deletions | additions       diff --git a/The combined likelihood test.tex b/The combined likelihood test.tex  index 5f10f43..497c218 100644  --- a/The combined likelihood test.tex  +++ b/The combined likelihood test.tex 
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The combined likelihood of both components of the model is:  \[  \textrm{L}\left(\alpha_h, \beta_h, \Phi_\bullet, \eta_j, \Upsilon_\bullet \eta_j  \left| D \right. \right) = \prod_i \left[ \Pr_{\mathrm{BNB}} \left( X = x_{ij} \left| \lambda_{hi}, \Phi_i, \eta_j \right. \right) \prod_k \Pr_{\mathrm{BB-mix}}\left(Y_{ik} = y_{ik} \left| p_{h}, n_{ik}, \Upsilon_i \right. \right) \right] \]  To test for an association with genotype we perform a likelhood ratio test that compares the alternative hypothesis $\alpha_h != \beta_h$ to the null hypothesis $\alpha_h = \beta_h$.