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$$  where $D$ is disease status ($D = 1$, case; $D=0$, control), and $G=(G_1, G_2,\ldots , G_{m})$ is an individual's multi-locus genotype at $m$ disease SNVs, with $G_j$ being the number of copies of the derived allele at the $j^{th}$ disease SNV. We obtained here $13$ risk SNVs by adding randomly sampled SNVs from the middle region one at a time, until the prevalence is between $9.5-10.5\%$ in the $1500$ individuals.(i.e. $m = 13$ for this simulation).     After assigning the disease status to the 1500 individuals, we sampled 50 case and 50 control individuals from all affected and unaffected individuals. In this case-control sample, we found that out of 4000 SNVs, 2630 were variable, and out of 13 risk SNVs, 8 were polymorphic. Table 1 summarizes the physical distances, number of recombinations and minor allele frequencies(MAF) of these 13 SNVs. We then extracted the data for these variable SNVs to examine the patterns of association in subsequent analyses. \begin{table}[!]  %\centering    \centering\caption{ Summaries for risk SNVs}  %\label{my-label}  \small  %\centering  \begin{tabular}{ccccccc}  \hline  Risk SNV & Physical dist(kbp) & \begin{tabular}[c]{@{}c@{}}Polymorphic \\ in sample\end{tabular} & \begin{tabular}[c]{@{}c@{}}No of \\ recombinations\\ in sample$^a$ \end{tabular} & \begin{tabular}[c]{@{}c@{}}MAF in \\ population\end{tabular} & \begin{tabular}[c]{@{}c@{}}MAF \\ in cases\end{tabular} & \begin{tabular}[c]{@{}c@{}}MAF\\ in controls\end{tabular} \\ \hline  rSNV1 & 953.0 & 1 & NA & 0.0140 & 0.05 & 0.02 \\   rSNV2 & 958.0 & 0 & NA & 0.0050 & 0.00 & 0.00 \\   rSNV3 & 969.0 & 1 & 15 & 0.0003 & 0.01 & 0.00 \\   rSNV4 & 992.5 & 0 & NA & 0.0003 & 0.00 & 0.00 \\   rSNV5 & 1007.0 & 1 & 28 & 0.0017 & 0.02 & 0.00 \\  rSNV6 & 1009.0 & 0 & NA & 0.0003 & 0.00 & 0.00 \\  rSNV7 & 1010.0 & 1 & 3 & 0.1250 & 0.33 & 0.17 \\  rSNV8 & 1011.0 & 1 & 0 & 0.0170 & 0.04 & 0.01 \\  rSNV9 & 1021.0 & 1 & 13 & 0.0743 & 0.21 & 0.05 \\  rSNV10 & 1022.0 & 0 & NA & 0.0003 & 0.00 & 0.00 \\  rSNV11 & 1034.0 & 1 & 17 & 0.0013 & 0.01 & 0.00 \\  rSNV12 & 1041.5 & 1 & 7 & 0.0140 & 0.00 & 0.00 \\  rSNV13 & 1046.5 & 0 & NA & 0.0020 & 0.00 & 0.00 \\ \hline  \end{tabular}      \end{table}