@@ -72,7 +72,7 @@ In traditional statistical theory, a wid
LRT=2(l_1-l_0)
\]
where $l_1$ is the maximum likelihood under the more parameter-rich, complex model (alternative hypothesis) and $l_0$ is the maximum likelihood under the less parameter-rich simple model (null hypothesis).
- When the models compared are nested (the null hypothesis is a special case of the alternative hypothesis) and the null hypothesis is correct, the LRT statistic is asymptotically distributed as a χ2 with q degrees of freedom, where q is the difference in number of free parameters between the two models \citep{Kendall-1979, Goldman-1993b}. Note that, to preserve the nesting of the models, the likelihood scores need to be estimated upon the same tree. When some parameter is fixed at its boundary (p-inv, α), a mixed χ2 is used instead \citep{Ohta-1992, Goldman-2000}. The behavior of the χ2 approximation for the LRT has been investigated with quite a bit of detail \citep{Goldman-1993a, Goldman-1993b, Yang-1995, Whelan-1999, Goldman-2000}.
+ When the models compared are nested (the null hypothesis is a special case of the alternative hypothesis) and the null hypothesis is correct, the LRT statistic is asymptotically distributed as a $\chi^2$ with q degrees of freedom, where q is the difference in number of free parameters between the two models \citep{Kendall-1979, Goldman-1993b}. Note that, to preserve the nesting of the models, the likelihood scores need to be estimated upon the same tree. When some parameter is fixed at its boundary (p-inv, $\alpha$), a mixed $\chi^2$ is used instead \citep{Ohta-1992, Goldman-2000}. The behavior of the $\chi^2$ approximation for the LRT has been investigated with quite a bit of detail \citep{Goldman-1993a, Goldman-1993b, Yang-1995, Whelan-1999, Goldman-2000}.
\subsection{Hierarchical Likelihood Ratio Tests (hLRT)}
Andreas Tille authored