The combined likelihood of both components of the model is:
\[\textrm{L}\left(\alpha_h, \beta_h, \phi_j \left| D \right. \right) = \prod_i \left[ \Pr_{\mathrm{BNB}} \left( X = x_{ij} \left| \lambda_{hi}, \Omega_i, \phi_j \right. \right) \prod_k \Pr_{\mathrm{BB-mix}}\left(Y = y_{ik} \left| p_{h}, n_{ik}, \Upsilon_i \right. \right) \right]\]
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 \(\Lambda = \frac{L(\hat{\theta}_1|D)}{L(\hat{\theta}_0|D)}\) 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.