deletions | additions       diff --git a/The combined likelihood test.tex b/The combined likelihood test.tex  index f785535..eb91ec6 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 $\alpha_h = \beta_h$. The CHT returns a likelihood ratio statistic $\Lambda = \frac{L(\theta_1|D)}{L(\theta_0|D)}$ \frac{L(\hat{\theta}_1|D)}{L(\hat{\theta}_0|D)}$  where $\hat{\theta}_1$ and $\hat{\theta}_0$ are maximum likelihood estimates of the parameters under the alternative and null hypotheses. P-values can be calculated from the the test statistic under the asymptotic assumption that $-2log(\Lambda)$ is $\chi^2$ distributed with one degree of freedom.