deletions | additions       diff --git a/The combined likelihood test.tex b/The combined likelihood test.tex  index 6d091c3..6235c34 100644  --- a/The combined likelihood test.tex  +++ b/The combined likelihood test.tex 
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\textrm{L}\left(\alpha_h, \beta_h, \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 = 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 $\alpha_h \neq \beta_h$ to the null hypothesis $\alpha_h = \beta_h$. We compuate a likelihood ratio statistic $\Lambda & = & \frac{L(\theta_1|D)}{L(\theta_0|D)}$ where $\theta_1$ and $\theta_0$ are maximum likelihood estimates of the parameters under the alternative and null hyptheses respectively. We calculate p-values from the the test statistic under the asymptotic assumption that it is $\chi^2$ distributed with one degree of freedom.