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\subsection{Estimating overdispersion parameters}  In order to estimate the genome-wide overdispersion parameters $\Phi_i$ and $\Upsilon_i$, we use the same likelihood equations as in the CHT, but assume that there are no genetic effects. This means that for the read depth part of the test the mean estimate, $\lambda$, $\lambda_{hi}$,  is equal to the expected counts, $T^*_{ij}$. For $T^*_{ij}$, and for  the allele specific allele-specific  part of the test, $p$ $p_h$  is equal to $0.5$. The fraction of reads comSince Since  the allele specific and read depth parts of the likelihood equation are independent, we can fit the overdispersion parameters separately. \subsubsection{Beta-Negative-Binomial}  In order to To  find the maximum likelihood estimate of $\Phi_i$ we need to sum the likelihoodequation  across all regions. This presents a problem, as $\eta_j$ must also be estimated for each region. We therefore interatively estimate $\eta_j$ using the equation \[  \textrm{L}\left(\eta_j %\[  %\textrm{L}\left(\eta_j  \left| D \right. \right) = \prod_i \left[ \Pr_{\mathrm{BNB}} \left( X = x_{ij} \left| %\left|  \lambda = T^*_{ij}, \Phi_i, \eta_j \right. \right) \right] \] %\]  and then $\Phi_i using the equation  \[  \textrm{L}\left(\Phi_i %\[  %\textrm{L}\left(\Phi_i  \left| D \right. \right) = \prod_j \left[ \Pr_{\mathrm{BNB}} \left( X = x_{ij} \left| %\left|  \lambda = T^*_{ij}, \Phi_i, \eta_j \right. \right) \right] \] %\]  \subsubsection{Beta-Binomial}  \subsubsection{Beta-Binomial}