailund_2006}): A fast method to localize the disease-causing variants.  \begin{itemize} 
...
\end{itemize}  \item True trees (MT-rank of the coalescent events, \cite{Burkett_2014}): Detect co-clustering of the disease trait and variants on genealogical trees.   \begin{itemize}  \item A previous simulation study of Burkett et al. established the optimality of tree based approach for detecting association. We therefore include two versions of Mantel test as a benchmark for comparison. Note that we define a 'phenotype' for each haplotype within an individual. Therefore, an individual has two phenotypes rather than one.  \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}