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I am performing likelihood ratio tests (LRTs).

The data set is not critical its the purity of the statistics that are important, but this is (phylo)genetics data.

I am examining (tree) hypothesis '1' (for example virus 1 and virus 2 share a common ancestor, $H_{1}$) and tree hypothesis '2' (virus 1 and virus 3 share a common ancestor, $H_{2}$ ) using the same data set. My 'global' hypothesis is that there is only one solution H1 or H2 for the data set, so H1 is not equal to $H_{2}$.

Two independent likelihood tests are performed comparing each hypothesis against the unconstrained solution $H_{0}$ (maximum likelihood). Each LRT comprises a null and alternative hypothesis, giving two separate results at 0.05% threshold. $H_{1}$ and $H_{2}$ are different constrained models (phylogenetic trees).

The 'global' hypothesis is true, if,:

$H_{1}$ is rejected and $H_{2}$ is accepted

OR

$H_{2}$ is rejected and $H_{1}$ is accepted

If both $H_{1}$ and $H_{2}$ are rejected the 'global hypothesis' is unresolved, if both are accepted the result is not significant, i.e. either not enough data or the difference is spurious.

Questions

  1. Is this a conditional statement or a nested hypothesis?
  2. How do I write the statement in notation form?

Could I write the above statement as,

$P(H_{1} | H_{2}) < 0.01$ or $P(H_{2} | H_{1}) < 0.01$

Is there a more succinct/alternative method?

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