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  \end{itemize}    \item Paragraph to discuss how we scored localization and signal detection for each of these methods.  \begin{itemize}  \item To address the localization, we scored the distance of the peak signal from the risk region based on the average distance across the entire genomic region.   \begin{itemize}  \item For each method on each dataset, we computed the average distance of the peak association signals from the risk region.  \item Under each method, we then plotted the empirical cumulative distribution function (ECDF) of the average peak signals for each dataset.  \item ECDF at point $x$ is the proportion of the samples with average distance less than or equal to $x$.  \item Therefore, a method with higher ECDF at a smaller value of $x$, has higher performance in localizing the signal.  \end{itemize}  \item Detect association: For a given simulated dataset and a given method, we used max. statistics across all the SNV as our global test statistics and determined its null distribution by permuting the case-control labels.   \begin{itemize}  \item For each of the test statistic evaluated, we used either maximum statistic or the maximum of $- log_{10}$ of p value as the score for the dataset.  \item We intended to compare the distribution of $200$ scores across different test statistics.  \item However, test statistics are not comparable since they are not on the same scale.  \item To make these score distributions comparable, we converted scores to their corresponding empirical p values.  \item We defined these p values as the proportion of observed values under the null model that are greater than or equal to the observed value under the alternative model.  \item We then compared the distribution of empirical p values for the different test statistics by plotting ECDF under each method.  \end{itemize}  \end{itemize}     \end{enumerate}